# Calculus of Variations/CHAPTER X

CHAPTER X: THE CRITERIA THAT HAVE BEEN DERIVED UNDER THE ASSUMPTION OF CERTAIN SPECIAL VARIATIONS ARE ALSO SUFFICIENT FOR THE ESTABLISHMENT OF THE FORMULAE HITHERTO EMPLOYED.

• 134 The process employed is one of progressive exclusion.
• 135 Summary of the three necessary conditions that have been derived.
• 136,137 Special variations. The total variation.
• 138 A theorem in quadratic forms.
• 139 Establishment of the conditions that have been derived from the second variation.
• 140,141,142,143,144 Application to the first four problems of Chapter I.

Article 134.
The methods which we have followed would indicate that the whole process of the Calculus of Variations is a process of progressive exclusion. We first exclude curves for which ${\displaystyle G}$ is different from zero and limit ourselves to curves which satisfy the differential equation ${\displaystyle G=0}$. From these latter curves we exclude all those along which ${\displaystyle F_{1}}$ does not retain the same sign. If, for any curves not yet excluded, ${\displaystyle F_{1}=O}$ at isolated points, we have simply a limiting case among those to which our conclusions apply. If ${\displaystyle F_{1}=0}$ for a stretch of curve not excluded by the above condition, we have to subject the curve to additional consideration in which the third and higher variations must be investigated. We further exclude all curves, in which conjugate points are found situated between the limits of integration, as being impossible generators of a maximum or a minimum. The cases in which no such pairs of points are to be found, or where such points are the limits of integration, require further investigation. This leads us to a fourth condition, a condition due to Weierstrass, which is discussed in Chapter XII. In this process of exclusion let us next see whether the variations admitted are sufficient for the general treatment under consideration.

Article 135.
As necessary conditions for the appearance of a maximum or a minimum, the following theorems have been established:

1) ${\displaystyle x,y}$ as functions of ${\displaystyle t}$ must be determined in such a – manner that they satisfy the differential equation ${\displaystyle G=0}$.

2) Along the curve that has been so determined the function ${\displaystyle F_{1}}$ cannot be positive for a maximum nor can it be negative for a minimum; moreover, the case that ${\displaystyle F=0}$ at isolated points or along a certain stretch, cannot in its generality be treated, but the problems that thus arise m,ust be subjected to special investigation.

3) The integration may extend at most from a point to its conjugate point, but not beyond this point.

The last two conditions, which were derived from the consideration of the second variation, require certain limitations. On the one hand, a proof has to be established that the sign of ${\displaystyle \Delta I}$ is in reality the same as the sign of ${\displaystyle \delta ^{2}I}$, if we choose for ${\displaystyle \xi }$, ${\displaystyle \eta }$, etc., the most general variations of all those special variations, for which the developments hitherto made were true ; it then remains to investigate whether and how far the criteria which have been established remain true for the case where the curve varies quite arbitrarily.

Article 136.
We return to the proof of the theorem proposed in the preceding article. We have, in the case of the investigations hitherto made, always assumed that ${\displaystyle \xi }$, ${\displaystyle \eta }$, ${\displaystyle \xi '}$, ${\displaystyle \eta '}$ were sufficiently small quantities, since only under this assumption can we develop the right-hand side of

${\displaystyle \Delta I=\int _{t_{0}}^{t_{1}}[F(x+\xi ,y+\eta ,x'+\xi ',y'+\eta ')-F(x,y,x',y')]{\text{d}}t}$

in powers of these quantities. This means not only that the curve which has been subjected to variation must lie indefinitely near the original curve, but also that the direction of the two curves can differ only a little from each other at corresponding points. We retain the same assumptions, and limit ourselves always to special variations.

We shall first prove that for all these variations the sign of ${\displaystyle \Delta I}$ and that of ${\displaystyle \delta ^{2}I}$ agree, so that for these variations the criteria already found are also sufficient. However, we no longer assume that the variations are expressible in the form ${\displaystyle \epsilon \xi }$, ${\displaystyle \epsilon \eta }$, where ${\displaystyle \epsilon }$ denotes a sufficiently small quantity.

