Calculus of Variations/CHAPTER VI

CHAPTER VI: THE FORM OF THE SOLUTIONS OF THE DIFFERENTIAL EQUATION $G=0$.

• 87 Another form of the differential equation $G=0$.
• 88 Another form of the function $F_{1}$.
• 89 Integration in power-series.
• 90 The solutions $x=\phi (t,\alpha ,\beta )$, $y=\psi (t,\alpha ,\beta )$ of the differential equation $G=0$.
• 91,92 The case when $F_{1}=0$ at the initial-point of the curve.
• 93 The form of the differential equation $G=0$ when $s$ is the independent variable.
• 94 The solution of this equation.
• 95 The curve can have no singular points within the interval in question. coordinates of any point of the curve expressed as power-series in $s$.

Article 87.
Before we proceed to the development of further conditions for the existence of a maximum or a minimum of the integral

$I=\int _{t_{0}}^{t_{1}}F(x,y,x',y')~{\text{d}}t{\text{,}}\qquad {\text{[1]}}$

we shall endeavor to investigate more closely the nature and form of the differential equation $G=0$.

We assume that a curve satisfying the differential equation

$G={\frac {\partial ^{2}F}{\partial x\partial y'}}-{\frac {\partial ^{2}F}{\partial y\partial x'}}+F_{1}\left(x'{\frac {{\text{d}}y'}{{\text{d}}t}}-y'{\frac {{\text{d}}x'}{{\text{d}}t}}\right)=0{\text{,}}\qquad {\text{[2]}}$

for which $F_{1}$ is different from zero, is known. Let $A$ be the initial point of the curve, and let $AA'$ be the direction of the curve at $A$. We suppose that the differential equation $G=0$ takes iits simplest form, if we regard one of the coordinates as a one-valued function of the other. In the integral $I$ above, the dependence of the quantities $x$,$y$ upon the quantity $t$ is subject only to the condition that a point is to traverse the curve from the beginningpoint to the end-point, when $t$, continuously increasing, goes from $t_{0}$ to $t_{1}$.

In an infinite number of ways we may introduce in the place of $t$ another variable $\tau$, where

$t=f(\tau )$

It is only necessary that the function $f$ be so formed, that with increasing $\tau$ also $\tau$ increases. In place of $\tau$ we may reciprocally introduce again

$\tau =\phi (t)$.

The form of the integral $I$ and of the differential equation $G=0$ does not change with these transformations.

Under certain conditions we may choose for $t$ the coordinate $x$ iself as independent variable, this being the case when on traversing the curve from the initial-point to the end-point, x continuously increases. In particular, we may take as the $X$-axis the tangent at the initial point $A$ and take the direction of the curve as the positive direction of the $X$-axis. Since we consider only regular curves or curves composed of regular portions it follows, if the point $P$ traverses the curve starting from the point $A$, that its distance from the normal at $A$ continuously increases for a certain portion of curve. Hence for this portion of curve, if we take the positive direction of the normal at ${\displaystyle }$ as the positive $Y$-axis, there is only one value of $y$ for every value of $x$. Consequently for a definite portion of curve we may always assume that, by a suitable choice of the system of coordinates, the second coordinate may be regarded as a one-valued function of the first. We have therefore only to mcike a transformation of coordinates.

Let the coordinates of the new origin of coordinates be $a$, $b$, and further let

$x=a+gu+g'v$, $y=b+hu+h'v$, $\qquad {\text{[3]}}$

where $u$ and $v$ are the new coordinates.

If $\lambda$ is the angle between the $X$-axis and the $u$-axis we have the well-known relations

$g=\cos(\lambda )$, $h=\sin(\lambda )$, $g'=-\sin(\lambda )$, $h'=\cos(\lambda )$. $\qquad {\text{[4]}}$

The integral $I$ becomes then since $u$ may be regarded as the independent variable,

$I=\int _{0}^{u}F\left(a+gu+g'v,b+hu+h'v,g+g'{\frac {{\text{d}}v}{{\text{d}}u'}},h+h'{\frac {{\text{d}}v}{{\text{d}}u}}\right)~{\text{d}}u$,

which we shall for brevity denote by

$I=\int _{0}^{u}f\left(u,v,{\frac {{\text{d}}v}{{\text{d}}u}}\right)~{\text{d}}u$. $\qquad {\text{[5]}}$

If we further write ${\frac {{\text{d}}v}{{\text{d}}u}}=v'$ then [5] becomes

$I=\int _{0}^{u}f(u,v,v')~{\text{d}}u$. $\qquad {\text{[}}5^{\text{a}}{\text{]}}$

We have a differential equation to determine $v$ as a function of $u$, if we apply to this integral the same methods as were used in Arts. 74-80 of the last Chapter.

