Calculus of Variations/CHAPTER XI

CHAPTER XI: THE NOTION OF A FIELD ABOUT THE CURVE WHICH OFFERS A MINIMUM OR A MAXIMUM VALUE OF THE INTEGRAL. THE GEOMETRICAL MEANING OF THE CONJUGATE POINTS.

• 145 Notion of a field.
• 146 Neighboring curves which belong to the family of curves ${\displaystyle G=0}$.
• 147 A general theorem in the reversion of series.
• 148 The coordinates of a neighboring curve expressed in power-series of ${\displaystyle k}$, where ${\displaystyle k}$ is the trigonometric tangent between the initial directions of the neighboring curve and the original curve.
• 149 A curve which satisfies the equation ${\displaystyle G=0}$ is determined as soon as its initial point and the direction of the tangent at this point are known.
• 150 Limits assigned to ${\displaystyle k}$. Extension of the notion of a field.
• 151 Intersection of two neighboring curves. Conjugate points.
• 152 A point cannot be its own conjugate. The derivative of ${\displaystyle \Theta (t,t')}$ does not vanish at a point which causes the function itself to vanish.

Article 145.
In Chapter IX we showed that the neighboring curves, which pass through a fixed point ${\displaystyle A}$ and belong to the family of curves ${\displaystyle G=0}$, intersect again in a point ${\displaystyle B}$, if ${\displaystyle B}$ is the point conjugate to ${\displaystyle A}$. We shall now consider more fully this property of conjugate points.

We may first introduce the notion of a field about the curve which is to cause the integral to have a maximum or a minimum value.

We have assumed for all points belonging to that portion of the curve for which the integral in question is to be a maximum or a minimum that the function ${\displaystyle F(x,y,x',y')}$ is regular in ${\displaystyle x,y,x'}$ and ${\displaystyle y'}$ and that ${\displaystyle F_{1}}$ along this portion of curve is neither zero nor infinite. Exceptions to these assumptions are left for special investigation. From this, in connection with the necessary conditions already established, it is seen that the portion of curve can have no singular points (see Art. 95). Such a portion of curve has therefore at every point only a single normal which cuts the curve, and the radii of curvature of all points have a lower limit which is different from zero (Art. 80). Consequently, we may determine on both sides of the normal drawn through the point ${\displaystyle C}$ of the curve ${\displaystyle AB}$ two points ${\displaystyle D}$ and ${\displaystyle D'}$ in such a way that the normal within the interval ${\displaystyle DD'}$ is not cut by any other normal to the curve in the neighborhood of the point ${\displaystyle C}$. Consider lengths similar to ${\displaystyle DD'}$ drawn for all points along the curve ; then the surface bounded by the points ${\displaystyle D}$ and ${\displaystyle D'}$, which follow one another and which envelop completely the curve ${\displaystyle AB}$, has the property that within it no two normals drawn through two points of the curve that lie very close together intersect.

Article 146.
We represent the curve ${\displaystyle AB}$, which satisfies the differential equation ${\displaystyle G=0}$, by the equations

${\displaystyle 1)\qquad x=\phi (t,\alpha ,\beta )\quad y=\psi (t,\alpha ,\beta )\quad (t=t_{0}\ldots t_{1})}$

and one of the neighboring curves, which also satisfies the differential equation ${\displaystyle G=0}$, by the equations

${\displaystyle 2)\qquad {\bar {x}}=\phi (t,\alpha +\alpha ',\beta +\beta ')\quad {\bar {y}}=\psi (t,\alpha +\alpha ',\beta +\beta ')}$

Both curves are to pass through the same point ${\displaystyle A}$. If for the first curve there corresponds to the point ${\displaystyle A}$ a definite value ${\displaystyle t_{0}}$ of ${\displaystyle t}$, there will correspondto the same point for the second curve another value, say ${\displaystyle t_{o}+\tau '}$.

