Calculus of Variations/CHAPTER XII

CHAPTER XII: A FOURTH AND FINAL CONDITION FOR THE EXISTENCE OF A MAXIMUM OR A MINIMUM, AND A PROOF THAT THE CONDITIONS WHICH HAVE BEEN GIVEN ARE SUFFICIENT.

• 153 The notion of a field continued from the preceding Chapter.
• 154 The function $ℰ(x,y,p,q,\overline{p},\overline{q})$.
• 155 The function $ℰ$ must have the same sign for every point of the curve.
• 156 The sufficiency of the above condition.
• 157 Another form of the function $ℰ$.
• 158 Still another form.
• 159,160 The signs of the function $ℰ$ and $F_1$.
• 161 Another proof of the sufficiency of the condition as given in Article 156.
• 162 The function $ℰ$ cannot be zero along an entire curve in the given field.
• 163 The envelope of conjugate points.
• 164 The curve may be composed of a finite number of regular traces.
• 165 Cases where the traces are not regular.
• 166 Generalizations in the Integral Calculus.
• 167,168,169,170,171,172 Applications to the four problems already considered.
• 173 When $F(x,y,x^{\prime },y^{\prime })$ is a rational function of $x^{\prime }$ and $y^{\prime }$, there can exist neither a maximum nor a minimum value of the integral.
• 174 General summary.
• 175 Extensions and generalizations: Instead of the determination of a structure of the first kind in the domain of two variables, it may be required to determine a structure of the first kind in the domain of $n$ quantities.
• 176 When equations of condition exist among the variables.
• 177 When the second and higher derivatives appear.
• 178 The Calculus of variations applied to the determination of structures of a higher kind. The minimal surfaces.

Article 153.
In the preceding Chapter we considered the family of curves that have the same initial point $A$ and satisfy the differential equation $G=0$. These deviate very little from one another in their initial direction. We saw that the curves again intersect only in the neighborhood of points that are the conjugates of $A$, the conjugate point along any curve being the limiting position of the point of intersection of this curve and a neighboring curve when the angle between their initial directions becomes infinitesimally small. All points that lie on these curves before the points that are conjugate to $A$ form a connected portion of surface ; that is, if $P_1$ is a point belonging to this collectivity of points, a boundary may be described about $P_1$ so that all points within this boundary also belong to the collectivity of points.

For, let

$x=\varphi (t,\alpha ,\beta )y=\psi (t,\alpha ,\beta )$

be the equations of a given curve which satisfies $G=0$, and let the coordinates of a point on this curve be

$x_1=\varphi (t_1,\alpha ,\beta )y_1=\psi (t_1,\alpha ,\beta )$

Further, let $x_1+\xi ,y_1+\eta$ be the coordinates of another point $P_2$ that lies in the neighborhood of $P_1,$ so that $\xi ,\eta$ are quantities arbitrarily small.

We may then (Art. 151) draw a curve between $A$ and $P_2$ which satisfies the differential equation $G=0$, if we can determine four quantities $\tau ,{\tau }^{\prime },{\alpha }^{\prime },{\beta }^{\prime }$ as power-series in $\xi ,\eta$ in such a way that the following equations are true:

$\begin{array}{cc}0& ={\varphi }^{\prime }(t_0){\tau }^{\prime }+{\varphi }_1(t_0){\alpha }^{\prime }+{\varphi }_2(t_0){\beta }^{\prime }+({\tau }^{\prime },{\alpha }^{\prime },{\beta }^{\prime }{)}_2\\ 0& ={\psi }^{\prime }(t_0){\tau }^{\prime }+{\psi }_1(t_0){\alpha }^{\prime }+{\psi }_2(t_0){\beta }^{\prime }+({\tau }^{\prime },{\alpha }^{\prime },{\beta }^{\prime }{)}_2\\ \xi & ={\varphi }^{\prime }(t_1){\tau }^{\prime }+{\varphi }_1(t_1){\alpha }^{\prime }+{\varphi }_2(t_1){\beta }^{\prime }+(\tau ,{\alpha }^{\prime },{\beta }^{\prime }{)}_2\\ \eta & ={\psi }^{\prime }(t_1){\tau }^{\prime }+{\psi }_1(t_1){\alpha }^{\prime }+{\psi }_2(t_1){\beta }^{\prime }+(\tau ,{\alpha }^{\prime },{\beta }^{\prime }{)}_2\end{array}$

Since the determinant of these equations (Art. 151) is $-\mathrm{\Theta }(t_1,t_0)$ and is different from zero, the point $t_1$ not being conjugate to $t_0$, it follows that the quantities $\tau ,{\tau }^{\prime },{\alpha }^{\prime },{\beta }^{\prime }$ may be developed in powerseries in $\xi ,\eta$ which are convergent for small values of these quantities.

Consequently, a curve may be drawn through $A$ and $P_2$ which satisfies the differential equation $G=0$, and this curve will be neighboring the first curve and will deviate as little as we wish in direction from its initial direction, if $\xi ,\eta$, and consequently also $\tau ,{\tau }^{\prime },{\alpha }^{\prime },{\beta }^{\prime }$ are sufficiently small.

If we form the determinant for the curve $A{P}_2$, which, when put equal to zero, is the equation for the determination of the point conjugate to $A$, it is seen that this determinant also may be developed as a power-series in $\xi ,\eta$, which becomes $-\mathrm{\Theta }(t_1,t_0)$ when $\xi =\eta =0$. The function $\mathrm{\Theta }(t_1,t_0)$ is different from zero when sufficiently small values are ascribed to $\xi ,\eta$. Consequently, within the interval $A{P}_2$, there is present no point which is conjugate to $A$.

We may therefore envelop the interval situated between two conjugate points of the original curve by a narrow surface area, which is of such a nattire that a curve, and only one, may be drawn from the point $A$ to any point within it, which satisfies the differential equation $G=0$, is neighboring the first curve and deviates in its initial direction only a little from it.

Article 154.
Let a portion of curve $P_0{P}_1$. satisfying the differential equation $G=0$, be given, which is of such a nature that for no point on it $F_1=0$ or $\mathrm{\infty }$, and suppose that the point conjugate to $P_0$ does not lie before $P_1$. Between $P_0$ and $P_1$ take an arbitrary point $P_2$ and draw through $P_2$ a regular curve. On this curve we choose a point $P_3$ so close to $P_2$ that a curve may be drawn through $P_0$ and $P_3$ which satisfies the differential equation $G=0$, and which lies entirely within the strip of surface defined above. Let us consider the change in the integral when we take it over $P_0{P}_3+P_3{P}_2$ instead of over $P_0{P}_2$. We may denote an integral taken over a curve that satisfies the differential equation by $I$, and one over an arbitrary curve by $\overline{I}$, and we may denote the direction of integration by added indices. We have therefore to compute the expression

$\mathrm{\Delta }I=I_{03}+{\overline{I}}_{32}-I_{02}$

or

$\mathrm{\Delta }I=(\underset{\overline{P_0{P}_3}}{\int }F\text{d}t-\underset{\overline{P_0{P}_2}}{\int }F\text{d}t)+\underset{\overline{P_0{P}_2}}{\int }F\text{d}t$

an expression which (Art. 79)

$=ϵ(\underset{\overline{P_0{P}_2}}{\int }Gw\text{d}s+{[\xi \frac{\partial F}{\partial x^{\prime }}+\eta \frac{\partial F}{\partial y^{\prime }}]}^{t_2})+({ϵ}^2)+\underset{\overline{P_0{P}_2}}{\int }F\text{d}t$

where $\xi ,\eta$ are measured in the direction from $P_2$ to $P_3$.

At the point $P_2$ and along the curve $P_3{P}_2$ in the direction $P_3{P}_2$ we have

$ϵ\xi =-{\overline{x}}_{2^{\prime }}\text{d}\overline{t}+(\text{d}\overline{t}{)}^2ϵ\eta =-{\overline{y}}_{2^{\prime }}\text{d}\overline{t}+(\text{d}\overline{t}{)}^2$

$\text{d}\overline{t}$ denoting that this differential is taken with respect to the curve $P_3{P}_2$.