Let the curve be subjected to a sliding in the direction only of the ordinate $v$, and therefore write $v+{\bar {v}}$ in the place of $v$, where ${\bar {v}}$ is a very small quantity which vanishes at the end-point and the initial-point of the portion of curve under consideration. The integral which has been subjected to variation is

$I+\Delta I=\int f\left(u,v+{\bar {v}},v'+{\frac {{\text{d}}{\bar {v}}}{{\text{d}}u}}\right)~{\text{d}}u$.

We next develop $\Delta I$ according to powers of ${\bar {v}}$ and ${\frac {{\text{d}}{\bar {v}}}{{\text{d}}u}}$. The aggregate of the terms of the first dimension is

$\delta I=\int _{0}^{u}\left({\frac {\partial f}{\partial v}}{\bar {v}}+{\frac {\partial f}{\partial v'}}{\frac {{\text{d}}{\bar {v}}}{{\text{d}}u}}\right)~{\text{d}}u{\text{,}}\qquad {\text{[6]}}$

or, since

${\frac {\partial f}{\partial v'}}{\frac {{\text{d}}{\bar {v}}}{{\text{d}}u}}={\frac {\text{d}}{{\text{d}}u}}\left({\bar {v}}{\frac {\partial f}{\partial v'}}\right)-{\bar {v}}{\frac {\text{d}}{{\text{d}}u}}{\frac {\partial f}{\partial v'}}$,

we have

$\delta I=\int _{0}^{u}\left({\frac {\partial f}{\partial v}}-{\frac {\text{d}}{{\text{d}}u}}{\frac {\partial f}{\partial v'}}\right){\bar {v}}~{\text{d}}u+\left[{\bar {v}}{\frac {{\text{d}}f}{{\text{d}}v'}}\right]_{0}^{u}$. $\qquad {\text{[6}}^{a}{\text{]}}$

The quantity in the square brackets vanishes, because ${\bar {v}}=0$ on the limits. Further, we must have

$\delta I=0$.

Since ${\bar {v}}$ is arbitrary, subjected only to the condition that it must vanish on the limits, it follows from the lemma of the preceding Chapter that

${\frac {\text{d}}{{\text{d}}u}}{\frac {\partial f}{\partial v'}}-{\frac {\partial f}{\partial v}}=0$,

or

${\frac {\partial ^{2}f}{\partial ^{2}v'^{2}}}{\frac {{\text{d}}v'}{{\text{d}}u}}+{\frac {\partial ^{2}f}{\partial v\partial v'}}v'+{\frac {\partial ^{2}f}{\partial u\partial v'}}-{\frac {\partial f}{\partial v}}=0$. $\qquad {\text{[2}}^{a}{\text{]}}$

Article 88.
If one of the coordinates can be regarded as a one-valued function of the other, the equation [$2^{\text{a}}$] may take the place of the form [2] for $G=0$.

We shall now show that the quantity ${\frac {\partial ^{2}f}{\partial v'^{2}}}$ which enters in [$2^{\text{a}}$] is identical with $F_{1}$, provided that in the function $F(x,y,x',y')$ $x$ and $y$ may be regarded as functions of $u$ alone.

Since

$f(u,v,v')\equiv F(a+gu+g'v,b+hu+h'v,g+g'v',h+h'v')\equiv F(x,y,x',y')$,

it follows that

${\frac {\partial f}{\partial v'}}={\frac {\partial F}{\partial x'}}g'+{\frac {\partial F}{\partial y'}}h'$,

and consequently

${\frac {\partial ^{2}f}{\partial v'^{2}}}={\frac {\partial ^{2}F}{\partial x'^{2}}}g'^{2}+2{\frac {\partial ^{2}F}{\partial x'\partial y'}}g'h'+{\frac {\partial ^{2}F}{\partial y'^{2}}}h'^{2}$.