The condition that the first curve shall cut the second curve is expressed by the two equations:

${\displaystyle 3\qquad \phi (t_{0}+\tau ',\alpha +\alpha ',\beta +\beta ')-\phi (t_{0},\alpha ,\beta )=0\quad \psi (t_{0}+\tau ',\alpha +\alpha ',\beta +\beta ')-\psi (t_{0},\alpha ,\beta )=0}$;

or, developed in powers of ${\displaystyle \tau '}$, ${\displaystyle \alpha '}$ and ${\displaystyle \beta '}$:

${\displaystyle 4)\qquad \phi '(t_{0})\tau '+\phi _{1}(t_{0})\alpha '+\phi _{2}(t_{0})\beta '+(\tau ',\alpha ',\beta ')_{2}=0\quad \psi '(t_{0})\tau '+\psi _{1}(t_{0})\alpha '+\psi _{2}(t_{0})\beta '+(\tau ',\alpha ',\beta ')_{2}=0}$

where ${\displaystyle (\tau ',\alpha ',\beta ')_{2}}$ denotes the terms of the second and higher powers of <math\tau'</math>, ${\displaystyle \alpha '}$ and ${\displaystyle \beta '}$.

Article 147.
We may solve the equations 4) with respect to ${\displaystyle \tau '}$, ${\displaystyle \alpha '}$ and ${\displaystyle /beta'}$ as follows. Suppose we have two equations

${\displaystyle ax+by+cz+(x,y,z)_{2}=0\quad a'x+b'y+c'z+(x,y,z)_{2}=0}$

where one of the three determinants ${\displaystyle ab'-a'b,ac'-a'c,bc'-b'c}$, is different from zero. It follows,<rev>See my lectures on the Theory of Maxima and Minima, etc., p. 102 and p. 21.</rev> then, that we may express all values ${\displaystyle x,y,z}$ which satisfy the two equations, and in which ${\displaystyle x,y,z}$ do not exceed certain limits through three power-series of a single quantity.

We may choose for this quantity ${\displaystyle s=c_{1}x+c_{2}y+c_{2}z}$, where ${\displaystyle c_{1}}$,c_{2}</math>, and ${\displaystyle c_{3}}$ need satisfy the only condition:

${\displaystyle {\begin{vmatrix}a&b&c\\a'&b'&c'\\c_{1}&c_{2}&c_{3}\\\end{vmatrix}}\neq 0}$,

For brevity, write (cf. Art. 126)

${\displaystyle \psi _{1}(t_{0})\phi _{2}(t_{0})-\phi _{1}(t_{0})\psi _{2}(t_{0})=\theta _{3}(t_{0})}$;

then the three expressions corresponding in equations 4) to the determinants

${\displaystyle ab'-a'b,ac'-a'c,bc'-b'c}$

are

${\displaystyle 5)}$

${\displaystyle \phi '(t_{0})\psi _{1}(t_{0})-\psi '(t_{0})\phi _{1}(t_{0})=-\theta _{1}(t_{0})}$

${\displaystyle \phi '(t_{0})\psi _{2}(t_{0})-\psi '(t_{0})\phi _{2}(t_{0})=-\theta _{2}(t_{0})}$

${\displaystyle \phi _{1}(t_{0})\psi _{2}(t_{0})-\psi _{1}(t_{0})\phi _{2}(t_{0})=-\theta _{3}(t_{0})}$

of which ${\displaystyle \theta _{1}(t_{0})}$ and ${\displaystyle \theta _{2}(t_{0})}$ cannot both simultaneously vanish (Art. 127).

We may accordingly write

${\displaystyle c_{1}=0\quad c_{2}\alpha '+c_{3}\beta '=k_{1}}$

and further impose upon the constants ${\displaystyle c_{2}}$ and ${\displaystyle c_{3}}$ the condition

${\displaystyle 6)\qquad {\begin{vmatrix}\phi '(t_{0})&\phi _{1}(t_{0})&\phi _{2}(t_{0})\\\psi '(t_{0})&\psi _{1}(t_{0})&\psi _{2}(t_{0})\\0&c_{2}&c_{3}\end{vmatrix}}=c_{2}\theta _{2}(t_{0})-c_{3}\theta _{1}(t_{0})=1}$

If we consider only the linear terms in equations 4), we have

${\displaystyle {\begin{aligned}\phi '(t_{0})\tau '+\phi _{1}(t_{0})\alpha '+\phi _{2}(t_{0})\beta '&=0\\\psi '(t_{0})\tau '+\psi _{1}(t_{0})\alpha '+\psi _{2}(t_{0})\beta '&=0\\c_{2}\alpha '+c_{3}\beta '&=k_{1}\end{aligned}}}$

From these equations we have as first approximations for ${\displaystyle \tau ',\alpha '}$ and ${\displaystyle /beta'}$ the values