If we consider the arguments in $F$ expressed as functions of $\overline{t}$ along the curve $P_3{P}_2$, it follows that

$\underset{\overline{P_0{P}_2}}{\int }F\text{d}t=F(x_2,y_2,{\overline{x}}_{2^{\prime }},{\overline{y}}_{2^{\prime }})\text{d}\overline{t}+(\text{d}\overline{t}{)}^2$

Hence at the point $P_2$, which is an arbitrary point of the curve $P_0{P}_1$, we have

$\mathrm{\Delta }I=(F(x_2,y_2,{\overline{x}}_{2^{\prime }},{\overline{y}}_{2^{\prime }})-[{\overline{x}}_{2^{\prime }}\frac{\partial }{\partial x_{2^{\prime }}}F(x_2,y_2,x_{2^{\prime }},y_{2^{\prime }})+{\overline{y}}_{2^{\prime }}\frac{\partial }{\partial y_{2^{\prime }}}F(x_2,y_2,x_{2^{\prime }},y_{2^{\prime }})])\text{d}\overline{t}+(\text{d}\overline{t}{)}^2$(a)

The function $F$ is homogeneous of the first order (Art. 68) with respect to its third and fourth arguments, so that (see Art. 72)

$F(x_2,y_2,{\overline{x}}_{2^{\prime }},{\overline{y}}_{2^{\prime }})={\overline{x}}_{2^{\prime }}{F}^{(1)}(x_2,y_2,{\overline{x}}_{2^{\prime }},{\overline{y}}_{2^{\prime }})+{\overline{y}}_{2^{\prime }}{F}^{(2)}(x_2,y_2,{\overline{x}}_{2^{\prime }},{\overline{y}}_{2^{\prime }})$

We define by $ℰ(x,y,x^{\prime },y^{\prime },{\overline{x}}^{\prime },{\overline{y}}^{\prime })$ the expression

$1)ℰ(x,y,x^{\prime },y^{\prime },{\overline{x}}^{\prime },{\overline{y}}^{\prime })={\overline{x}}^{\prime }(F^{(1)}(x_2,y_2,{\overline{x}}_{2^{\prime }},{\overline{y}}_{2^{\prime }})-F^{(1)}(x_2,y_2,x_{2^{\prime }},y_{2^{\prime }}))+{\overline{y}}^{\prime }(F^{(2)}(x_2,y_2,{\overline{x}}_{2^{\prime }},{\overline{y}}_{2^{\prime }})-F^{(2)}(x_2,y_2,x_{2^{\prime }},y_{2^{\prime }}))$

Hence at the point $P2$ it follows that

$\mathrm{\Delta }I=ℰ(x,y,x^{\prime },y^{\prime },{\overline{x}}^{\prime },{\overline{y}}^{\prime })\text{d}\overline{t}+(\text{d}\overline{t}{)}^2$

when in the function $ℰ$ we have substituted for the arguments those values that belong to the point $P_2$. The direction-cosines of the curve $P_0{P}_2$ at $P$ are denoted by

$p_2=\frac{x_{2^{\prime }}}{\sqrt{x_2{\text{'}}^2+x_2{\text{'}}^2}}\text{and}{q}_2=\frac{y_{2^{\prime }}}{\sqrt{x_2{\text{'}}^2+x_2{\text{'}}^2}}$

and those of the curve $P_3{P}_2$ at $P_2$ by ${\overline{p}}_2$ and ${\overline{q}}_2$. It is evident from a consideration of the right-hand side of the formula defining $ℰ$ above (and cf. Art. 68) that

$\frac{ℰ(x,y,x^{\prime },y^{\prime },{\overline{x}}^{\prime },{\overline{y}}^{\prime })}{\sqrt{{\overline{x}}_2{\text{'}}^2+{\overline{y}}_2{\text{'}}^2}}=ℰ(x,y,p,q,\overline{q},\overline{q})$

Article 155.
If further we denote by $\sigma$ the differential of arc $P_3{P}_2$, we have finally

$2)\mathrm{\Delta }I=ℰ(x,y,p,q,{\overline{p}}^{\prime },\overline{q})\sigma +(\sigma {)}^2$

Accordingly, if we take $\sigma$ sufficiently small; that is, if we choose the point $P_3$ very close to $P_2$, then we may always bring it about that the change in the integral has the same sign as that of the function $ℰ$.

The point $P_2$ was an arbitrary point on the curve $P_0{P}_1$, and $P_2{P}_3$ also represented an arbitrary direction.

It follows that if for any point $P_2$ and for any direction at $P_2$ the function $ℰ$ were negative, and for any other point and direction positive, then the given curve could vary in such a manner that the change in the integral is at one time positive and at another time negative. We have, therefore, the following theorem :

If the integral taken over the curve $P_0{P}_1$ which satisfies the differential equation $G=0$ is to be a maximum or a minimum, then the function $ℰ$ must have the same sign for every point of the curve, and at every point of the curve for any direction, and this sign m.ust be negative for a maximum and positive for a minimum,.

Article 156.
That the above condition is sufficient to assure the existence of a maximum or a minimum may be shown as follows : Let $P_0(I)P_1$ be a curve which satisfies the four conditions already established (and recapitulated in Art. 174), and let $P_0(II)P_1$ be any arbitrary curve that lies in the field about the curve $P_0(I)P_1$. It is subject only to the condition that it must be a regular curve and lie wholly in the given field.

By varying the parameters $\alpha$ and $\beta$ we can construct a system of curves as near as we like to one another, all satisfying the differential equation $G=0$. These curves cut the curve $P_0(II)P_1$ in two (or perhaps more) points. They do not cut the curve $P_0(I)P_1$ or intersect among themselves within the field in question. The function $ℰ$ must have the same sign along each of these curves as it has along the curve $P_0(I)P_1$. For, take an arbitrary point $P$ on any of these curves. Then on the curve $P_0(I)P_1$ there is a point for which the quantities $x,y,p,q$ differ only a little from the quantities that belong to the point $P$, and consequently $ℰ$ has the same sign for both points.

Consider now the variation in our integrals as we pass from $P_0(I)P_1$ to $P_0{P}_2{P}_3{P}_1$ and from $P_0{P}_2{P}_3{P}_1$ to $P_0{P}_4{P}_5{P}_1$, etc. As we saw in the preceding article, the variation caused by passing from $P_0(I)P_1$ to $P_0{P}_2{P}_3{P}_1$

$=\underset{t_0}{\overset{t_2}{\int }}F\text{d}t+ϵ{[\frac{\partial F}{\partial x^{\prime }}\xi +\frac{\partial F}{\partial y^{\prime }}\eta ]}_{t_2}^{t_3}+\underset{t_3}{\overset{t_1}{\int }}F\text{d}t$

$=\underset{t_0}{\overset{t_2}{\int }}F\text{d}t-{[\overline{p}\frac{\partial F}{\partial x^{\prime }}+\overline{q}\frac{\partial F}{\partial y^{\prime }}]}^{t_2}\sigma +\underset{t_3}{\overset{t_1}{\int }}F\text{d}t-{[\overline{p}\frac{\partial F}{\partial x^{\prime }}+\overline{q}\frac{\partial F}{\partial y^{\prime }}]}^{t_3}\sigma$

$\overline{p},\overline{q}$ being the direction cosines of the tangent to the curve $P_0(II)P_1$ at the points $P_2$ and $P_3$, which, we notice, have opposite signs at these points.

If we denote the integration along the curves by the curves themselves, it is seen at once that the variation in these integrals may be expressed by

$P_0{P}_2{P}_{3}1P_1-P_0(I)P_1=[ℰ{]}^{t_0}\sigma +[ℰ{]}^{t_1}\sigma +(\sigma {)}^2$

where the first is the length from $P_0$ to $P_2$ and the second from $P_3$ to $P_1$.

Similarly the differences in the integrals along

$P_0{P}_4{P}_{5}1P_1-P_0{P}_2{P}_3{P}_1=[ℰ{]}^{t_2}\sigma +[ℰ{]}^{t_3}\sigma +(\sigma {)}^2$

$P_0{P}_6{P}_{7}1P_1-P_0{P}_4{P}_5{P}_1=[ℰ{]}^{t_4}\sigma +[ℰ{]}^{t_5}\sigma +(\sigma {)}^2$

$...................................$

$P_0{P}_{2\nu }{P}_{2\nu +1}1P_1-P_0{P}_{2\nu -2}{P}_{2\nu -1}{P}_1=[ℰ{]}^{t_{2\nu -2}}\sigma +[ℰ{]}^{t_{2\nu -1}}\sigma +(\sigma {)}^2$

$P_0(II)P_1-P_0{P}_{2\nu }{P}_{2\nu +1}{P}_1=[ℰ{]}^{t_{2\nu }}\sigma +[ℰ{]}^{t_{2\nu +1}}\sigma +(\sigma {)}^2$

Adding these results together, we have the difference in the integrals along

$P_0(II)P_1-P_0(II)P_1=\underset{P_0(II)P_1}{\int }ℰ\sigma +(\sigma {)}^2$

$\sigma$ being a differential of arc along the curve $P_0(I)P_1$. This is a verification of the theorem stated at the end of the last Article.

We also see that, if we had not assured ourselves that none of the intermediary curves intersect, the signs of the $\sigma$'s would not all have been alike, and consequently the sum total of all these $\sigma$'s would not have constituted the curve $P_{)}(II)P_1$.

Article 157.
Another form of the function $ℰ$.

We have seen in the Integral Calculus that

$f(p_1,q_1)-f(p_0,q_0)=\underset{p_0,q_0}{\overset{p_1,q_1}{\int }}\text{d}f(p,q)$

$=\underset{p_0,q_0}{\overset{p_1,q_1}{\int }}(\frac{\partial f(p,q)}{\partial p}\text{d}p+\frac{\partial f(p,q)}{\partial q}\text{d}q)$

$=\underset{k=0}{\overset{k=1}{\int }}(f^{(1)}[p_0+k(p_1-p_0),q_0+k(q_1-q_0)](p_1-p_0)+f^{(2)}[p_0+k(p_1-p_0),q_0+k(q_1-q_0)](q_1-q_0))\text{d}k$

Hence, if we write

$p_k=p+k(\overline{p}-p)=(1-k)p+k\overline{p}$

$q_k=q+k(\overline{q}-q)=(1-k)q+k\overline{q}$

it is seen that

$F^{(1)}(x,y,\overline{p},\overline{q})-F^{(1)}(x,y,p,q)=\underset{k=0}{\overset{k=1}{\int }}(F^{(11)}(x,y,p_k,q_k)(\overline{p}-p)+F^{(12)}(x,y,p_k,q_k)(\overline{q}-q))\text{d}k$

$F^{(2)}(x,y,\overline{p},\overline{q})-F^{(2)}(x,y,p,q)=\underset{k=0}{\overset{k=1}{\int }}(F^{(21)}(x,y,p_k,q_k)(\overline{p}-p)+F^{(22)}(x,y,p_k,q_k)(\overline{q}-q))\text{d}k$

Note that (see Art. 73)

$F^{(11)}=q_k^2{F}_1{F}^{(12)}=-p_k{q}_k{F}_1{F}^{(22)}=p_k^2{F}_1$

and further that $F^{(12)}=F^{(21)}$.