Since the curvature of the original curve does not become infinitely large at any point (see Art. 95), and since further the original curve and the curve which has been subjected to variation deviate only a little from each other at corresponding points both in their position and the direction of their tangents, it follows that with each point of the original curve is associated the point of the curve that has been varied, in which the latter curve is cut by the normal drawn through the point on the first curve.

The equation of the normal at the point ${\displaystyle x,y}$ is

${\displaystyle (X-x)x'+(Y-y)y'=0}$;

and from the remarks just made, the point ${\displaystyle x+\xi }$, ${\displaystyle y+\eta }$ is to lie on this normal so that

${\displaystyle \xi x'+\eta y'=0}$.

If we consider this equation in connection with the definition of ${\displaystyle w}$:

${\displaystyle w=\xi y'-\eta x'}$,

it follows that the variations may be represented in the form

${\displaystyle 1)\qquad \xi ={\frac {w}{x'^{2}+y'^{2}}}y',\qquad \eta =-{\frac {w}{x'^{2}+y'^{2}}}x'}$.

In these expressions ${\displaystyle {\frac {w}{x'^{2}+y'^{2}}}}$ is an indefinitely small quantity, since ${\displaystyle x'}$ and ${\displaystyle y'}$ cannot both vanish at the same time (Art. 95), and it varies in a continuous manner with ${\displaystyle t}$. Likewise the derivative of this quantity with respect to ${\displaystyle t}$ is an indefinitely small quantity which, however, may not be everywhere continuous.

Under the assumption that ${\displaystyle \xi }$, ${\displaystyle \eta }$, ${\displaystyle \xi '={\frac {{\text{d}}\xi }{{\text{d}}t}}}$, ${\displaystyle \eta '={\frac {{\text{d}}\eta }{{\text{d}}t}}}$ are sufficiently small quantities, we may develop the total variation of the integral

${\displaystyle \Delta I=\int _{t_{0}}^{t_{1}}[F(x+\xi ,y+\eta ,x'+\xi ',y'+\eta ')-F(x,y,x',y')]{\text{d}}t}$

with respect to the powers of ${\displaystyle \xi }$, ${\displaystyle \eta }$, ${\displaystyle \xi '}$, ${\displaystyle \eta '}$; and, if we make use of Taylor's Theorem in the form

${\displaystyle F(x_{1}+\xi _{1},\ldots ,x_{n}+\xi _{n})=F(x_{1},\ldots ,\xi _{n})+\sum _{i}{\frac {\partial F}{\partial x_{i}}}\xi _{i}+\int _{0}^{1}(1-\epsilon ){\text{d}}\epsilon \sum _{i,j}[F_{i,j}(x_{1}+\epsilon \xi _{1},\ldots ,x_{n}+\epsilon \xi _{n})\xi _{i}\xi _{j}]}$,

where ${\displaystyle F_{i,j}={\frac {\partial ^{2}F}{\partial x_{i}\partial x_{j}}}}$, we have, since the terms of the first dimension vanish, a development of the form

${\displaystyle 2)\qquad \Delta I=\int _{t_{0}}^{t_{1}}\int _{0}^{1}(1-\epsilon )[F_{1,1}(x+\epsilon \xi ,y+\epsilon \eta ,x'+\epsilon \xi ',y'+\epsilon \eta ')\xi ^{2}+\cdots ]{\text{d}}\epsilon {\text{d}}t}$.

Article 137.
If we further develop ${\displaystyle F_{1,1}}$, etc., with respect to powers of ${\displaystyle \epsilon }$, it is found that the aggregate of terms that do not contain ${\displaystyle \epsilon }$ is identical with ${\displaystyle \delta ^{2}F}$ which was obtained in Chapter VIII.

Integrating with respect to ${\displaystyle \epsilon }$, we may represent the other terms as a quadratic form in ${\displaystyle \xi }$, ${\displaystyle \eta }$, ${\displaystyle \xi '}$, ${\displaystyle \eta '}$, whose coeflcients also contain these quantities and in such a way that they become indefinitely small with these quantities.