${\displaystyle 7)\qquad \tau '=-k_{1}\theta _{3}(t_{0})\quad \alpha '=+k_{1}\theta _{2}(t_{0})\quad \beta '=-k_{1}\theta _{1}(t_{0})}$

and therefore, finally,

${\displaystyle 7^{a})\qquad \tau '=-k_{1}\theta _{3}(t_{0})+k_{1}^{2}P_{1}(k_{1},t_{0})\quad \alpha '=+k_{1}\theta _{2}(t_{0})+k_{1}^{2}P_{2}(k_{1},t_{0})\quad \beta '=-k_{1}\theta _{1}(t_{0})+k_{1}^{2}P_{3}(k_{1},t_{0})}$

where ${\displaystyle P_{1}(k_{1},t_{0}),P_{2}(k_{1},t_{0})}$ and ${\displaystyle P_{3}(k_{1},t_{0})}$ are power-series in ${\displaystyle k_{1}}$ and ${\displaystyle t_{0}}$.

If we write these expressions in equations 2), or, what is the same thing, in

${\displaystyle 8)\qquad {\bar {x}}=\phi (t+\tau ',\alpha +\alpha ',\beta +\beta ')\quad {\bar {y}}=\psi (t+\tau ',\alpha +\alpha ',\beta +\beta ')}$

where, now, ${\displaystyle t}$ may take values less than ${\displaystyle t_{0}}$, then we have

${\displaystyle 9)}$

${\displaystyle {\bar {x}}=x-k_{1}[\phi '(t)\theta _{3}(t_{0})-\phi _{1}(t)\theta _{2}(t_{0})+\phi _{2}(t)\theta _{1}(t_{0})]+k_{1}(t,k_{1})}$

${\displaystyle {\bar {y}}=y-k_{1}[\psi '(t)\theta _{3}(t_{0})-\psi _{1}(t)\theta _{2}(t_{0})+\psi _{2}(t)\theta _{1}(t_{0})]+k_{1}(t,k_{1})Intheseequationsthesymbol<math>(t,k_{1})}$ is used to represent quantities which for every value of ${\displaystyle t}$ become indefinitely small at the same time as ${\displaystyle k_{1}}$.

When ${\displaystyle k_{1}=0}$, the curve represented by equations 9) becomes the original curve, and we see that ${\displaystyle k_{1}}$ can be taken so small that the two curves at corresponding points, that is, at points that belong to the same value of ${\displaystyle t}$, may come as near to each other as we wish. We shall show in the following Article that by this process we have derived all the curves that satisfy the differential equation ${\displaystyle G=0}$, which go through the point ${\displaystyle A}$ and are neighboring the first curve.

Article 148.
Instead of the quantity ${\displaystyle k_{1}}$ we may substitute a power series in ${\displaystyle \tau ',\alpha ',\beta '}$ which is subjected only to the condition that if ${\displaystyle c_{1},c_{2},c_{3}}$ are the coeflcients of the linear terms in ${\displaystyle \tau ',\alpha ',\beta '}$, the determinant

${\displaystyle {\begin{vmatrix}\phi '(t_{0})&\phi _{1}(t_{0})&\phi _{2}(t_{0})\\\psi '(t_{0})&\psi _{1}(t_{0})&\psi _{2}(t_{0})\\0&c_{2}&c_{3}\end{vmatrix}}\neq 0}$

This condition is satisfied by the power-series which expresses the trigonometric tangent of the angle which the initial directions of the two curves at the point ${\displaystyle A=t_{0}}$ include with each other.

For, denoting this tangent by ${\displaystyle k}$, we have

${\displaystyle k={\frac {{\frac {{\text{d}}y_{0}}{{\text{d}}x_{0}}}-{\frac {{\text{d}}{\bar {y}}_{0}}{{\text{d}}{\bar {x}}_{0}}}}{1+{\frac {{\text{d}}y_{0}}{{\text{d}}x_{0}}}{\frac {{\text{d}}{\bar {y}}_{0}}{{\text{d}}{\bar {x}}_{0}}}}}={\frac {{\bar {x}}_{0}'y_{0}'-{\bar {y}}_{0}'x_{0}'}{{\bar {x}}_{0}'x_{0}'+{\bar {y}}_{0}'y_{0}'}}={\frac {{\begin{vmatrix}\phi ''(t_{0})&\psi ''(t_{0})\\\phi '(t_{0})&\psi '(t_{0})\end{vmatrix}}\tau '+{\begin{vmatrix}\phi _{1}'(t_{0})&\psi _{1}'(t_{0})\\\phi '(t_{0})&\psi '(t_{0})\end{vmatrix}}\alpha '+{\begin{vmatrix}\phi _{2}'(t_{0})&\psi _{2}'(t_{0})\\\phi '(t_{0})&\psi '(t_{0})\end{vmatrix}}\beta '+(\tau ',\alpha ',\beta ')_{2}}{\phi '^{2}(t_{0})+\psi '^{2}(t_{0})+(\tau ',\alpha ',\beta ')_{1}}}}$