By substituting these values in the above expressions, and in turn the resulting quantities in the expression for $ℰ$, we have

$ℰ(x,y,p,q,\overline{p}\overline{q})=\overline{p}[F^{(1)}(x,y,\overline{p},\overline{q})-F^{(1)}(x,y,p,q)]+\overline{q}[F^{(2)}(x,y,\overline{p},\overline{q})-F^{(2)}(x,y,p,q)]$

$=\underset{k=0}{\overset{k=1}{\int }}{F}_1(x,y,p_k,q_k)([q_k(\overline{p}-p)-p_k(\overline{q}-q)]q_k\overline{p}+[-q_k(\overline{p}-p)+p_k(\overline{q}-q)]p_k\overline{q})\text{d}k$

The expression in the square brackets is

$(q_k\overline{p}-p_k\overline{q})[q_k(\overline{p}-p)-p_k(\overline{q}-q)]=(1-k)(q\overline{p}-p\overline{q}{)}^2$

and consequently

$3)ℰ(x,y,p,q,\overline{p},\overline{q})=(q\overline{p}-p\overline{q}{)}^2\underset{k=0}{\overset{k=1}{\int }}{F}_1(x,y,p_k,q_k)\text{d}k$

This expression for $ℰ$ in the form of a definite integral is defective, in that it has a meaning only when $F_1$ remains finite for all values of $p_k$ and $q_k$, as $k$ varies between 0 and 1. For example, if $k=1/2$, then $p_{1/2}=p+(\overline{p}-p)/2=(p+\overline{p})/2$, andi if $\overline{p}=-p$, then $p_{1/2}=0$; in the same way for $k=1/2$ and $\overline{q}=-q$, then also $q_{1/2}=0$. These two arguments being zero, $F_1$ becomes infinite (cf. Art. 73). Further, if the two directions $p,q$ and $\overline{p},\overline{q}$ coincide, then $ℰ$ becomes zero of the second order.

If $OP$ and $O\overline{P}$ are vectors of unit length with components $p,q$ and $\overline{p},\overline{q}$, then the components of $O{P}^{\prime }$, when $P^{\prime }$ travels along the line $P\overline{P}$, are $p_k,q_k,k$ varying between 0 and 1.

Article 158.
Another form was given by Weierstrass to the expression $ℰ$, in which he avoided the defect mentioned above, by integrating along the arc of a circle instead of along the straight line $P\overline{P}$. If we integrate along the arc of a circle of unit radius from the point $P$ to the point $\overline{P}$ we obtain an expression for $ℰ$ which is universally true.

We have as before, if $POX=\tau$, $\overline{P}OX=\overline{\tau }$, $\omega =\overline{\tau }-\tau (\text{mod }2\pi )$, and $-\pi \le \omega \le =\pi$,

$ℰ(x,y,p,q,\overline{p},\overline{q})=\overline{p}[F^{(1)}(x,y,\overline{p},\overline{q})-F^{(1)}(x,y,p,q)]+\overline{q}[F^{(2)}(x,y,\overline{p},\overline{q})-F^{(2)}(x,y,p,q)]$

$=\cos \overline{\tau }[F^{(1)}(x,y,\cos \overline{\tau },\sin \overline{\tau })-F^{(1)}(x,y,\cos \tau ,\sin \tau )]+\sin \overline{\tau }[F^{(2)}(x,y,\cos \overline{\tau },\sin \overline{\tau })-F^{(2)}(x,y,\cos \tau ,\sin \tau )]$

$=\cos \overline{\tau }\underset{\lambda =0}{\overset{\lambda =\omega }{\int }}{\text{d}}_{\lambda }{F}^{(1)}[x,y,\cos (\tau +\lambda ),\sin (\tau +\lambda )]+\sin \overline{\tau }\underset{\lambda =0}{\overset{\lambda =\omega }{\int }}{\text{d}}_{\lambda }{F}^{(2)}[x,y,\cos (\tau +\lambda ),\sin (\tau +\lambda )]$

But, if $F^{(1)}$ denotes the derivative of $F$ with respect to its third argument, etc.,

${\text{d}}_{\lambda }{F}^{(1)}[x,y,\cos (\tau +\lambda ),\sin (\tau +\lambda )]$

$=[F_{\cos ,\cos }^{(11)}\sin (\tau +\lambda )+F_{\cos ,\sin }^{(12)}\cos (\tau +\lambda )]\text{d}\lambda$

$=[-{\sin }^3(\tau +\lambda )-\sin (\tau +\lambda ){\cos }^2(\tau +\lambda )]F_1\text{d}\lambda$

$=-\sin (\tau +\lambda )F_1[x,y,\cos (\tau +\lambda ),\sin (\tau +\lambda )]\text{d}\lambda$

similarly,

${\text{d}}_{\lambda }{F}^{(2)}[x,y,\cos (\tau +\lambda ),\sin (\tau +\lambda )]=\cos (\tau +\lambda )F_1[x,y,\cos (\tau +\lambda ),\sin (\tau +\lambda )]\text{d}\lambda$

Hence, it follows that

$ℰ(x,y,p,q,\overline{p},\overline{q})=\underset{\lambda =0}{\overset{\lambda =\omega }{\int }}[-\cos \overline{\tau }\sin (\tau +\lambda )+\sin \overline{\tau }\cos (\tau +\lambda )]F_1\text{d}\lambda$

$=\underset{\lambda =0}{\overset{\lambda =\omega }{\int }}\sin (\overline{\tau }-\tau -\lambda )F_1\text{d}\lambda =\underset{\lambda =0}{\overset{\lambda =\omega }{\int }}\sin (\omega -\lambda )F_1[x,t,\cos (\tau +\lambda ),\sin (\tau +\lambda )]\text{d}\lambda$

If we write

$\omega -\lambda ={\lambda }^{\prime }$

the integral just written is

$\underset{{\lambda }^{\prime }=0}{\overset{{\lambda }^{\prime }=\omega }{\int }}\sin {\lambda }^{\prime }{F}_1[x,y,\cos (\overline{\tau }-{\lambda }^{\prime }),\sin (\overline{\tau }-{\lambda }^{\prime })]\text{d}{\lambda }^{\prime }=F_1[x,y,\cos (\overline{\tau }-{\lambda }_{2^{\prime }}),\sin (\overline{\tau }-{\lambda }_{2^{\prime }})]\underset{{\lambda }^{\prime }=0}{\overset{{\lambda }^{\prime }=\omega }{\int }}\text{d}\cos {\lambda }^{\prime }$

where ${\lambda }_{2^{\prime }}$ is intermediary between 0 and $\omega$.

We therefore have finally

$4)ℰ(x,y,p,q,\overline{p},\overline{q})=(1-\cos \omega )F_1[x,y,cos(\overline{\tau }-{\lambda }_{2^{\prime }}),\sin (\overline{\tau }{\lambda }_{2^{\prime }})]$

If then $F_1[x,y,cos(\overline{\tau }-{\lambda }^{\prime }),\sin (\overline{\tau }{\lambda }_{\text{'}})]$ has a constant sign between 0 and $\omega$ it follows also that $ℰ(x,y,p,q,\overline{p},\overline{q})$ has this sign, since ${\lambda }_{2^{\prime }}$ is one of the values of ${\lambda }^{\prime }$ within this interval.

The above formula is true for all values of $\omega$ situated between $-\pi$ and $+\pi$, and since $\cos (\overline{\tau }-{\lambda }_{2^{\prime }})$ and $\sin (\overline{\tau }-{\lambda }_{2^{\prime }})$ cannot both be zero at the same time, it is seen that

$F_1[x,y,\cos (\overline{\tau }-{\lambda }_{2^{\prime }}),\sin (\overline{\tau }-{\lambda }_{2^{\prime }})]\ne \mathrm{\infty }$

and consequently the expression 4 for $ℰ$ has not the same defect as the one given in the preceding article.

Article 159.
For any displacement of the curve $\omega \ne 0$, and consequently $1-\cos \omega$ is a positive quantity. Hence $ℰ$ has the same sign as $F_1$. If $F_1[x,y,\cos (\overline{\tau }-\lambda ),\sin (\overline{\tau }-\lambda )]$ is found by examination to have always the same sign independently of $\cos (\overline{\tau }-\lambda )$, $\sin (\overline{\tau }-\lambda )$ for every point of the curve within the interval in question, then we may be convinced that there is a maxim.um, or a minimumof the integral without the derivation and examination of the function $ℰ$. By this process, however, we have shown without the second variation that the function $F_1(x,y,p,q)$ can change its sign for no point on the curve, and for no direction of the tangent to the curve at a point.