On the other hand by its definition $F_{1}$ was determined by any of the relations :

${\frac {\partial ^{2}F}{\partial x'^{2}}}=F_{1}y'^{2}$; ${\frac {\partial ^{2}F}{\partial x'}}{\partial y'}=-F_{1}x'y'$; ${\frac {\partial ^{2}F}{\partial y'^{2}}}=F_{1}x'^{2}$. $\qquad {\text{[7]}}$

From this it follows that

${\frac {\partial ^{2}f}{\partial v'^{2}}}=(g'^{2}y'^{2}-2g'h'x'y'+h'^{2}x'^{2})F_{1}=(g'y'-h'x')^{2}F_{1}=(g'h-h'g)^{2}F_{1}=(-\sin ^{2}(\lambda )-\cos ^{2}(\lambda ))^{2}F_{1}$;

or finally,

${\frac {\partial ^{2}f}{\partial v'^{2}}}=F_{1}$. $\qquad {\text{[8]}}$

Hence [$2^{\text{a}}$] may be written

$F_{1}{\frac {{\text{d}}v'}{{\text{d}}u}}+{\frac {\partial ^{2}f}{\partial v\partial v'}}v'+{\frac {\partial ^{2}f}{\partial u\partial v'}}-{\frac {\partial f}{\partial v}}=0$.

Since we have

$v'={\frac {{\text{d}}v}{{\text{d}}u}}$,

and

$f(u,v,v')\equiv F(x,y,x',y')$,

where $x$,$y$ are determined in terms of $u$,$v$ from [3], it follows that

$F_{1}{\frac {{\text{d}}^{2}v}{{\text{d}}u^{2}}}+{\frac {\partial ^{2}F}{\partial v\partial v'}}{\frac {{\text{d}}v}{{\text{d}}u}}+{\frac {\partial ^{2}F}{\partial u\partial v'}}-{\frac {\partial F}{\partial v}}=0$. $\qquad {\text{[2}}^{\text{b}}{\text{]}}$

Article 89.
In the theory of differential equations it is known that every differential equation of the form [$2^{\text{b}}$] may be integrated in the form of a power-series of the independent variable $u$.

As a special case we have the following:

Suppose 1) that at the initial point of the curve represented by the power-series which is to be formed, we have

$u=0$, $v=b_{0}$,

where $b_{0}$ is an arbitrary constant;

2) that the direction of the curve at the initial point is determined by the arbitrary constant

${\frac {{\text{d}}v}{{\text{d}}u}}=v_{0}'$;

then $v$ for sufficiently small values of $u$ may be expressed in the power-series

$v=b_{0}+v_{0}'u+\cdots$, $\qquad {\text{[9]}}$

where we have assumed that $F_{1}$ is different from zero at the initial point $(0,b_{0})$.

The second and higher derivatives of $v$ on the position $(0,b_{0},v_{0}')$ may all be derived from the differential equation [$2^{\text{b}}$]. Hence, in [9] we have $v$ as a power-series in $u$, whose coefficients contain besides the constants had in each problem, only the two arbitrary constants $b_{0}$ and $v_{0}'$, which change from curve to curve.

Article 90.
If we substitute the expression for $v$ given by equation [9] in the formulae [3] , we have $x$ and $y$ expressed in terms of $u$. In these expressions there appear the constants $g$, $g'$, $h$, $h'$, which depend upon and also upon the coordinates $a$, $b$ of the origin of the $u$, $v$ system of coordinates and the two constants of integration $b_{0}$ and $v_{0}'$, defined in Art. 89. These latter constants vary from curve to curve. In these formulae, just as in Art. 89, we can ascribe only small values to the quantity $u$.

We know, however as is seen in the theory of functions that if a curve is given only in a small portion its continuation is thereby completely determined. We therefore need to know the curve only for indefinitely small $u$'s in order to be able to follow its trace at pleasure.