Next, writing in ${\displaystyle \Delta I}$ the values of ${\displaystyle \xi }$, ${\displaystyle \eta }$ given in 1) and the following values of ${\displaystyle \xi '}$, ${\displaystyle \eta '}$ also derived from 1):

${\displaystyle 3)\qquad \xi '={\frac {y'}{x'^{2}+y'^{2}}}{\frac {{\text{d}}w}{{\text{d}}t}}+w{\frac {\text{d}}{{\text{d}}t}}\left({\frac {y'}{x'^{2}+y'^{2}}}\right)\qquad \eta '=-{\frac {x'}{x'^{2}+y'^{2}}}{\frac {{\text{d}}w}{{\text{d}}t}}-w{\frac {\text{d}}{{\text{d}}t}}\left({\frac {x'}{x'^{2}+y'^{2}}}\right)}$,

we have

${\displaystyle 4)\qquad \Delta I=\int _{t_{0}}^{t_{1}}\left[F_{1}\left({\frac {{\text{d}}w}{{\text{d}}t}}+F_{2}w^{2}\right)^{2}\right]{\text{d}}t+\int _{t_{0}}^{t_{1}}\left[fw^{2}+2gw{\frac {{\text{d}}w}{{\text{d}}t}}+h\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}\right]{\text{d}}t}$,

where ${\displaystyle f}$, ${\displaystyle g}$, ${\displaystyle h}$ denote functions which still contain ${\displaystyle w}$ and ${\displaystyle {\frac {{\text{d}}w}{{\text{d}}t}}}$, and in such a way that they become indefinitely small at the same time as these quantities.

Article 138.
After a known theorem[1] in quadratic forms,

${\displaystyle fw^{2}+2gw{\frac {{\text{d}}w}{{\text{d}}t}}+h\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}}$,

may always, through linear substitutions not involving imaginaries, be brought to the form

${\displaystyle f_{1}u_{1}^{2}+f_{2}u_{2}^{2}}$,

in such a way that at the same time the relation

${\displaystyle u_{1}^{2}+u_{2}^{2}=w^{2}+\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}}$

is true, and where ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$ are roots of the quadratic equation in ${\displaystyle x}$:

${\displaystyle (f-x)(h-x)=g^{2}\equiv x^{2}-x(f+h)+fh-g^{2}=0}$.

Since the coefficients in this equation become simultaneously small with ${\displaystyle w}$ and ${\displaystyle {\frac {{\text{d}}w}{{\text{d}}t}}}$, the same must also be true of ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$, the roots this equation.

If ${\displaystyle l}$ is the mean value between ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$, which also becomes indefinitely small with ${\displaystyle w}$ and ${\displaystyle {\frac {{\text{d}}w}{{\text{d}}t}}}$, we may bring the expression

${\displaystyle f_{1}u_{1}^{2}+f_{2}u_{2}^{2}}$

to the form

${\displaystyle f_{1}u_{1}^{2}+f_{2}u_{2}^{2}=l(u_{1}^{2}+u_{2}^{2})=l\left(w^{2}+\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}\right)}$,

and consequently we have for ${\displaystyle \Delta I}$ the expression

${\displaystyle \Delta I=\int _{t_{0}}^{t_{1}}\left[F_{1}\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}+F_{2}w^{2}\right]{\text{d}}t+\int _{t_{0}}^{t_{1}}l\left[w^{2}+\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}\right]{\text{d}}t}$,

or finally

${\displaystyle \Delta I=\int _{t_{0}}^{t_{1}}\left[(F_{1}+l)\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}+(F_{2}+l)w^{2}\right]{\text{d}}t}$;

and thus we have for ${\displaystyle \Delta I}$ the same form as we had before for ${\displaystyle \delta ^{2}I}$ (Art. 115).

Article 139.
We assume now that the necessary conditions for the existence of a maximum or a minimum are satisfied; that therefore along the whole curve ${\displaystyle G=0}$, the function ${\displaystyle F_{1}}$ is different from zero or infinity, and always retains the same sign; that a function ${\displaystyle u}$ may be determined which satisfies the equation

${\displaystyle {\frac {\text{d}}{{\text{d}}t}}\left(F_{1}{\frac {{\text{d}}u}{{\text{d}}t}}\right)-F_{2}u=0}$,

and nowhere vanishes within the interval ${\displaystyle t_{0}\ldots t_{1}}$ or upon the boundaries of this interval.