It is assumed that the curve is regular at the point ${\displaystyle A}$, so that the quantities ${\displaystyle \phi '(t_{0})}$ and ${\displaystyle \psi '(t_{0})}$ are not simultaneously zero, and consequently ${\displaystyle \phi '^{2}(t_{0})+\psi '^{2}(t_{0})}$ is different from zero.

Hence, the determinant of the equations 4) and 10) is

${\displaystyle {\frac {1}{\phi '^{2}(t_{0})+\psi '^{2}(t_{0})}}{\begin{vmatrix}\phi '(t_{0})&\phi _{1}(t_{0})&\phi _{2}(t_{0})\\\psi '(t_{0})&\psi _{1}(t_{0})&\psi _{2}(t_{0})\\{\begin{vmatrix}\phi ''(t_{0})&\psi ''(t_{0})\\\phi '(t_{0})&\psi '(t_{0})\end{vmatrix}}&{\begin{vmatrix}\phi _{1}'(t_{0})&\psi _{1}'(t_{0})\\\phi '(t_{0})&\psi '(t_{0})\end{vmatrix}}&{\begin{vmatrix}\phi _{2}'(t_{0})&\psi _{2}'(t_{0})\\\phi '(t_{0})&\psi '(t_{0})\end{vmatrix}}\end{vmatrix}}}$

Multiply the first horizontal row by ${\displaystyle \psi ''(t_{0})}$, the second by ${\displaystyle -\phi ''(t_{0})}$, and add them both to the third row, which then becomes

${\displaystyle 0\qquad {\begin{vmatrix}\phi _{1}'(t_{0})&\psi _{1}'(t_{0})\\\phi '(t_{0})&\psi '(t_{0})\end{vmatrix}}+{\begin{vmatrix}\phi _{1}(t_{0})&\psi _{1}(t_{0})\\\phi ''(t_{0})&\psi ''(t_{0})\end{vmatrix}}\qquad {\begin{vmatrix}\phi _{2}'(t_{0})&\psi _{2}'(t_{0})\\\phi '(t_{0})&\psi '(t_{0})\end{vmatrix}}+{\begin{vmatrix}\phi _{2}(t_{0})&\psi _{2}(t_{0})\\\phi ''(t_{0})&\psi ''(t_{0})\end{vmatrix}}}$

or, what is the same thing,

${\displaystyle 0\qquad \theta _{1}'(t_{0})\qquad \theta _{2}'(t_{0})}$

Hence, the above determinant is

${\displaystyle 10)\qquad {\frac {1}{\phi '^{2}(t_{0})+\psi '^{2}(t_{0})}}[\theta _{1}'(t_{0})\theta _{2}(t_{0})-\theta _{2}'(t_{0})\theta _{1}(t_{0})]={\frac {1}{\phi '^{2}(t_{0})+\psi '^{2}(t_{0})}}{\frac {C}{F_{1}(t_{0})}}\quad }$ (see Art. 129),

an expression which (loc. cit.) is different from zero.

We may accordingly write ${\displaystyle k}$ in the place of ${\displaystyle k_{1}}$, and find in the same way as above :

${\displaystyle 11)\qquad {\bar {x}}=x+kf_{1}(t,k)\quad {\bar {y}}=y+kf_{2}(t,k)}$

Article 149.
In Art. 89 the form of the solution of the differential equation ${\displaystyle G=0}$ was given. It follows that a curve which satisfies the equation ${\displaystyle G=0}$ is completely determined as soon as its initial point and the direction of the tangent at this point are known.