The coordinates $x$, $y$ of the curve may be represented as functions of $t$ and two arbitrary constants $\alpha$ and $\beta$. Instead of $u$, we may introduce an arbitrary function of another quantity if only this quantity increases in a continuous manner, when the curve is traversed from the beginning-point to the end-point. As already mentioned, the two constants of integration vary from curve to curve. If we determine suitably these constants, we can compel the curve, which satisfies the differential equation $G=0$, to pass through two prescribed points.

In this manner we have a clear representation of the manner in which the analytical expressions giving $x$ and $y$ are derived; $x$ and $y$ are found in general from the equation $G=0$ in the form

$x=\phi (t,\alpha ,\beta )$, $y=\psi (t,\alpha ,\beta )$. $\qquad {\text{[10]}}$

At the same time it is seen that up to a certain point, at least, $x$ and $y$ are one-valued and regular functions of $t$ and of the two constants of integration $\alpha$ and $\beta$, so that eventually we can also differentiate with respect to these two constants.

Article 91.
It seems desirable here in connection with what was given in Arts. 89, 90 to consider the exceptional case, viz., the one in which

${\frac {\partial ^{2}f}{\partial v'^{2}}}=F_{1}={\frac {\partial ^{2}F}{\partial x'^{2}}}\sin ^{2}(\lambda )-2{\frac {\partial ^{2}f}{\partial x'\partial y'}}\sin(\lambda )\cos(\lambda )+{\frac {\partial ^{2}F}{\partial y'^{2}}}\cos ^{2}(\lambda )\qquad {\text{[8]}}$

is equal to zero for the origin $(y=0,v-b_{0})$ of the curve which satisfies the equation $G=0$.

We shall see that this is only an exceptional case by showing the following :

If we draw around the point $(u=0,v=b_{0})$ a small circle, then this circle may be so distributed into sectors that within each sec tor ${\frac {\partial ^{2}f}{\partial v'^{2}}}$ is not equal to zero. For we may regard the radius of a sufficiently small circle about the initial point $(u=0,v=b_{0})$ of the curve in question as the initial direction of this curve. If for $v$ we write in [8] the power-series given in [9] , we have, by putting $F_{1}=0$, an equation for the determination of $v_{0}'$, that is, the quantity which fixes the initial direction. This equation has either no real roots, and then there will exist no curve starting from the point $(u=0,v-b_{0})$, or $F_{1}$ vanishes for single $v_{0}'$'s, and then the radii determine separate sectors. Within these sectors curves may be drawn starting from the point $(u=0,v-b_{0})$ in every direction, for which $F_{1}$ is different from zero. Consequently one can always assign limits for $v_{0}'$ within which the corresponding curves, satisfying the equation $G=0$ and starting from the point $(u=0,v-b_{0})$ have at the origin, at least, an $F_{1}$ different from zero.

Article 92.
Finally we shall show that the curves starting from the same point $(u=0,v-b_{0})$ which satisfy the equation $G=0$ lie completely separated from one another at their initial point.

If we draw a small circle around the point $(u=0,v=b_{0})$, then on its periphery we can easily determine the point $u$, $v$ in which it is cut by one of the curves in question. For let $\rho$ be the radius of the small circle, so that

$u^{2}+(v-b_{0})^{2}=\rho ^{2}$. $\qquad {\text{[11]}}$

Writing for $v$ the power-series [9], we have

$\rho ^{2}=(1+v_{0}'^{2})u^{2}+C_{3}u^{3}+\cdots$,

or

$\rho ={\sqrt {1+v_{0}'^{2}}}u+(u)_{2}+\cdots$.

We may revert this series and have $u$ expressed as a function of $\rho$, so that

$u={\frac {1}{\sqrt {1+v_{0}'^{2}}}}\rho +(\rho )_{2}'+\cdots \qquad {\text{[12.1]}}$

and therefore

$v-b_{0}={\frac {v_{0}'}{\sqrt {1+v_{0}'^{2}}}}\rho +(\rho )_{2}''+\cdots \qquad {\text{[12.2]}}$

These series are convergent for all $\rho$'s within a certain limit $\rho _{0}$, so that therefore $u$ and $v$, the coordinates of the point to be determined upon the periphery of the circle with the radius $\rho$, are uniquely found for all values of $\rho <\rho _{0}$. Consequently at the beginning, at least, the curves which belong to a sector in reality lie completely separated from one another.