If we therefore understand by ${\displaystyle k}$ a positive quantity, and write

${\displaystyle l=-k+l+k}$,

then the expression for ${\displaystyle \Delta I}$ above becomes

${\displaystyle \Delta I=\int _{t_{0}}^{t_{1}}\left[(F_{1}-k)\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}+(F_{2}-k)w^{2}\right]{\text{d}}t+\int _{t_{0}}^{t_{1}}(l+k)\left[\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}+w^{2}\right]{\text{d}}t}$.

If ${\displaystyle k}$ is given a fixed value, then we may choose ${\displaystyle \xi }$, ${\displaystyle \eta }$ so small that the absolute value of the quantity ${\displaystyle l}$ that depends upon them is less than ${\displaystyle k}$. The quantity ${\displaystyle k+l}$ is therefore positive, and consequently also the second integral of the above expression. We have yet to show that the first integral is also positive, if ${\displaystyle F_{1}>0}$.

After a known theorem in differential equations it is always possible, as soon as the equation

${\displaystyle {\frac {\text{d}}{{\text{d}}t}}\left(F_{1}{\frac {{\text{d}}u}{{\text{d}}t}}\right)-F_{2}u=0}$

is integrated through a continuous function ${\displaystyle u}$ of ${\displaystyle t}$, which within and on the boundaries of the interval ${\displaystyle t_{0}\ldots t_{1}}$ nowhere vanishes, also to integrate the differential equation

${\displaystyle {\frac {\text{d}}{{\text{d}}t}}\left[(F_{1}-k)){\frac {{\text{d}}{\bar {u}}}{{\text{d}}t}}\right]-(F_{2}-k){\bar {u}}=0}$

through a continuous function of ${\displaystyle t}$, which, if ${\displaystyle k}$ does not exceed certain limits, deviates indefinitely little throughout its whole trace from ${\displaystyle u}$, and may therefore be represented in the form

${\displaystyle {\bar {u}}=u+(t,k)}$,

where ${\displaystyle (t,k)}$ becomes indefinitely small at the same time as ${\displaystyle k}$ for all values of ${\displaystyle t}$ that come into consideration.

The function ${\displaystyle {\bar {u}}}$ will therefore vanish nowhere within the interval ${\displaystyle t_{0}\ldots t_{1}}$. In this manner a certain limit has also been established for ${\displaystyle k}$, which it cannot exceed ; but if the condition is also added that ${\displaystyle k}$ must be so small that ${\displaystyle F_{1}-k}$ has the same sign as ${\displaystyle F_{1}}$, then ${\displaystyle \xi }$, ${\displaystyle \eta }$ may always be chosen so small that ${\displaystyle |l|.

The first integral may then be transformed in a manner similar to that in which the integral 8) of Art. 115 was transformed into 14) of Art. 119, and we thus have

${\displaystyle \Delta I=\int _{t_{0}}^{t_{1}}(F_{1}-k)\left({\frac {{\text{d}}w}{{\text{d}}t}}-{\frac {{\text{d}}{\bar {u}}}{{\text{d}}t}}{\frac {w}{\bar {u}}}\right)^{2}{\text{d}}t+\int _{t_{0}}^{t_{1}}(l+k)\left(\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}+w^{2}\right){\text{d}}t}$,

which shows that ${\displaystyle \Delta I}$ for all indefinitely small variations of the curve which have been brought about under the given assumptions, is positive if ${\displaystyle F_{1}}$ is positive. If ${\displaystyle F_{1}}$ is negative, the same determinations regarding ${\displaystyle k}$ remain ; only ${\displaystyle k}$ must be chosen negative and ${\displaystyle |l|<-k}$. Both integrals on the right of the above equation are then negative, and consequently ${\displaystyle \Delta I}$ is itself negative.

We have therefore proved the assertion made above : If in the interval ${\displaystyle t_{0}\ldots t_{1}}$ the necessary conditions which were derived from the consideration of the second variation of the integral for the existence of a maximum or a m-inimum, are satisfied, then the sign of the total variation will be the same as the sign of the second variation for all variations of the curve which have been so chosen, that not only the distances between corresponding points on the original curve and the curve subjected to variation are arbitrarily small, but also the directions of both curves at corresponding points deviate from, each other by an arbitrarily small quantity.