Article 160.
It is evident that if $F_1$, considered as a function of its third and fourth arguments, has a definite sign, then $ℰ$ has also the same sign; but if $ℰ$ retains a definite sign, $p$ and $q$ being fixed while $\overline{p}$ and $\overline{q}$ are varied, it does not then follow that $F_1$ always has a definite sign. This is illustrated in the following example, due to Schwarz :

Let

$F(x,y,x^{\prime },y^{\prime })=\alpha \sqrt{x{\text{'}}^2+y{\text{'}}^2}+beta\frac{x^{\prime }y{\text{'}}^2}{x{\text{'}}^2+y{\text{'}}^2}=(\alpha +\beta \cos \tau {\sin }^2\tau )\sqrt{x{\text{'}}^2+y{\text{'}}^2}$

It follows that

$F_1(x,y,x^{\prime },y^{\prime })=\frac{\alpha }{(x{\text{'}}^2+y{\text{'}}^2{)}^{3/2}}+2\beta \frac{x^{\prime }(x{\text{'}}^2-3y{\text{'}}^2)}{(x{\text{'}}^2+y{\text{'}}^2{)}^3}=(\alpha +2\beta \cos 3\tau )\frac{1}{(x{\text{'}}^2+y{\text{'}}^2{)}^{3/2}}$

and, since $x{\text{'}}^2+y{\text{'}}^2={\cos }^2\lambda +{\sin }^2\lambda =1$,

$ℰ(x,y,p,q,\overline{p},\overline{q})=\underset{\lambda =0}{\overset{\lambda =\omega }{\int }}\sin (\omega -\lambda )F_1[x,y,\cos (\tau +\lambda ),\sin (\tau +\lambda )]\text{d}\lambda$

$=\underset{\lambda =0}{\overset{\lambda =\omega }{\int }}\sin (\omega -\lambda )(\alpha +2\beta \cos 3\lambda )\text{d}\lambda$

where we have written $\tau +\lambda =\lambda$ or $\tau =0$; i.e., we have taken the $X$-axis as the initial direction, from which $\omega$ is measured.

Noting that

$\sin (\omega +2\lambda )+\sin (\omega -4\lambda )=2\sin (\omega -\lambda )\cos 3\lambda$

it is seen that

$ℰ(x,y,p,q,\overline{p},\overline{q})=(1-\cos \omega )[\alpha +\beta (\cos \omega +{\cos }^2\omega )]$

The greatest and least values that $\cos \omega +{\cos }^2\omega$ can have are 2 and $-1/4$, the corresponding values of $\omega$ being 0 and $2\pi /3$. Hence, if we we make $\alpha =l$ and $/beta=1$, the function $ℰ$ is situated between the values

$\frac{3}{4}(1-\cos \omega )\text{and}3(1-\cos \omega )$

and can consequently vanish only for $\omega =0$, and is never negativeOn the other hand, $1+2\cos 3\tau$ changes sign repeatedly, for example, when $\tau =40^{\circ }$.

Article 161.
The proof stated at the end of Art. 155 is of paramount importance in the determination whether there exists a true maximum or minimum. The proof of the sufficiency of this theorem, as illustrated in Art. 156, was given in a somewhat different form by Prof. Schwarz. Owing to its importance we add another proof, taken from the lectures of Weierstrass.

Let $O{O}_{1}1$ be the curve which satisfies the differential equation $G=0$, and let $0_{1}31$ be the arbitrary curve in the field, as defined in Art. 156. Let 3 be any point on the arbitrary curve, whose coordinates we consider as functions of length of arc $s$ (instead of $t$, as before). The point $O_1$ is taken between 0 and 1 so that the curve $0_{1}31$ may lie wholly within the field, since the field might terminate in a point at 0. From the point 0 we draw a curve to 3 which satisfies the differential equation $G=0$. We consider the sum of integrals $I_{03}+{\overline{I}}_{31}$ as a function of $s$. This function we denote by $\int (s)$. Further, take on the arbitrary curve a point 2 in the neighborhood of the point 3 and before it. Join the points 0 and 2 by a curve which satisfies the differential equation $G=0$. Then, if we denote the increment of $s$ by $\sigma$, it is seen that

$5)\int (s-\sigma )-\int (s)=I_{02}+{\overline{I}}_{21}-I_{03}-{\overline{I}}_{31}=I_{02}-I_{03}+{\overline{I}}_{23}=ℰ(x_3,y_3,p_3,q_3,{\overline{p}}_3,{\overline{q}}_3)\sigma +(\sigma {)}^2$

In the same manner take a point 4 immediately after the point 3 on the arbitrary curve and join this point with the point 0 by a curve which satisfies the differential equation $G=0$. Then we have

$6)\int (s-\sigma )-\int (s)=I_{04}-I_{03}-{\overline{I}}_{34}=-ℰ(x_3,y_3,p_3,q_3,{\overline{p}}_3,{\overline{q}}_3)\sigma +(\sigma {)}^2$

It therefore follows that

$7)\underset{\sigma =0}{\lim }\frac{\int (s-\sigma )-\int (s)}{-\sigma }=\underset{\sigma =0}{\lim }\frac{\int (s+\sigma )-\int (s)}{\sigma }=-ℰ(x_3,y_3,p_3,q_3,{\overline{p}}_3,{\overline{q}}_3)$

that is, the quantity $-ℰ(x_3,y_3,p_3,q_3,{\overline{p}}_3,{\overline{q}}_3)$ is the differential quotient of the function $\int (s)$ at the point 3.

If, then, along the curve $O_{1}31$ the function $ℰ$ is nowhere positive, the function $\int (s)$ continuously diminishes when the point 3 slides from $O_1$ toward the point 1.

Let the point </math>O_{1}</math>, which was taken very near the point 0, coincide with this point; then we can say :

If the function $ℰ$ is nowhere positive and is not zero at every point of the arbitrary curve 031 the integral taken over the original curve is always greater than the integral extended over the curve 031 ; and if the function $ℰ$ is not negative and not zero at evevy point of the curve 031 then the integral taken over the original curve 01 is continuously less than the integral extended over the arbitrary curve 031.

Article 162.
It remains yet to see if it is possible for the function $ℰ$ to vanish along the whole curve 031. It appears from the formula 3) that this is possible only when along the whole curve we have

$(p\overline{q}-q\overline{p}{)}^2=0\text{ or }p\overline{q}-q\overline{p}=0$

In this case every curve 03 which satisfies the differential equation $G=0$ has a common tangent at the point 3 with the curve 031.

We shall show that the curve $MN$ which is formed of the points conjugate to the point 0 has this property, and that no curve having this property can be drawn from 0 within the region that is bounded by $MN$. In other words, $ℰ$ is equal to zero along the curve $MN$, but is not equal to zero for all the points of any other curve that can be drawn within the region that is enveloped by $MN$.

All the curves that satisfy the differential equation $G=0$, which pass through one point, and whose initial directions differ from one another by very small quantities, may be represented (Art. 148) in the form

$x=\varphi (t,k)y=\psi (t,k)$

where the values of $k$ are within certain limits.

To each curve corresponds a different value of $k$. If, therefore, we fix a value of $k$ and take a second value $k+k^{\prime }$ the curve which corresponds to this value may be expressed by the equations

$x+\xi =\varphi (t+{\tau }^{\prime },k+k^{\prime })y+\eta =\psi (t+{\tau }^{\prime },k+k^{\prime })$

where the same value of $t$ corresponds to the initial directions of both curves.

If the latter curve is cut by the former we must have

$0={\varphi }^{\prime }(t){\tau }^{\prime }+\frac{\partial \varphi }{\partial k}{k}^{\prime }+({\tau }^{\prime },k^{\prime }{)}_{2}0={\psi }^{\prime }(t){\tau }^{\prime }+\frac{\partial \psi }{\partial k}{k}^{\prime }+({\tau }^{\prime },k^{\prime }{)}_2$

The determinant of the linear terms of the equations just written gives, when put equal to zero, the equation for the determination of the point conjugate to the initial point, i. e.,

${\varphi }^{\prime }(t)\frac{\partial \psi }{\partial k}-{\psi }^{\prime }(t)\frac{\partial \varphi }{\partial k}=0$ \qquad (A)

The smallest root of this equation, which is greater than the value $t_0$ of $t$, gives the value of $t$, which belongs to the conjugate point. If this value is $t_1$, then the coordinates of the point are

$\overline{x}=\varphi (t_1,k)\overline{y}=\psi (t_1,k)$

If we consider $t_1$ as a function of $k$, defined through the equation (A), and if we give to $k$ a series of values, the two equations just written represent the curve that is constituted of the points conjugate to 0.

The direction-cosines of the tangent to this curve are proportional to the quantities $\frac{\partial \overline{x}}{\partial k},\frac{\partial \overline{y}}{\partial k}$. But we also have

$\frac{\partial \overline{x}}{\partial k}=\frac{\partial \varphi (t_1,k)}{\partial t_1}\frac{\text{d}{t}_1}{\text{d}k}+\frac{\partial \varphi (t_1,k)}{\partial k}\frac{\partial \overline{y}}{\partial k}=\frac{\partial \psi (t_1,k)}{\partial t_1}\frac{\text{d}{t}_1}{\text{d}k}+\frac{\partial \psi (t_1,k)}{\partial k}$

Multiply the first of these equations by $\frac{\partial \psi (t_1,k)}{\partial t_1}={\psi }^{\prime }(t_1)$, and subtract from it the second after it has been multiplied by $\frac{\partial \varphi (t_1,k)}{\partial t_1}={\varphi }^{\prime }(t_1)$. We have then, with the aid of (A),

${\varphi }^{\prime }(t_1)\frac{\text{d}\overline{y}}{\text{d}k}-{\psi }^{\prime }(t_1)\frac{\text{d}\overline{x}}{\text{d}k}=0$

Since ${\varphi }^{\prime }(t_1),{\psi }^{\prime }(t_1)$ are proportional to the direction-cosines of the tangent at a point $t_1$ of the curve through $t_0$ and $t_1$, which satisfies the differential equation $G=0$, it follows from the above equation that the tangents to both curves at the point $t_1$ coincide. Hence, the locus of the conjugate points to is the envelope of the curves through 0, which satisfy the differential equation $G=0$.