Article 93.
The form of the differential equation $G=0$, where $s$ is introduced as the independent variable instead of $t$. If we introduce instead of $t$ another variable $\tau$ by writing $t$ equal to a function of $\tau$, we arrive at the same differential equation $G=0$. It is usually advantageous to introduce the length of arc $s$ as the independent variable.

Since (Art. 68) the derivatives of the function $F$ with. respect to its third and fourth arguments are invariantive, we have (writing $k={\frac {1}{\sqrt {x'^{2}+y'^{2}}}}={\frac {{\text{d}}t}{{\text{d}}s}}$ in the formulae of Art. 68)

${\frac {\partial }{\partial x'}}F(x,y,x',y,)={\frac {\partial }{\partial \left({\frac {{\text{d}}x}{{\text{d}}s}}\right)}}F\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)$,

${\frac {\partial }{\partial y'}}F(x,y,x',y,)={\frac {\partial }{\partial \left({\frac {{\text{d}}y}{{\text{d}}s}}\right)}}F\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)$.

From this it is seen that ${\frac {\partial F}{\partial x'}}$, ${\frac {\partial F}{\partial y'}}$ are independent of the manner ox ay in which $y$ and $y$ are expressed as functions of $t$, and depend only upon the point in question of the curve and the direction of the tangent at this point. We have at once

${\frac {\partial ^{2}F}{\partial x'\partial y'}}={\frac {\partial ^{2}}{\partial {\frac {{\text{d}}x}{{\text{d}}s}}\partial {\frac {{\text{d}}y}{{\text{d}}s}}}}F\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right){\frac {1}{\frac {{\text{d}}s}{{\text{d}}t}}}$;

and since

$F_{1}=-{\frac {1}{x'y'}}{\frac {\partial ^{2}F(x,y,x',y')}{\partial x'\partial y'}}$,

it follows that

$F_{1}\left({\frac {{\text{d}}s}{{\text{d}}t}}\right)^{3}=-{\frac {1}{{\frac {{\text{d}}x}{{\text{d}}s}}{\frac {{\text{d}}y}{{\text{d}}s}}}}{\frac {\partial ^{2}}{\partial {\frac {{\text{d}}x}{{\text{d}}s}}\partial {\frac {{\text{d}}y}{{\text{d}}s}}}}F\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)$.

If further we write

${\frac {\partial ^{2}}{\partial {\frac {{\text{d}}x}{{\text{d}}s}}\partial {\frac {{\text{d}}y}{{\text{d}}s}}}}F\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)=-{\frac {{\text{d}}x}{{\text{d}}s}}{\frac {{\text{d}}y}{{\text{d}}s}}F_{1}\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)$,

we have

$F\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)\left({\frac {{\text{d}}s}{{\text{d}}t}}\right)^{3}=F_{1}\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)$.

Hence the equation $G=0$ becomes

${\frac {\partial ^{2}}{\partial x\partial {\frac {{\text{d}}y}{{\text{d}}s}}}}F\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)-{\frac {\partial ^{2}}{\partial y\partial {\frac {{\text{d}}x}{{\text{d}}s}}}}F\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)+\left({\frac {{\text{d}}x}{{\text{d}}s}}{\frac {{\text{d}}^{2}y}{{\text{d}}s^{2}}}-{\frac {{\text{d}}y}{{\text{d}}s}}{\frac {{\text{d}}^{2}x}{{\text{d}}s^{2}}}\right)F_{1}\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)=0$.

Article 94.
From the above equation the second and all higher derivatives of $x$ and $y$ with respect to $s$ may be explicitly expressed in terms of $x$,$x$,${\frac {{\text{d}}x}{{\text{d}}s}}$,${\frac {{\text{d}}y}{{\text{d}}s}}$.