Let ${\displaystyle a,b}$ be the coordinates of ${\displaystyle A}$ and ${\displaystyle X}$ (see Fig. of Art. 87) the angle which the initial direction makes with the ${\displaystyle X}$-axis ; further, take instead of the coordinates ${\displaystyle x,y}$. new system of coordinates ${\displaystyle t,v}$ with a new origin at ${\displaystyle A}$ in such a way that

${\displaystyle 12)\qquad t=(x-a)\cos \lambda -(y-b)\sin \lambda \quad v=(x-a)\sin \lambda +(y-b)\cos \lambda }$

or

${\displaystyle 13)\qquad x=a+t\cos \lambda +v\sin \lambda \quad y=b-t\sin \lambda +v\cos \lambda }$

Now if we choose ${\displaystyle t}$ as the independent variable, then is

${\displaystyle x'=\cos \lambda +{\frac {{\text{d}}v}{{\text{d}}t}}\sin \lambda \qquad {\frac {{\text{d}}x'}{{\text{d}}t}}={\frac {{\text{d}}^{2}v}{{\text{d}}t^{2}}}\sin \lambda }$

${\displaystyle y'=-\sin \lambda +{\frac {{\text{d}}v}{{\text{d}}t}}\cos \lambda \qquad {\frac {{\text{d}}y'}{{\text{d}}t}}={\frac {{\text{d}}^{2}v}{{\text{d}}t^{2}}}\cos \lambda }$

and consequently

${\displaystyle x'{\frac {{\text{d}}y'}{{\text{d}}t}}-y'{\frac {{\text{d}}x'}{{\text{d}}t}}={\frac {{\text{d}}^{2}v}{{\text{d}}t^{2}}}}$

The differential equation ${\displaystyle G=0}$, i.e.,

${\displaystyle F_{1}\left(x'{\frac {{\text{d}}y'}{{\text{d}}t}}-y'{\frac {{\text{d}}x'}{{\text{d}}t}}\right)+H(x,y,x',y')=0}$

becomes then

${\displaystyle 14)\qquad 0={\frac {{\text{d}}^{2}v}{{\text{d}}t^{2}}}F_{1}\left(a+t\cos \lambda +v\sin \lambda ,b-t\sin \lambda +v\cos \lambda ,\cos \lambda +{\frac {{\text{d}}v}{{\text{d}}t}}\sin \lambda ,-\sin \lambda +{\frac {{\text{d}}v}{{\text{d}}t}}\cos \lambda \right)+H\left(t,v,{\frac {{\text{d}}v}{{\text{d}}t}}\right)\quad }$ (Art. 94)

Following the method of integration given in Chapter VI, we solve the above equation in such a way that when ${\displaystyle t=0}$, both ${\displaystyle v=0}$ and ${\displaystyle {\frac {{\text{d}}v}{{\text{d}}t}}=0}$, the ${\displaystyle v}$-axis being the direction of the tangent at the point ${\displaystyle A}$.

Hence, if ${\displaystyle F_{1}(a,b,\cos \lambda ,-\sin \lambda )}$ has a finite value different from zero and if ${\displaystyle H\left(t,v,{\frac {{\text{d}}v}{{\text{d}}t}}\right)}$ does not become infinitely large at the point ${\displaystyle A}$, as we have assumed was the case, since ${\displaystyle F_{1}}$ together with its derivatives, of which ${\displaystyle H}$ consists, is a regular function of its arguments, it follows that there is only one power-series of ${\displaystyle t}$ that satisfies the differential equation, and which with its first derivative vanishes for ${\displaystyle t=0}$.

This power-series has the form

${\displaystyle v=t^{2}P(t)}$

Writing this value of ${\displaystyle v}$ in the equations 13), they become

${\displaystyle 15)\qquad x=a+t\cos \lambda +A_{1}t^{2}+\ldots \quad y=b-t\sin \lambda +B_{1}t^{2}\ldots }$

where the constants ${\displaystyle A_{1},B_{1},\ldots }$ are definitely determined.

Thus the equations 15) completely determine the curve which satisfies the differential equation ${\displaystyle G=0}$, where ${\displaystyle a,b}$ are the coordinates of its initial point and the angle which its initial direction makes with the ${\displaystyle X}$-axis.

From this it follows at once that through equations 11) we have all the neighboring curves of the original curve which pass through the same initial point and satisfy the differential equation ${\displaystyle G=0}$.

Article 150.
We may therefore give ${\displaystyle k}$ an upper limit in such a way that all curves belonging to a value of ${\displaystyle k}$ below this limit and satisfying the differential equation ${\displaystyle G=0}$ lie completely in the surface which envelops the original curve.

This makes it possible to bring about a one-valued relation between both curves in such a way that, corresponding to every point of the original curve, we may determine the point of the neighboring curve at which this curve is cut by the normal at a point on the first curve.

Let ${\displaystyle x,y}$ be the coordinates of a point ${\displaystyle P}$ on the original curve, and ${\displaystyle x+\xi ,y+\eta }$ the coordinates of the corresponding point on the neighboring curve.