Article 163.
Let $\overline{x}=f(u)$ and $\overline{y}=g(u)$ be an arbitrary curve 031, which passes through the point 0, and is situated entirely within the region bounded by the envelope. Further, suppose that 031 does not coincide throughout its whole extent with any of the curves passing through 0, which satisfy the differential equation $G=0$. Suppose, however, that 031 is touched by the curves that pass through and satisfy the differential equation $G=0$. At the point of contact we must have

$\varphi (t,k)=f(u)\psi (t,k)=g(u)$

and

$\frac{\partial \varphi }{\partial t}\frac{\text{d}g}{\text{d}u}-\frac{\partial \psi }{\partial t}\frac{\text{d}f}{\text{d}u}=0$ \qquad (B)

The values of $t$ and $u$, which belong to the point of contact, are determined as functions of $k$ through the first two equations.

These equations, being true for sufficiently small values of $k$, may be differentiated with respect to $k$, and we thus have:

$\frac{\partial \varphi }{\partial t}\frac{\text{d}t}{\text{d}k}+\frac{\partial \varphi }{\partial k}=\frac{\text{d}f}{\text{d}u}\frac{\text{d}u}{\text{d}k}\frac{\partial \psi }{\partial t}\frac{\text{d}t}{\text{d}k}+\frac{\partial \psi }{\partial k}=\frac{\text{d}g}{\text{d}u}\frac{\text{d}u}{\text{d}k}$

If we multiply the first of these equations by $\frac{\text{d}g}{\text{d}u}$ and the second by $-\frac{\text{d}f}{\text{d}u}$ and add we have with the aid of (B)

$\frac{\partial \varphi }{\partial k}\frac{\text{d}g}{\text{d}u}-\frac{\partial \psi }{\partial k}\frac{\text{d}f}{\text{d}u}=0$

If between this equation and the equation (B) we eliminate the quantities $\frac{\text{d}g}{\text{d}u}$ and $\frac{\text{d}f}{\text{d}u}$, we have

$\frac{\partial \varphi }{\partial t}\frac{\partial \psi }{\partial k}-\frac{\partial \psi }{\partial t}\frac{\partial \varphi }{\partial k}=0$

an equation, which served for the determination of the point conjugate to the initial point. Consequently the point of contact of the curve, that passes through 0 and satisfies the differential equation $G=0$, with the arbitrary curve must be the point conjugate to 0.

But this is possible only if the curve $\overline{x}=f(u),\overline{y}=g(u)$ coincides with the envelope ; while according to our supposition the curve 031 is to lie entirely within the region that is bounded by the envelope. It follows that there can be within the region no curve 031 such that each of the curves which satisfies the differential equation $G=0$, and which joins the point 0 with a point of 031, touches 031 at the same time.

Hence, the quantity $q\overline{p}-p\overline{q}$ can be everywhere zero only when the arbitrary curve between 0 and 1 coincides throughout its whole extent with one of the curves that passes through 0 and satisfies the differential equation $G=0$. But since, within the strip of surface inclosing the field as we have defined it, there can be only one curve drawn through 0 and 1 which satisfies the differential equation $G=0$, it follows that the arbitrary curve 031 can coincide only with the original curve 01, and then it is not a variation of that curve. It therefore follows that the function $ℰ$ cannot vanish for all the points of the curve that has been subjected to variation.

Article 164.
It is not necessary that the curve 031 be a single trace of a regular curve in its whole extent. If we assume that 031 is composed of an arbitrary number of regular portions of curve, the integral may be regarded as the sum of the integrals over the single portions, and the conclusions made above are also applicable.

It may happen that one of the portions of curve coincides throughout its whole extent with a portion of one of the curves that goes through 0 and satisfies the differential equation $G=O$. If this is the case for 23, for example, so that $ℰ$ is equal to zero along 23, then we may replace this portion of curve by an arbitrary portion of curve $2^{\prime }3$, which lies very near 23. Then the theorem proved above is true for the curve $02^{\prime }31$, viz., that

$I_{03}\stackrel{>}{<}{I}_{02^{\prime }}+{\overline{I}}_{2^{\prime }3}$

according as the function $ℰ$ is nowhere positive or nowhere negative along the curve $02^{\prime }31$. Now, if we bring the curve $2^{\prime }3$ as near to the curve 23 as we wish, the absolute value of the difference $I_{03}-I_{02^{\prime }}-{\overline{I}}_{2^{\prime }3}$ can be made smaller than any arbitrarily small quantity $\delta$; and, in accordance with what was proved above, in the first case the difference $I_{03}-I_{02^{\prime }}-{\overline{I}}_{2^{\prime }3}$ is certainly not negative, and in the second case it is not positive.

If we shove the point 3 further along the arbitrary curve toward 1, then, when 3 takes a position in the neighborhood of 4, it follows again that $I_{04}-I_{03}-{\overline{I}}_{34}$ is greater or less than zero, and, as above, we see that the integral $I_{01}$, extended over the curve that satisfies the differential equation $G=0$, is greater or less than the integral taken over the arbitrary curve 0231, according as the function $ℰ$ is nowhere negative or nowhere positive.

Article 165.
Further, it is not necessary that the single portions of the curve which has been subjected to variation be regular in order that our conclusions be correctly drawn, if only the coordinates can be expressed as functions of some quantity, and if these functions have derivatives. Finally, if we consider the variation made quite arbitrary, so that only the positions of the points are given, while it is not known whether their coordinates have derivatives, then indeed the integral taken over this curve has no longer any meaning. But the meaning of the integral may be extended so that it has a signification even in this case. For if at first we assume that the coordinates of the curve, which has been subjected to variation, are expressible through functions that have derivatives, then the integral taken over the curve is

$\underset{t_0}{\overset{t_1}{\int }}F[f(t),g(t),f^{\prime }(t),g^{\prime }(t)]\text{d}t$

This integral distributed into a sum of integrals (corresponding to the intervals $t_0\dots {\tau }_1,{\tau }_1\dots {\tau }_2,\dots ,{\tau }_n\dots t_1$ is equal to

$\underset{t_0}{\overset{{\tau }_1}{\int }}F\text{d}t+\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}F\text{d}t+\cdots +\underset{{\tau }_n}{\overset{t_1}{\int }}F\text{d}t$ \qquad (C)

We assume that the points $x_0,y_0;x_1,y_1;\dots x_n,y_n;x_{n+1},y_{n+1}$ correspond to the values $t_{0}m{\tau }_1,\dots {\tau }_n,t_1$.

We then have:

$x_1-x_0=f^{\prime }(t_0)({\tau }_1-t_0)+({\tau }_1-t_0)[{\tau }_1-t_0]$

$..........................................$

$x_{n+1}-x_n=f^{\prime }({\tau }_n)(t_1-{\tau }_n)+(t_1-{\tau }_n)[t_1-{\tau }_n]$

$y_1-y_0=g^{\prime }(t_0)({\tau }_1-t_0)+({\tau }_1-t_0)[{\tau }_1-t_0]$

$..........................................$

$y_{n+1}-y_n=g^{\prime }({\tau }_n)(t_1-{\tau }_n)+(t_1-{\tau }_n)[t_1-{\tau }_n]$

where $[{\tau }_v-{\tau }_{v-1}]$ denotes a quantity which becomes indefinitely small at the same time with ${\tau }_v-{\tau }_{v-1}$.

For the first of the integrals in the expression (C) we write:

$x=x_0+x_{0^{\prime }}(t-t_0)+(t-t_0)[t-t_0]$

$y=y_0+y_{0^{\prime }}(t-t_0)+(t-t_0)[t-t_0]$

$x^{\prime }=x_{0^{\prime }}+x_{0^{″}}(t-t_0)+(t-t_0)[t-t_0]$

$y^{\prime }=y_{0^{\prime }}+y_{0^{″}}(t-t_0)+(t-t_0)[t-t_0]$

for the second integral we write

$x=x_1+x_{1^{\prime }}(t-{\tau }_1)+(t-{\tau }_1)[t-{\tau }_1]$

$y=y_1+y_{1^{\prime }}(t-{\tau }_1)+(t-{\tau }_1)[t-{\tau }_1]$

and similarly for the other integrals.

These expressions we write in the sum of integrals (C), and, developing them in power-series, we have through integration

$({\tau }_1-t_0)F(x_0,y_0,x_{0^{\prime }},y_{0^{\prime }})+({\tau }_2-{\tau }_1)F(x_1,y_1,x_{1^{\prime }},y_{1^{\prime }})+\cdots +(t_1-{\tau }_n)F(x_n,y_n,x_{n^{\prime }},y_{n^{\prime }})\dots$

plus a similar number of terms, which become indefinitely small of the second order with respect to the quantities ${\tau }_v-{\tau }_{v-1}$.