It has thus been shown that the three conditions given in Art. 135 are necessary for the existence of a maximum or a minimum. A further examination will give a fourth condition (Weierstrass's condition, see Chapter XII) whose fulfillment is also sufficient. This condition, if fulfilled, is then decisive, after we have first assured ourselves that the other three conditions are satisfied.

APPLICATION OF THE ESTABLISHED CRITERIA TO THE PROBLEMS I, II, III AND IV, WHICH WERE PROPOSED IN CHAPTER I AND FURTHER DISCUSSED IN CHAPTER VII.

Article 140.
Problem I. The problem of the minimal surface of rotation.

As the solution of the equation ${\displaystyle G=0}$, we found (Art. 100) the two simultaneous equations of the catenary :

${\displaystyle 1)\qquad x=\alpha +\beta t=\phi (t,\alpha ,\beta )\qquad y={\frac {\beta }{2}}(e^{t}+e^{-t})=\psi (t,\alpha ,\beta )}$.

We have, therefore (Art. 125),

${\displaystyle 2)}$
${\displaystyle \phi '(t)=\beta \quad \phi _{1}(t)=1\quad \phi _{2}(t)}$
${\displaystyle \psi '(t)={\frac {\beta }{2}}(e^{t}-e^{-t})\quad \psi _{1}(t)=0\quad \psi _{2}(t)={\frac {1}{2}}(e^{t}+e^{-t})}$;
and consequently,
${\displaystyle \theta _{1}(t)=\psi '(t)\phi _{1}(t)-\phi '(t)\psi _{1}(t)={\frac {\beta }{2}}(e^{t}-e^{-t})=y'\quad \theta _{2}(t)=\psi '(t)\phi _{2}(t)-\phi '(t)\psi _{2}(t)=ty'-y}$.

If, now, ${\displaystyle x_{0},y_{0},x_{0}',y_{0}'}$ are the values of ${\displaystyle x,y,x',y'}$ which correspond to the value ${\displaystyle t_{0}}$, then is

${\displaystyle 3)\qquad \Theta (t,t_{0})=\theta _{1}(t_{0})\theta _{2}(t)-\theta _{2}(t_{0})\theta _{1}(t)=y_{0}'(ty'-y)-(t_{0}y_{0}'-y_{0})t'}$;

or, since

${\displaystyle t={\frac {x-\alpha }{\beta }}\qquad t_{0}={\frac {x_{0}-\alpha }{\beta }}\qquad \beta =x'=x_{0}'\qquad }$ [cf. 2)],

we have

${\displaystyle 4)\qquad \Theta (t,t_{0})={\frac {1}{\beta }}[y_{0}'(xy'-yx')-y'(x_{0}y_{0}'-y_{0}x_{0}')]}$.

In order to find the point conjugate to ${\displaystyle t_{0}}$ we have to write in this expression for ${\displaystyle x,y,x',y'}$ their values in terms of ${\displaystyle t}$ and then solve the equation ${\displaystyle \Theta (t,t_{0})=0}$.

To avoid this somewhat complicated calculation, however, we may make use of a geometrical interpretation (Art. 58). The equation of the tangent to the catenary at the point ${\displaystyle x_{0},y_{0}}$ is

${\displaystyle y_{0}'(X-x_{0})-x_{0}'(Y-y_{0})=0}$.

Therefore, the tangent cuts the ${\displaystyle X}$-axis in the point determined through the equation

${\displaystyle y_{0}'X=xy'-yx'}$.

The tangent at any point of the catenary cuts the ${\displaystyle X}$-axis at a point determined by the equation

${\displaystyle y'X=xy'-yx'}$.

If, now, the point ${\displaystyle x,y}$ is to be conjugate to ${\displaystyle x_{0},y_{0}}$, then its coordinates must satisfy 4), which becomes

${\displaystyle y_{0}y'(X-X_{0})=0}$.

Hence, since ${\displaystyle y_{0}'}$ and ${\displaystyle y'}$ do not vanish (Art. 101), we have

${\displaystyle X=X_{0}}$;

that is, the conjugate points of the catenary have the property that the tangents drawn through them cut each other on the ${\displaystyle X}$-axis. We thus have an easy geometrical method of determining the point conjugate to any point on the catenary.