If ${\displaystyle P'}$ is the point corresponding to ${\displaystyle P}$, its coordinates are

${\displaystyle x+\xi =\phi (t+\tau ,\alpha +\alpha ',\beta +\beta ')\qquad y+\eta =\psi (t+\tau ,\alpha +\alpha ',\beta +\beta ')}$

and besides, since ${\displaystyle (X-x)x'+(Y-y)y'=0}$ is the equation of the normal, and ${\displaystyle X=x+\xi ,Y=y+\eta }$ is a point on it, we have

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

Hence, ${\displaystyle \xi ,\eta }$ and ${\displaystyle \tau }$ are to be determined from the equations

${\displaystyle 16)\qquad \xi =\phi '(t)\tau +\phi _{1}(t)\alpha '+\phi _{2}(t)\beta '+\ldots \quad \eta =\psi '(t)\tau +\psi _{1}(t)\alpha '+\psi _{2}(t)\beta '+\ldots \quad 0=\phi '(t)\xi +\psi '(t)\eta }$

The last of these equations combined with the first and second gives

${\displaystyle 17)\qquad 0=[\phi '^{2}(t)+\psi '^{2}(t)]\tau +kf(t)+(t,k)}$

when for ${\displaystyle \alpha ',\beta '}$ we have written from ${\displaystyle 7^{a})}$ their power-series in ${\displaystyle k}$.

Since the portion of curve ${\displaystyle AB}$ has no singularity, and consequently ${\displaystyle \phi '^{2}(t)+\psi '^{2}(t)}$ nowhere vanishes, we may from equation 17) express ${\displaystyle \tau }$ and therefore also ${\displaystyle \xi }$ and ${\displaystyle \eta }$ as power-series in ${\displaystyle k}$. If, then, we limit ourselves to curves with which ${\displaystyle k}$ remains within a certain limit, we may always determine the point where such a curve is cut by a normal of the original curve.

Article 151.
We ask if it is possible for the second curve to intersect the first curve. For this to be the case the length ${\displaystyle PP'}$ must be zero ; that is, ${\displaystyle \xi ,\eta }$ must for some value of ${\displaystyle t}$ be equal to zero. Hence we have so to choose the quantities ${\displaystyle t,\tau ,\tau ',\alpha ',\beta '}$ that the equations 4) and 16), when in 16) ${\displaystyle \xi }$ and ${\displaystyle \eta }$ are put equal to zero, are satisfied.

The terms of like dimension in 4) and 16) are homogeneous functions of ${\displaystyle \tau ',\alpha ',\beta '}$ and of ${\displaystyle \tau ,\alpha ',\beta '}$ respectively; these equations may be written :

${\displaystyle (\phi '(t_{0})+v)\tau '+(\phi _{1}(t_{0})+p)\alpha '+(\phi _{2}(t_{0})+q)\beta '=0}$

${\displaystyle (\psi '(t_{0})+v_{1})\tau '+(\psi _{1}(t_{0})+p_{1})\alpha '+(\psi _{2}(t_{0})+q_{1})\beta '=0}$

${\displaystyle (\phi '(t_{0})+v_{2})\tau '+(\phi _{1}(t_{0})+p_{2})\alpha '+(\phi _{2}(t_{0})+q_{2})\beta '=0}$

${\displaystyle (\psi '(t_{0})+v_{3})\tau '+(\psi _{1}(t_{0})+p_{3})\alpha '+(\psi _{2}(t_{0})+q_{2})\beta '=0}$

where ${\displaystyle v,p,q,v_{1},p_{1},q_{1}}$ represent functions of ${\displaystyle \tau ',\alpha ',\beta '}$ and ${\displaystyle v_{2},p_{2},q_{2},v_{3},q_{3},q_{3}}$ are functions of ${\displaystyle \tau ,\alpha ',\beta '}$, which with these functions and therefore also with ${\displaystyle k}$, become infinitely small.

The first two of these equations express that the two neighboring curves pass through the initial point ${\displaystyle A}$, and the last two that they are to go through another point.