We may therefore write the integral in the form

$\underset{n\to \mathrm{\infty }}{\lim }(\sum _{v=1}^{n+1}F(x_{v-1},y_{v-1},x_{v^{\prime }-1},y_{v^{\prime }-1}))$

where we must understand by ${\tau }_0$ the value $t_0$, and by $t_{n+1}$ the value $t_1$.

Since ${\tau }_v-{\tau }_{v-1}$ are positive quantities, and the functions $F$ in regard to $x_{1^{\prime }},y_{1^{\prime }}\dots$ are homogeneous of the first degree, we may write the above limit in the form

$\underset{n\to \mathrm{\infty }}{\lim }(\sum _{v=1}^{n+1}F(x_{v-1},y_{v-1},({\tau }_v-{\tau }_{v-1})x_{v^{\prime }-1},({\tau }_v-{\tau }_{v-1})y_{v^{\prime }-1}))$

or, since

$x_v-x_{v-1}=({\tau }_v-{\tau }_{v-1})x_{v^{\prime }-1}+({\tau }_v-{\tau }_{v-1})[{\tau }_v-{\tau }_{v-1}]$

the above expression is

$\underset{n\to \mathrm{\infty }}{\lim }(F(x_0,y_0,x_1-x_0,y_1-y_0)+\cdots +F(x_n,y_n,x_{n+1}-x_n,y_{n+1}-y_n))$

Article 166.
The integral in the above form has a more general meaning than the one hitherto employed, with which, however, it coincides in every particular where that one has a meaning. We may assume, with respect to any arbitrary variation, a series of points $x_0,y_0;x_1,y_1;\dots x_n,y_n;x_{n+1},y_{n+1}$ of such a nature that the distance between, say, two successive points does not exceed a certain quantity $\delta$.

We then form the sum

$F(x_0,y_0,x_1-x_0,y_1-y_0)+\cdots +F(x_n,y_n,x_{n+1}-x_n,y_{n+1}-y_n)$

If we make $\delta$ smaller and smaller by increasing the number of points, it may happen that this sum approaches a definite limit. We call this limit the value of the integral taken over the curve. It may also happen that the limit does not approach a definite value; for example, it may vacillate between two values. We then say the integral taken over this curve has no meaning.

If we think of the series of points that are taken upon the curve, joined together successively by a broken line, the integral taken over this broken line will approach the same limit as will the integral taken over the curve, if the integral has a meaning.

If, therefore, a curve 01 is given, which satisfies all the conditions that have hitherto been made for a maximum or a minimum, and if this curve varies in an arbitrary manner, then if the integral taken over the curve, which has been subjected to variation, has a meaning as defined above, we ma)' draw a broken line, the integral over which deviates as little as we wish from the integral taken over the curve that has been caused to vary and to which the theorem of Art. 161 is applicable. Consequently, we may say, in the case of a maximum, the integral taken over the curve subjected to variation cannot be greater than the integral taken over the original curve, and in the case of a m,inimum,, it cannot be less than the integral taken over the original curve.

Since we may make the region as narrow as we wish within which all the variations are to lie, we ma)' assume that upon the curve which has been varied a point 3 lies so near to 01 (but not upon it) that two curves 03, 31 can be drawn between the points and 3 and between 3 and 1, which also satisfy all the conditions of the problem.

For the sake of brevity, let us assume that we have to do with a maximum. Then, as we have just seen, the integrals over 03 and 31 cannot at all events be smaller than the integrals over the corresponding parts of the curve which has been varied ; but, after the preceding theorems, the integral taken over 01 is greater than the sum of the integrals taken over 03 and 31, and consequently also greater than the integral over the curve that has been varied. A maximum is therefore in reality present.

Article 167.
We may now investigate the behavior of the function $ℰ$ in the case of the four problems which we last considered in Arts. 140–144.

The problem of the surface of rotation of minimum area.

We saw that the catenary between limits, within which were situated no pair of conjugate points, was the curve that described a surface of minimum area when rotated around the axis of the half-plane. From the point $P_0$ we may draw in any direction a curve which satisfies the differential equation $G=0$ (a catenary); the function $F_1$ is positive for each of these curves as soon as we limit ourselves to the half-plane in which $y$ is positive. A true minimum will therefore in reality enter. For if $p,q$ are the direction-cosines of the tangent to the catenary at any point, $\overline{p},\overline{q}$ those of the tangent to any arbitrary curve through the same point, then, owing to the relations

$F^{(1)}(x,y,x^{\prime },y^{\prime })=\frac{y{x}^{\prime }}{\sqrt{x{\text{'}}^2+y{\text{'}}^2}}{F}^{(2)}(x,y,x^{\prime },y^{\prime })=\frac{y{y}^{\prime }}{\sqrt{x{\text{'}}^2+y{\text{'}}^2}}$

it follows that

$F^{(1)}(x,y,x^{\prime },y^{\prime })=yp{F}^{(2)}(x,y,x^{\prime },y^{\prime })=yq$

since

$p^2+q^2=1$

and consequently

$ℰ(x,y,p,q,\overline{p},\overline{q})=y((\overline{p}-p)\overline{p}+\overline{q}-q)\overline{q})=y(1-(p\overline{p}+q\overline{q}))$

The expression $p\overline{p}+q\overline{q}$ is the cosine of the angle between the two tangents. Hence we see that the function $ℰ$ is negative for no point which comes under consideration, and for no two directions $p,q$ and $\overline{p},\overline{q}$.

If, therefore, $y=0$ for no point of the curve, our former conclusions are applicable, and a true minimum of the integral has, in reality, been found.

Article 168.
The Brachistochrone. We saw that this curve is the cycloid

$x=g+r(1-\sin t)y+a=r(1-\cos t)$

We assume that the point $A$, from which the moving point starts, having an initial velocity proportional to the quantity $\sqrt{a}$, is the origin of coordinates, and that the $Y$-axis is the direction of gravity. We saw that the cycloid could then be generated by a point described by a circle which rolls upon the straight line $y=-a$. If $a$ is different from zero, an arc of a cycloid may be constructed through $A$ in any direction. If the curve passes through a singular point it does not minimize the integral, as was shown in Art. 104. If $A$ and $B$ are not singular points, the function $F_1$ has a positive value different from zero everywhere along this curve and in the neighborhood of it in every direction.

Between two arbitrary points (see Art. 105), when the quantity $a$ is given, there can alwaj'-s be drawn one, and only one, arc of a cycloid which has no singular points between these two points. If, therefore, $a$ is different from zero, and consequently $A$ and $B$ are not singular points, then (see Art. 159) it follows that the curve, in reality, causes the integral to have a minimum value. Suppose that $A$ or $B$ is a singular point; then at this point $F_1$ becomes infinite, a case which we consider in the next Article.

Article 169.
Suppose $A$ is a singular point and $a=0$. Draw an arbitrary curve between $A$ and $B$. Take upon this curve in the neighborhood of $A$ a point $A_1$, and through $A_1$ and $B$ draw a cycloid which cuts the $X$-axis at $A_{1^{\prime }}$. The material point under the action of gravity passes through $A_1$ with the same velocity which it would have at an equal distance below the $X$-axis if it traversed the cycloid drawn through $A$ and $B$.

The following notation may be introduced :

$I_{01}$ to denote the time of falling between $A_{1^{\prime }}$ and $B$ upon the cycloid $A_{1^{\prime }}B$,

$I_{0^{\prime }1}$ to denote the time of falling between $A$ and $A_1$ upon the arbitrary curve $AB$,

$I$ to denote the time of falling between $A_1$ and $B$ upon the cycloid $A_{1}B$,

$I^{\prime }$ to denote the time of falling between $A_1$ and $B$ upon the arbitrary curve $A_{1}B$.

We proved that

$I^{\prime }>I$

and therefore, if we write

$\overline{I}=I_{0^{\prime }1}+I^{\prime }$

it follows that

$\overline{I}>I+I_{0^{\prime }1}$

Now, let the point $A_1$ approach nearer and nearer the point $A$, so that the integral $I$ approaches the limit $I_{01}$, while $I_{0^{\prime }1}$ becomes indefinitely small. We must then have

$\overline{I}\ge I_{01}$

That $\overline{I}$ is greater than $I_{01}$ may be seen as follows: As soon as $G\ne 0$ along a portion of curve, we may always vary it in such a way that the increment in the corresponding integral may have any sign. If, then, $G\ne 0$ along the whole curve $A{A}_{1}B$, we may substitute another curve, for which, if $I^{″}$ is the value of the integral which belongs to it,

$I^{″}<\overline{I}$

But since we also have

$I^{″}\ge I_{01}$

it follows that

$\overline{I}>I_{01}$

If, on the other hand, $G=0$ along the whole curve $A{A}_{1}B$, then this curve must consist of several cycloidal arcs ; since, if it were only one, the curves $A{A}_{1}B$ and $AB$ would be identical. These arcs must have different tangents at the point where they come together ; for, since this point cannot lie on the $X$-axis, a consecutive point having the same direction must lie on the same cycloidal arc. If corners were present, however, they could be so rounded off that there would be a shorter path between the two points, and consequently, the velocity being the same, the time of falling would be shorter.