For, from the relation

$\left({\frac {{\text{d}}x}{{\text{d}}s}}\right)^{2}+\left({\frac {{\text{d}}y}{{\text{d}}s}}\right)^{2}=1$,

it follows, through differentiation, that

${\frac {{\text{d}}x}{{\text{d}}s}}{\frac {{\text{d}}^{2}x}{{\text{d}}s^{2}}}+{\frac {{\text{d}}y}{{\text{d}}s}}{\frac {{\text{d}}^{2}y}{{\text{d}}s^{2}}}=0$. $\qquad {\text{[i]}}$

If, for brevity, we write

${\frac {\partial ^{2}}{\partial x\partial {\frac {{\text{d}}y}{{\text{d}}s}}}}F\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)-{\frac {\partial ^{2}}{\partial y\partial {\frac {{\text{d}}x}{{\text{d}}s}}}}F\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)=H\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)$,

we may write the differential equation of the last article in the form:

$F_{1}\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)\left({\frac {{\text{d}}x}{{\text{d}}s}}{\frac {{\text{d}}^{2}y}{{\text{d}}s^{2}}}-{\frac {{\text{d}}y}{{\text{d}}s}}{\frac {{\text{d}}^{2}x}{{\text{d}}s^{2}}}\right)=H\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)$

With the aid of [i], we have

${\frac {{\text{d}}^{2}x}{{\text{d}}s^{2}}}F_{1}\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)={\frac {{\text{d}}y}{{\text{d}}s}}H\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)$, ${\frac {{\text{d}}^{2}y}{{\text{d}}s^{2}}}F_{1}\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)=-{\frac {{\text{d}}y}{{\text{d}}s}}H\left(x,y,{\frac {{\text{d}}x}{{\text{d}}s}},{\frac {{\text{d}}y}{{\text{d}}s}}\right)$. $\qquad {\text{[ii]}}$

It requires no further explanation to show how from these relations one can express the third and higher derivatives of $x$ and $y$ with respect to $s$ in terms of $x$, $y$, ${\frac {{\text{d}}x}{{\text{d}}s}}$ and ${\frac {{\text{d}}y}{{\text{d}}s}}$.

Article 95.
If, then, $F_{1}$ nowhere vanishes, and like $H$ is a continuous function of its arguments, and if $H$ never becomes infinite (see Art. 149), it follows that ${\frac {{\text{d}}^{2}x}{{\text{d}}s^{2}}}$ and ${\frac {{\text{d}}^{2}y}{{\text{d}}s^{2}}}$ can never become infinitely large, and are also continuous functions of the arc.

It follows that the curve has no singular point within the interval in question and that the curvature nowhere becomes infinitely large. This may be shown in the following manner : Let the points $x_{0}$, $y_{0}$ of the curve correspond to the value $s_{0}$ of $s$ then owing to the equations [ii] of the preceding article, the curve in the neighborhood of this point may be represented by the equations

$x=x_{0}+A(s-s_{0})^{\mu }+\cdots$,

$y=y_{0}+B(s-s_{0})^{\mu }+\cdots$,

where the constants $A$ and $B$ do not vanish simultaneously. When the values of ${\frac {{\text{d}}x}{{\text{d}}s}}$ and ${\frac {{\text{d}}y}{{\text{d}}s}}$ derived from these equations are substituted in

$\left({\frac {{\text{d}}x}{{\text{d}}s}}\right)^{2}+\left({\frac {{\text{d}}y}{{\text{d}}s}}\right)^{2}=1$,

it follows that

$1=\mu ^{2}(A^{2}+B^{2})(s-s_{0})^{2\mu -2}+A_{1}(s-s_{0})^{2\mu -3}+\cdots$,

and since this equation is true for all points in the neighborhood of $x_{0}$, $y_{0}$ it is seen that

$A_{1}=A_{2}=\cdots =0$,

and that further

$\mu =1$, $A^{2}+B^{2}=1$.

Hence the coordinates of every point of the curve situated in the neighborhood of $x_{0}$, $y_{0}$ may be represented through the regular functions

$x=x_{0}+A(s-s_{0})+\cdots$,

$y=y_{0}+B(s-s_{0})+\cdots$,

where $A$ and $B$ do not simultaneously vanish. Since this is true for every point $x_{0}$, $y_{0}$, it follows that the curve can have no singular points. Hence also the quantities $x'$ and $y'$ can not both vanish at the same time.