In order that these four equations exist simultaneously, their determinant must vanish. This determinant, when in it we make ${\displaystyle k=0}$, is :

${\displaystyle {\begin{vmatrix}\phi '(t_{0})&0&\phi _{1}(t_{0})&\phi _{2}(t_{0})\\\psi '(t_{0})&0&\psi _{1}(t_{0})&\psi _{2}(t_{0})\\0&\phi '(t)&\phi _{1}(t)&\phi _{2}(t)\\0&\psi '(t)&\psi _{1}(t)&\psi _{2}(t)\end{vmatrix}}}$

and this is nothing other than the function ${\displaystyle -\Theta (t,t_{0})}$. Hence the determinant of the above system of equations may be brought to the form ${\displaystyle -\Theta (t,t_{0})-k(t,t_{0},k)}$; and, as this determinant is to vanish, we must have

${\displaystyle 18)\qquad \Theta (t,t_{0})+k(t,t_{0},k)=0}$

If, now, ${\displaystyle C}$ is a point of the original curve for which ${\displaystyle t=t'}$ and which is not conjugate to ${\displaystyle A}$, then ${\displaystyle \Theta (t',t_{0})}$ is different from zero, and we may therefore fix a limit for ${\displaystyle k}$ so that for all values of ${\displaystyle k}$ under this limit the expression ${\displaystyle \Theta (t',t_{0})+k(t',t_{0},k)}$ is different from zero ; that is, none of the curves which lie very near the original curve can cut this curve at the point ${\displaystyle t'}$ or in the neighborhood of it, since we can always find a limit ${\displaystyle h}$ of such a nature that for every value of ${\displaystyle t}$ within the interval ${\displaystyle t'-h\ldots t'+h}$ the expression is different from zero. And, reciprocally, every curve that lies very near the original curve will cut this curve in the neighborhood of ${\displaystyle C}$, as soon as there is a point ${\displaystyle C}$ in the interval ${\displaystyle AB}$ which is conjugate to ${\displaystyle A}$. For one can then always find for ${\displaystyle k}$ a value sufficiently small that, with very small values of ${\displaystyle h}$, the sign of ${\displaystyle \Theta (t'-h,t_{0})-k(t'-h,t_{0},k)}$ is the same as the sign of ${\displaystyle \Theta (t'-h,t_{0})}$, and the sign of ${\displaystyle Theta(t'+h,t_{0})+k(t'+h,t_{0},k)}$ is the same as that of ${\displaystyle \Theta (t'+h,t_{0})}$. But when the function ${\displaystyle \Theta (t',t_{0})}$ passes through the value zero it changes its sign, as is seen in the following Article. Hence, it follows, as ${\displaystyle \Theta (t',t_{0})}$ is to be zero, that the expression ${\displaystyle \Theta (t,t_{0})+k(t,t_{0},k)}$ must vanish once within the interval ${\displaystyle t'-h\ldots t'+h}$; or, in other words: If, in the interval ${\displaystyle AB}$ of the original curve, there is a point ${\displaystyle t=t'}$ conjugate to the initial point, then all the curves which lie very close to the first curve, which satisfy the differential equation ${\displaystyle G=0}$ and which have the same initial point ${\displaystyle A}$, will cut again the first curve in the neighborhood of the point ${\displaystyle t'}$. Consequently the conjugate point is nothing other than the limiting position which the points of intersection of a neighboring curve with the original curve approach, if we make smaller and smaller the angle which the initial directions of the two curves make with each other.

If there is no such limiting position within the interval ${\displaystyle AB}$, then there is no conjugate point within this interval.

Article 152.
It remains yet to show that the point ${\displaystyle A}$ cannot itself be this limiting position; that is, of all the neighboring curves there cannot be one which cuts the original curve as close as we wish to ${\displaystyle A}$. Analytically this case may be expressed in the following manner : If at the point ${\displaystyle t}$ the original curve is cut by a neighboring curve, we have the equation

${\displaystyle {\begin{vmatrix}0&\phi '(t_{0})+v&\phi _{1}(t_{0})+p&\phi _{2}(t_{0})+q\\0&\psi '(t_{0})+v_{1}&\psi _{1}(t_{0})+p_{1}&\psi _{2}(t_{0})+q_{1}\\\phi '(t)+v_{2}&0&\phi _{1}(t)+p_{2}&\phi _{2}(t)+q_{2}\\\psi '(t)+v_{3}&0&\psi _{1}(t)+p_{3}&\psi _{2}(t)+q_{3}\end{vmatrix}}=0}$