Hence the arc of a cycloid also minimizes the time of falling between $A$ and $B$ in the case where $A$ is a singular point ; that is, when the material point starts from $A$ with an initial velocity that is zero.

The conclusions just made are also applicable, if $B$ is a singular point ; for it makes no difference whether the material point ascends from $B$ to $A$ or falls from $A$ to $B$, if we allow the material point to go back with the same initial velocity with which it arrived at $B$. On the way back it will reach $A$ with its original velocity. Its velocity will be the same in both cases at all points of the curve, but directed toward opposite directions. The integral taken over the curve has the same value in both cases ; and consequently the curve which caused the integral to have a minimum value will also, in the second case, minimize the integral.

Article 170.
The problem of the geodesic line on a sphere offers here nothing of special interest. It is found that the function $ℰ$ retains a positive sign along the arc of a great circle situated between two poles.

Article 171.
Problem of the surface of revolution which offers the least resistance.

In this problem

$F()x,y,x^{\prime },y^{\prime })=\frac{xx{\text{'}}^3}{x{\text{'}}^2+y{\text{'}}^2}$

and since

$p^2+q^2=1$

it follows that

$F(x,y,p,q)=\frac{x{p}^3}{p^2+q^2}=x{p}^3$

$\frac{\partial F}{\partial p}=x(p^4+3p^2{q}^2)\frac{\partial F}{\partial q}=-2x{p}^{3}q$

Substituting these values in

$ℰ(x,y,p,q,\overline{p},\overline{q})=F(x,y,\overline{p},\overline{q})-\overline{p}\frac{\partial F}{\partial p}-\overline{q}\frac{\partial F}{\partial q}$

we have

$ℰ(x,y,p,q,\overline{p},\overline{q})=x[{\overline{p}}^3-\overline{p}{p}^2-2\overline{p}{p}^2{q}^2+2\overline{q}{p}^{3}q]$

$=x[\overline{p}({\overline{p}}^2-p^2)+2p^{2}q(p\overline{q}-\overline{p}q)]$

$=x[\overline{p}({\overline{p}}^2(p^2+q^2)+2p^{2}q(p\overline{q}-\overline{p}q))+2p^{2}q(p\overline{q}-\overline{p}q)]$

$=x(\overline{p}q-p\overline{q})[\overline{p}(\overline{p}q+p\overline{q})-2p^{2}q]$

$=x(\overline{p}q-p\overline{q})[\overline{q}(\overline{q}q-p\overline{q})(p^2+q^2)+2p\overline{p}\overline{q}(p^2+q^2)-2p^{2}q({\overline{p}}^2+{\overline{q}}^2)]$

$=x(\overline{p}q-p\overline{q})[\overline{p}(\overline{p}q-p\overline{q})(p^2+q^2)-2p^2\overline{p}(\overline{p}q-p\overline{q})+2pq\overline{q}(\overline{p}q-p\overline{q})]$

$=x(\overline{p}q-p\overline{q})[\overline{p}(p^2+q^2)-2p^2\overline{p}+2pq\overline{q}]$

$=x(\overline{p}q-p\overline{q})[\overline{p}(q^2-p^2)+2pq\overline{q}]$

Writing

$\cos \overline{\tau }=\overline{p}\sin \overline{\tau }=\overline{q}\cos \tau =p\sin \tau =q$

we have

$ℰ(x,y,p,q,\overline{p},\overline{q})=-x\sin (\overline{\tau }-\tau {)}^2\cos (\overline{\tau }+2\tau )$

Therefore, the sign of $ℰ$ is the same as that of $-\cos (\overline{\tau }+2\tau )$, and may be either positive or negative by properly choosing $\overline{\tau }$, an angle which depends upon $\overline{p},\overline{q}$.

At every point of the curve for which $x\ne 0$ the function $ℰ$ can have different signs, and consequently a maximum or a minimum value of the integral does not exist. We saw in Art. 109 that $x$ must be different from zero for all points of the arc.

Article 172.
Legendre (Mimoire sur la manihre de distinguer les maxima des minima dans le Calcul des Variations) showed that by taking a zigzag line for the generating curve, the resistance could be made as small as we wish.

Suppose that the arc $P_1{P}_2$, had the desired property of generating a surface of least resistance, and suppose that the tangent to this curve is nowhere parallel to the $X$-axis. Writing $p=\frac{\text{d}x}{\text{d}y}$, it follows that $p\ne 0$ along the arc $P_1{P}_2$.

We have then (Art. 108)

$I_{1,2}=\underset{t_1}{\overset{t_2}{\int }}\frac{xx{\text{'}}^3}{x{\text{'}}^2+y{\text{'}}^2}\text{d}t=\underset{x_1}{\overset{x_2}{\int }}\frac{p^{2}x}{1+p^2}\text{d}x$

Since $p$ is finite and continuous along the arc in question, it follows that $\frac{p^2}{1+p^2}$ has the same properties along the arc, and therefore

$I_{1,2}=\frac{p_0^{2}x}{1+p_0^2}\underset{x_1}{\overset{x_2}{\int }}x\text{d}x=\frac{p_0^{2}x}{1+p_0^2}\frac{x_2^2-x_1^2}{2}$

where $p_0$ is a mean value of $p$, lying between the points $P_1$ and $P_2$ of the curve.

Between the ordinates at $P_1$ and $P_2$ draw a line parallel to the $Y$-axis, and on this line take a point $P_3$ whose ordinate is longer than those of the points $P_1$ and $P_2$. Draw the straight lines $P_1{P}_3$ and $P_2{P}_3$, and let $p_1$ and $p_2$ be the values of $\frac{\text{d}x}{\text{d}y}$ for these lines. The integral $\int F\text{d}t$ taken over the broken line $P_1{P}_3{P}_2$ may be denoted by $I_{13}+I_{32}$, where

$I_{13}=\underset{x_1}{\overset{x_3}{\int }}\frac{p_1^{2}x}{1+p_1^2}\text{d}x=\frac{p_1^2}{1+p_1^2}\frac{x_3^2-x_1^2}{2}$

and

$I_{32}=\underset{x_3}{\overset{x_2}{\int }}\frac{p_2^{2}x}{1+p_2^2}\text{d}x=\frac{p_2^2}{1+p_2^2}\frac{x_2^2-x_3^2}{2}$

We have then

$I_{132}-I_{12}=I_{13}+I_{32}-I_{12}=\frac{p_1^2(x_3^2-x_1^2)}{2(1+p_1^2)}+\frac{p_2^2(x_2^2-x_3^2)}{2(1+p_2^2)}-\frac{p_0^2(x_2^2-x_1^2)}{2(1+p_0^2)}$

The first two terms of this expression may be made as small as we choose by sufficiently diminishing the quantities $p_1$ and $p_2$, which is done by removing indefinitely the point $P_3$ along its ordinate. Hence, their sum is less than the third term, so that, consequently,

$I_{132}<I_{12}$

This result may also be derived as follows :

$\frac{I_{13}+I_{32}}{I_{12}}=\frac{p_1^2(1+p_0^2)}{p_0^2(1+p_1^2)}\frac{x_3^2-x_1^2}{x_2^2-x_1^2}+\frac{p_2^2(1+p_0^2)}{p_0^2(1+p_2^2)}\frac{x_2^2-x_3^2}{x_2^2-x_1^2}<\frac{p_1^2(1+p_0^2)}{p_0^2(1+p_1^2)}+\frac{p_2^2(1+p_0^2)}{p_0^2(1+p_2^2)}$

since $x_1<x_3<x_2$.

Hence also for a greater reason

$\frac{I_{13}+I_{32}}{I_{12}}=<(p_1^2+p_2^2)\frac{1+p_0^2}{p_0^2}$

From this it is seen that the ratio $\frac{I_{13}+I_{32}}{I_{12}}$ may be indefinitely dimmished by properly choosing $p_1$ and $p_2$. There is then no limit to the least possible resistance.

The method just given does not replace the $ℰ$-criterion which shows that no surface of minimal resistance exists. It shows simply that no rotational surface exists, which gives an absolute minimum of resistance—a resistance less than any other neighboring surface. The $ℰ$-criterion shows that no minimum exists in the sense of giving a resistance less than that given any neighboring curve within a limited neighborhood.

Article 173.
In the general case, when $F(x,y,x^{\prime },y^{\prime })$ is a rational function of $x^{\prime }$ and $y^{\prime }$, neither a maximum nor a minimum can exist. For in this case

$ℰ=F(x,y,\overline{p},\overline{q})-\overline{p}\frac{\partial F}{\partial p}-\overline{q}\frac{\partial F}{\partial q}$

is also a rational function of $\overline{p}$ and $\overline{q}$ and homogeneous in these quantities of the first degree. Consequently

$ℰ(x,y,k\overline{p},k\overline{q})=kℰ(x,y,\overline{x},\overline{y})$

and therefore

$ℰ(x,y,-\overline{p},-\overline{q})=-ℰ(x,y,\overline{p},\overline{q})$

It is thus seen that we have only to reverse the direction of the displacement to effect a change of sign in the function $ℰ$.

Article 174.
We have now completely solved the four problems that were proposed in Chapter I, and at the same time one of the principal parts of the Calculus of Variations has been finished. After stating succinctly the four criteria that have been established, we shall take up the second part, which has as its object the theoretical and practical solution of problems, a general type of which were the Problems V and VI of Chapter I.