Further we have

${\displaystyle F_{1}={\frac {y}{({\sqrt {x'^{2}+y'^{2}}})^{3}}}}$,

and since ${\displaystyle y}$ is always positive, and ${\displaystyle x',y'}$ cannot simultaneously vanish, it follows that ${\displaystyle F_{1}}$ is always positive and different from zero and infinity. Hence, the portion of a catenary that is situated between two conjugate points, when rotated about the ${\displaystyle X}$-axis, generates a surface of smallest area (cf. Art. 167).

At the same time in this problem it is seen how small a role the condition regarding ${\displaystyle F_{1}}$, has played in the strenuous proof relative to the existence of a minimum.

Article 141.
Problem II. Problem of the brachistochrone.

In this problem the expression for ${\displaystyle F_{1}}$ is found to be

${\displaystyle 1)\qquad F_{1}={\frac {1}{({\sqrt {x'^{2}+y'^{2}}})^{3}}}{\frac {1}{\sqrt {4gy+\alpha ^{2}}}}}$.

We assumed from certain a priori reasons that between the points ${\displaystyle A}$ and ${\displaystyle B}$ of the curve there could be present no cusp (see also Art. 104); that is no point for which ${\displaystyle x'}$ and ${\displaystyle y'}$ are both equal to zero simultaneously. For such an arc of the curve ${\displaystyle F_{1}}$ is then always positive and different from zero and infinity, since the quantities under the square root sign are always finite and different from zero (see also Art. 95).

We obtained (Art. 103) the solution of the equation ${\displaystyle G=0}$ in the form

${\displaystyle 2)\qquad x=\alpha +\beta (t-\sin t)=\phi (t,\alpha ,\beta )\qquad y+a=\beta (1-\cos t)=\psi (t,\alpha ,\beta )}$,

where here ${\displaystyle t}$ is written in the place of ${\displaystyle \phi }$, and ${\displaystyle \alpha }$ in the place of ${\displaystyle -x_{0}}$, and ${\displaystyle \beta }$ instead of ${\displaystyle 1/(2c^{2})}$; ${\displaystyle a}$ is a given quantity which is determined through the initial velocity.

We consequently have

${\displaystyle 3)}$
${\displaystyle \phi '(t)=\beta (1-\cos t)\quad \phi _{1}(t)=1\quad \phi _{2}(t)=t-\sin t}$;
${\displaystyle \psi '(t)=\beta \sin t\qquad \phi _{1}(t)=0\qquad \psi _{2}(t)=1-\cos t}$;
${\displaystyle \theta _{1}(t)=\beta \sin t}$,
${\displaystyle \theta _{2}(t)=\beta \sin t(t-\sin t)-\beta (1-\cos t)^{2}=2\beta \sin(t/2)[t\cos(t/2)-2\sin(t/2)]}$;

and therefore

${\displaystyle \Theta (t,t_{0})=4\beta ^{2}\sin {\frac {t_{0}}{2}}\sin {\frac {t}{2}}\left[\cos {\frac {t_{0}}{2}}\left(t\cos {\frac {t}{2}}-2\sin {\frac {t}{2}}\right)-\cos {\frac {t}{2}}\left(t_{0}\cos {\frac {t_{0}}{2}}-2\sin {\frac {t_{0}}{2}}\right)\right]}$.

With the positions which we have assumed for ${\displaystyle A}$ and ${\displaystyle B}$ both ${\displaystyle t_{0}}$ and ${\displaystyle t}$ are different from ${\displaystyle 0}$ and ${\displaystyle 2\pi }$, and consequently the equation for the determination of the point conjugate to ${\displaystyle t_{0}}$ has the form

${\displaystyle \cos {\frac {t_{0}}{2}}\left(t\cos {\frac {t}{2}}-2\sin {\frac {t}{2}}\right)-\cos {\frac {t}{2}}\left(t_{0}\cos {\frac {t_{0}}{2}}-2\sin {\frac {t_{0}}{2}}\right)=0}$,

or

${\displaystyle 4)\qquad t-2\tan {\frac {t}{2}}=t_{0}-2\tan {\frac {t_{0}}{2}}}$,

which is a transcendental equation for the determination of ${\displaystyle t}$.