a determinant which becomes ${\displaystyle \Theta (t,t_{0})=0}$, when ${\displaystyle k=0}$. If for ${\displaystyle \tau ,\tau ',\alpha ',\beta '}$ expressed as power-series in ${\displaystyle k}$, their values be substituted in the determinant, it becomes an equation in ${\displaystyle t}$ and ${\displaystyle k}$. Further, since ${\displaystyle \phi '(t),\phi _{1}(t),\ldots \psi _{2}(t)}$ are power-series in ${\displaystyle t}$ which are regular functions in the neighborhood of ${\displaystyle t_{0}}$, the determinant may be developed in a power-series in ${\displaystyle t-t_{0}}$ and ${\displaystyle k}$, which converges for sufficiently small values of ${\displaystyle t-t_{0}}$ and ${\displaystyle k}$. If in the neighborhood of the original curve there exist curves which cut this curve as near as we wish to ${\displaystyle A}$, then, after sufficiently small limits have been given to ${\displaystyle t-t_{0}}$ and ${\displaystyle k}$, it is possible to find values for these quantities within the given limits for which the equation is satisfied.

If we write ${\displaystyle t-t_{0}}$, the quantities ${\displaystyle v,p,q}$ are respectively equal to ${\displaystyle v_{2},p_{2},q_{2}}$ and the quantities ${\displaystyle v_{1},p_{1},q_{1}}$ to ${\displaystyle v_{3},p_{3},q_{3}}$.

When this is the case, the determinant has the form

${\displaystyle {\begin{vmatrix}0&a&b&c\\0&a_{1}&b_{1}&c_{1}\\a&0&b&c\\a_{1}&0&b_{1}&c_{1}\end{vmatrix}}}$

which is identically zero. Therefore the power-series in ${\displaystyle t-t_{0}}$ and ${\displaystyle k}$ will vanish for ${\displaystyle t-t_{0}}$, whatever be the value of ${\displaystyle k}$; and consequently this series is divisible by ${\displaystyle t-t_{0}}$.

The determinant, then, when divided by ${\displaystyle t-t_{0}}$ is for the value ${\displaystyle t=t_{0}}$:

${\displaystyle 19)\qquad \left[{\frac {{\text{d}}\Theta (t,t_{0})}{{\text{d}}t}}\right]_{t=t_{0}}+(t-t_{0},k)_{t=t_{0}}=0}$

We saw in Arts. 128 and 129 that

${\displaystyle \Theta (t,t_{0})=\theta _{1}(t_{0})\theta _{2}(t)-\theta _{2}(t_{0})\theta _{1}(t)}$

and that

${\displaystyle \theta _{1}(t)\theta _{2}'(t)-\theta _{2}(t)\theta _{1}'(t)={\frac {C}{F_{1}(t)}}}$

If ${\displaystyle t'}$ is a conjugate point to ${\displaystyle t_{0}}$, so that

${\displaystyle \Theta (t,t_{0})=\theta _{1}(t_{0})\theta _{2}(t')-\theta _{2}(t_{0})\theta _{1}t')=0}$

it follows that

${\displaystyle \theta _{1}(t')=\lambda \theta _{1}(t_{0})\qquad \theta _{2}(t')=\lambda \theta _{2}(t_{0})}$

where ${\displaystyle \lambda }$ is a constant different from zero.

We further have, since

${\displaystyle {\frac {\text{d}}{{\text{d}}t}}\Theta (t,t_{0})=\theta _{1}(t_{0})\theta _{2}'(t)-\theta _{2}(t_{0})\theta _{1}'(t)}$

the relation

${\displaystyle \left[{\frac {\text{d}}{{\text{d}}t}}\Theta (t,t_{0})\right]_{t=t'}={\frac {1}{\lambda }}{\frac {C}{F_{1}(t')}}}$

which is different from zero.

It is thus seen that the derivative of ${\displaystyle \Theta (t,t_{0})}$ does not vanish on the positions at which the function itself vanishes.

At the same time it is shown that the equation 19) is not satisfied, so long as ${\displaystyle k}$ and ${\displaystyle t-t_{0}}$ remain within finite limits; and consequently a neighboring curve cannot intersect the original curve a second time indefinitely near the initial point through which both curves pass.

As there is a great range of choice regarding the variable ${\displaystyle t}$, and as the constants ${\displaystyle \alpha }$ and ${\displaystyle /beta}$ may be chosen in many ways, it is possible to give many forms to the function ${\displaystyle \Theta }$. To be strictly rigorous, it would yet remain to prove that the solution of the equation ${\displaystyle \Theta (t,t_{0})}$ leads always to the same conjugate point, whatever be the form of ${\displaystyle \Theta }$; the geometrical significance of these points, however, make such a proof superfluous.