These criteria may be summarized as follows (cf. Art. 125): There exists a minimum or a maximum value of the integral

$I=\underset{t_0}{\overset{t_1}{\int }}F(x,y,x^{\prime },y^{\prime })\text{d}t$

where $F$ is a one-valued, regular function of its four arguments and homogeneous of the first degree in $x^{\prime }$ and $y^{\prime }$, if

1) the differential equation $G=0$ is satisfied for every point of the curve;

2) $F_1$ is positive or negative throughout the whole interval $t_0\dots t_1$;

3) there are no conjugate points of the curve within the interval $t_0\dots t_1$ (limits included);

4) the function $ℰ$ is positive or negative throughout the whole interval $t_0\dots t_1$

In this discussion we have excluded the cases where

1) the extremities of the curve are conjugate points;

2) $F_1=0$ for some point of the curve;

3) $F_1=0$ for some stretch of the curve;

4) $ℰ=0$ for some point or stretch of the curve.

A general treatment of the first three cases would require the extension of the theory to variations of a higher order. Otherwise particular devices must be employed in every example in which one of the above exceptional cases is found.

Article 175.
Before we begin the consideration of Relative Maxima and Minima, we may, at least, indicate the natural extensions and generalizations of the theory which has already been presented : Instead of the determination of a structure of the first kind in the domain of two quantities, it may be required to determine a structure of the first hind in the domain of $n$ quantities.

If a structure of the first kind is determined in the domain of the $n$ quantities $x_1,x_2,\dots ,x_2$, then $n-1$ of these quantities may be expressed as functions of the remaining one, say, $x_1$.

Writing

$u=\int F(x_1,x_2,\dots ,x_n,\frac{\text{d}{x}_2}{\text{d}{x}_1},\frac{\text{d}{x}_3}{\text{d}{x}_1},\dots ,\frac{\text{d}{x}_n}{\text{d}{x}_1})\text{d}{x}_1$

it is seen that $u$ is so connected with the $n-1$ functions that

$\frac{\text{d}u}{\text{d}{x}_1}=F(x_1,x_2,\dots ,x_n,\frac{\text{d}{x}_2}{\text{d}{x}_1},\frac{\text{d}{x}_3}{\text{d}{x}_1},\dots ,\frac{\text{d}{x}_n}{\text{d}{x}_1})$

The difference of the values of $u$ at the initial-point and at the end-point of the structure is expressed by a definite integral.

This integral takes the form, when we consider the $x$'s expressed as functions of $t$, say, $x_1=x_1(t),x_2=x_2(t),\dots ,x_n=x_n(t)$,

$I=\underset{t_0}{\overset{t_1}{\int }}F(x_1,x_2,\dots ,x_n,x_{1^{\prime }},x_{2^{\prime }},\dots ,x_{n^{\prime }})\text{d}t$

The function $F$ must be a one-valued, regular function of its arguments in the whole or a limited portion of the fixed domain.

The value of the integral $I$ is independent of the manner in which the variables $x_1,x_2,\dots ,x_n$ have been expressed as functions of $t$. It therefore follows after the analogon of Art. 68 that the function $F$ is subjected to the further restriction :

$kF(x_1,x_2,\dots ,x_n,x_{1^{\prime }},x_{2^{\prime }},\dots ,x_{n^{\prime }})=F(x_1,x_2,\dots ,x_n,k{x}_{1^{\prime }},k{x}_{2^{\prime }},\dots ,k{x}_{n^{\prime }})$

where $k$ is a positive constant.

The indicated generalization of the problem given in Art. 13 may accordingly be expressed as follows :

The $n$ quantities $x_1,x_2,\dots ,x_n$ are to be determined as functions of a quantity $t$ in such a manner that for the analytical structure that is defined through the equations

$x_1-x_1(t),x_2=x_2(t),\dots ,x_n=x_n(t)$

the value of the integral

$I=\underset{t_0}{\overset{t_1}{\int }}F(x_1,x_2,\dots ,x_n,x_{1^{\prime }},x_{2^{\prime }},\dots ,x_{n^{\prime }})\text{d}t$

is a maximum or a m.inim,um,; in other words, if one causes the above analytical structure to vary indefinitely little, the change in the integral thereby produced must in the case of a maximum be constantly negative, and in the case of a minimum it must be constantly positive. Further, the function $F$ is to be considered a one-valued, regular function of its arguments, and indeed, with respect to $x_{1^{\prime }},x_{2^{\prime }},\dots ,x_{n^{\prime }}$, a homogeneous function of the first degree.

Article 176.
The treatment of the above problem is found to be the complete analogon of the problem given in Art. 13. A greater complication arises when there are present equations of condition among the variables $x_1,x_2,\dots ,x_n$. An example of this kind we had in Problem III of Chapter I.

This problem may be expressed thus : Among all the curves in space which belong to the surface

$f(x,y,z)=0$

determine that one for which the integral

$\underset{t_0}{\overset{t_1}{\int }}\sqrt{x{\text{'}}^2+y{\text{'}}^2+z{\text{'}}^2}\text{d}t$

is a minimum.

The general problem may be formulated as follows : Among the structures of the first kind in the domain of the quantities $x_1,x_2,\dots ,x_n$, for which the $m$ equations

$f_{\mu }(x_1,x_2,\dots ,x_n)=0\mu =1,2,\dots ,m;m<n-1$

exist, that one is to be determined for which the integral

$I=\underset{t_0}{\overset{t_1}{\int }}F(x_1,x_2,\dots ,x_n,x_{1^{\prime }},x_{2^{\prime }},\dots ,x_{n^{\prime }})\text{d}t$

is a maximum or a minimum.

This problem may be reduced to the one of the preceding Article in an analogous manner as is done in Art. 10 for the case of the shortest line upon a surface. The $m$ equations of condition may be satisfied by introducing for the variables $x_1,x_2,\dots ,x_n$ functions of $n-m$ new variables after the method given in the Lectures on the Theory of Maxima and Minima, etc., Chapter I, Art. 15. The new variables are independent of one another, so that the above integral may be replaced by one in which the variables are free from extraneous conditions ; or we may proceed as was done in the Theory of Maxima and Minima where the variables are subject to subsidiary conditions (loc. cit., p. 54).

Article 177.
The more general problem of the Calculus of Variations, in so far as it has to do with the structures of the first kind, may be stated as follows:

Among the structures of the first kind in the domain of the $n$ quantities $x_1,x_2,\dots ,x_n$, for which definite equations of condition exist, not only among the $n$ quantities themselves, but also among their first derivatives, that structure is to be determined for which the integral

$I=\underset{t_0}{\overset{t_1}{\int }}F(x_1,x_2,\dots ,x_n,x_{1^{\prime }},x_{2^{\prime }},\dots ,x_{n^{\prime }})\text{d}t$

becomes a maximum or a minimum

It may be easily shown that the apparently more general case in which $F$ is a function of $x_1,x_2,\dots ,x_n$ and of the first and higher derivatives of these quantities, is contained in the problem just stated. For the sake of simplicity, take the case where only two variables are involved and write

$u=\underset{x_1}{\overset{x_2}{\int }}F(x,y,\frac{\text{d}y}{\text{d}x},\frac{{\text{d}}^{2}y}{\text{d}{x}^2},\dots )\text{d}x$

If in this integral we express $x$ and $y$ as functions of $t$, we have

$\frac{\text{d}y}{\text{d}t}=\frac{\text{d}y}{\text{d}x}{x}^{\prime }\frac{{\text{d}}^{2}y}{\text{d}{t}^2}=\frac{{\text{d}}^{2}y}{\text{d}{x}^2}x{\text{'}}^2+\frac{\text{d}y}{\text{d}x}{x}^{″}$

We may consequently change the integral $u$ into

$u=\underset{t_0}{\overset{t_1}{\int }}F(x,y,x^{\prime },y^{\prime },x^{″},y^{″},\dots )\text{d}t$

We further have

$\frac{\text{d}{x}^{\prime }}{\text{d}t}-x^{″}=0\frac{\text{d}{y}^{\prime }}{\text{d}t}-y^{″}=0$

We may therefore write

$u=\underset{t_0}{\overset{t_1}{\int }}F(x,y,x_1,y_1,\frac{\text{d}{x}_1}{\text{d}t},\frac{\text{d}{y}_1}{\text{d}t})\text{d}t$

with the equations of condition :

$\frac{\text{d}x}{\text{d}t}=x_1\frac{\text{d}y}{\text{d}t}=y_1$

If, then, there appear in $F$ only the first and second derivatives, it is seen that $F$ depends upon the four functions $x,y,x_1,y_1$ which are to be determined, while at the same time the two equations of condition just written must be satisfied. One of the classes of problems belonging to the general problem just stated is the one which was formulated in Art. 17 and which is treated in the following Chapters.

Article 178.
It may be mentioned finally that the problem of the Calculus of Variations may be further generalized, if we require the determination of structures of a higher kind. For example, in the simplest case the three quantities $x,y,z$ may be determined as functions of two independent variables $u$ and $v$. We have then instead of the single integral the double integral

$\int \int F(x,y,z,\frac{\partial x}{\partial u},\frac{\partial y}{\partial u},\frac{\partial z}{\partial u},\frac{\partial x}{\partial v},\frac{\partial y}{\partial v},\frac{\partial z}{\partial v})\text{d}u\text{d}v$

which must be a maximum or a minimum.

The treatment of this problem would give a theory of Minimal Surfaces.