We easily see that there is no other real root within the interval ${\displaystyle 0\ldots 2\pi }$ except ${\displaystyle t=t_{0}}$, since the derivative of ${\displaystyle t-2\tan(t/2)}$, namely, ${\displaystyle 1-{\frac {1}{\cos ^{2}(t/2)}}}$ is negative, so that ${\displaystyle t-2\tan(t/2)}$ continuously decreases, if ${\displaystyle t}$ deviates from ${\displaystyle t_{0}}$, and can never again take the value ${\displaystyle t_{0}-2\tan(t_{0}/2)}$.

Consequently there is no point conjugate to the point ${\displaystyle t_{0}}$ on the arc of the cycloid upon which ${\displaystyle t_{0}}$ lies, and therefore every arc of the cycloid situated between two cusps of this curve has the property that a material point which slides along it from a point ${\displaystyle A}$ reaches another point ${\displaystyle B}$ of the curve in the shortest time (Art. 168).

In this problem we see that the condition ${\displaystyle F_{1}>0}$ was sufficient to establish the existence of a minimum. The case where the initial velocity is zero and the point ${\displaystyle A}$ is situated at one of the cusps will be discussed later (Art. 169).

Article 142.
Problem III. Problem of the shortest line on the surface of a sphere.

In this problem we find that

${\displaystyle 1)\qquad F_{1}={\frac {\sin ^{2}u}{({\sqrt {u'^{2}+v'^{2}\sin ^{2}u}})^{3}}}}$.

This expression cannot become infinitely large, since ${\displaystyle u'}$ and ${\displaystyle v'}$ cannot simultaneously vanish.

However, the function ${\displaystyle F_{1}}$, will vanish if ${\displaystyle \sin u=0}$; that is, when ${\displaystyle u=0}$ or ${\displaystyle \pi }$. Consequently, in this case, we must so choose the system of coordinates that ${\displaystyle u}$ nowhere along the trace of the curve becomes equal to zero or to ${\displaystyle \pi }$. If this has been done, then ${\displaystyle F_{1}}$ for the whole stretch from ${\displaystyle A}$ to ${\displaystyle B}$ is positive, and does not become zero or infinitely large.

The equation ${\displaystyle G=0}$ furnishes the arc of a great circle, whose equations are (see Art. 106):

${\displaystyle 2)}$
${\displaystyle \cos u=\cos c\cos(s-b)}$
${\displaystyle \cot(v-\beta )=\sin c\cot(s-b)}$;
or,
${\displaystyle u=\arccos(\cos c\cos(s-b))=\phi (s,\alpha ,\beta )}$
${\displaystyle v=\beta +\operatorname {arccot}(\sin c\cot(s-b))=\psi (s,\alpha ,\beta )}$.

Accordingly, we have

${\displaystyle \phi '(s)={\frac {\cos c\sin(s-b)}{\sqrt {1-cos^{2}c\cos ^{2}(s-b)}}}\qquad \phi _{1}(s)={\frac {\sin s\cos(s-b)}{\sqrt {1-\cos ^{2}c\cos ^{2}(s-b)}}}\qquad \phi _{2}(s)=0}$
${\displaystyle \psi '(s)={\frac {\sin c}{1-\cos ^{2}c\cos ^{2}(s-b)}}\qquad \psi _{1}(s)={\frac {-\cos c\sin(s-b)\cos(s-b)}{1-\cos ^{2}c\cos ^{2}(s-b)}}\qquad \psi _{2}(s)=1}$

and consequently

${\displaystyle \theta _{1}(s)={\frac {\cos(s-b)}{\sqrt {1-\cos ^{2}c\cos ^{2}(s-b)}}}\qquad \theta _{2}(s)={\frac {-\cos c\sin(s-b)}{\sqrt {1-\cos ^{2}c\cos ^{2}(s-b)}}}}$.

Hence, since for the point ${\displaystyle A}$ we have ${\displaystyle s=s_{0}=0}$, it follows that

${\displaystyle 3)\qquad \Theta (s,s_{0})=-{\frac {\cos c\sin s}{{\sqrt {1-\cos ^{2}c\cos ^{2}b}}{\sqrt {1-\cos ^{2}c\cos ^{2}(s-b)}}}}}$.

Therefore, in order to find the point conjugate to the point ${\displaystyle s_{0}=0}$, we have to solve the equation ${\displaystyle \Theta (s,s_{0})=0}$ with respect to ${\displaystyle s}$.

Since the denominator of 3) cannot become infinite, the conjugate point is to be determined from the equation ${\displaystyle \sin s=0}$. We consequently have ${\displaystyle s=\pi }$ as the point conjugate to ${\displaystyle s=0}$; that is, the point conjugate to ${\displaystyle A}$ is the other end of the diameter of the circle drawn through ${\displaystyle A}$.

Hence the arc of a great circle through the points ${\displaystyle A}$ and ${\displaystyle B}$, measured in a direction fixed as positive, is the shortest distance upon the surface of the sphere only when these points are not at a distance of ${\displaystyle 180^{\circ }}$ or more from each other, a result which is of itself geometrically clear.

We may remark that the condition that ${\displaystyle F_{1}}$ cannot vanish is clearly in this case unnecessary ; since the arc of a great circle possesses the property of a minimum independently of the choice of the system of coordinates with respect to which ${\displaystyle F_{1}}$, say, at some point of the curve vanishes.

Article 143.
From the figure in Art. 107 it is clear that when ${\displaystyle A}$ is the pole of the sphere, the family of curves passing through ${\displaystyle A}$ and satisfying the differential equation ${\displaystyle G=0}$ (i.e., arcs of great circles) intersect again only at the other pole. In the next Chapter it will appear that the two poles are conjugate points. This, together with what was given in the preceding article, may be taken as a proof that the arcs of great circles can meet only at opposite poles.

Article 144.
Problem IV. Problem of the surface offering the least resistance.

In this problem let us write (Art. 110)

${\displaystyle \alpha =-C/2\qquad \beta =-C_{1}}$

so that

${\displaystyle x=\alpha [t+2t^{-1}+t^{-3}]=\phi (t,\alpha ,\beta )\qquad y=\alpha \left[\ln t+t^{-2}+{\frac {3}{4}}t^{-4}\right]-\beta =\psi (t,\alpha ,\beta )}$.

Hence,

${\displaystyle \phi '(t)=x'\qquad \phi _{1}(t)={\frac {x}{\alpha }}\qquad \phi _{2}(t)=0}$
${\displaystyle \psi '(t)=y'\qquad \psi _{1}(t)={\frac {y+\beta }{\alpha }}\qquad \psi _{2}(t)=-1}$
${\displaystyle \theta _{1}(t)=y'{\frac {x}{\alpha }}-x'{\frac {y+\beta }{\alpha }}\qquad \theta _{2}(t)=x'}$

and

${\displaystyle \Theta (t,t_{0})={\frac {x'(y_{0}'x_{0}-x_{0}'y_{0})-x_{0}'(y'x-x'y)}{\alpha }}}$.

Now the tangent to the curve at any point ${\displaystyle x_{0},y_{0}}$ is

${\displaystyle y_{0}'(X-x_{0})-x_{0}'(Y-y_{0}))=0}$,

and the intercept on the ${\displaystyle Y}$-axis is

${\displaystyle x_{0}'Y_{0}=x_{0}'y_{0}-y_{0}'x_{0}}$.

The tangent to the curve at any point ${\displaystyle x,y}$ cuts the ${\displaystyle Y}$-axis where

${\displaystyle x'Y=x'y-y'x}$.

We therefore have for the determination of the point conjugate to ${\displaystyle x_{0},y_{0}}$ the equation

${\displaystyle x'x_{0}'Y_{0}=x_{0}x'Y\quad }$ or ${\displaystyle \quad Y_{0}=Y}$.

As in Art. 140, this gives an easy geometrical construction for conjugate points.

1. Such substitutions are called by Cayley orthogonal (Crelle, bd. 32, p. 119); see also Euler, Nov. Comm. Petrop., IS, p. 275; 20, p. 217; Cauchy, Exerc. de Math., 4, p. 140; Jacobi, Crelle, bd. 12, p. 7; bd. 30, p. 46; Baltzer, Theoiie und Anwendungen der Determinanten, 1881, p. 187; Rodrigves, Liouv. Journ., t. S, p. 405; Hesse, Crelle, bd. 57, p. 175.