The same reasoning may be applied to partial differential equations of higher orders, of which we have obtained the complete first integral: the solutions obtained will be of an order next inferior to that of the given equation.

(70.) Let it be required to determine whether a particular value ≈ = U is or is not comprised in the equation

z=f{x,y,p(P), ¥(Q)},

the general solution of the partial differential equation, Z=0. Let the general solution be put under the form

z=U+au,

in which it is necessary that u should contain as many arbitrary functions as the general solution, and also have a finite value when a=0.

We have p=dU+adu, q=d,U+adu,

r=d2U+ad2u, s=d ̧dU+addu, t=d2U+adžu, &c. and since ≈ = U will satisfy the equation Z=0, we obtain d ̧Z.u+d,Z.d ̧u + d2 Z. du

+d,Z.d2 u+d ̧Z.d ̧du+d,Z.d2u + &c.=0.

If this substitution renders all the differential coefficients of Z respectively equal to nothing, except dZ, this last equation cannot be satisfied unless u=0, in which case no arbitrary functions can be introduced in the equation = U. If d,Z, and d. Z do not vanish, the function u, depending on the equation d. Z. u+d2Z.d ̧u + d2Z. d,u=0,

will involve one arbitrary constant.

(L. C. D. 793-5.)

APPLICATION OF THE DIFFERENTIAL. AND

INTEGRAL CALCULUS TO GEOMETRY.

THE CONTACT OF LINES.

(1.) Let y=(x), y=4(x), be the equations of two curves having the co-ordinates a, y common to both, then

Dx (Dx)2 d2 p(x) + &c.

(x+Dx)=y+ d2p(x)+
1

1.2

(Dx)

1.2

d2 (x) + &c.

If d ̧p(x)=d ̧¥(x), they have a contact of the first order, and do not intersect each other. If d p(x)=d2↓(x), they have a contact of the second order, and intersect each other at the point (x, y): and generally, if drp(x)=d" \(x), they have a contact of the nth order, and intersect or not, according as n is even or odd.

Let x, y, be the co-ordinates of a curve, 1, y1, those of its tangent, then the equation of the tangent is

The distances from the origin at which the tangent cuts the axes of x and y respectively are

The subnormal = ydy.

The part of the normal intercepted between the axis and the curve = y.1+(d ̧y)o1.

(2.) The circle of curvature, or osculating circle, is one which has a contact of the second order with any proposed curve. Let ρ be its radius, x, y, the co-ordinates of its centre, then

If x and y are functions of a third variable, t, then

If the arc s is considered the independent variable, then

=

= { (d2 y)2 + (d2x)2 } − §.

(Laplace, Mec. Cel. Liv. 1. Ch. ii.)

When the radius of curvature is a maximum or minimum, there is a contact of the third order, and in that case the circle of curvature does not cut the curve.

The curve y=4(x) is convex or concave towards the axis of x, according as p(x+Dx) and dy have the same or different signs. (L. D. C. 151.) (3.) The evolute of a plane curve is the locus of all centres of curvature: its equation may be found by substituting for dy and day their values obtained from the equation of the proposed curve, p(x,y) =0, in the equations

and eliminating x and y between these, and p(x,y)=0: the resulting equation between a, and y, will be the equation of the

evolute.

X1

be the equation of a plane curve, in which the quantities n,, n-1, &c. form a decreasing series, of which n_, is the first negative term, then the curve whose equation is

19

-1

y=a,x2r + a„−1x^r-1 + &c. + α1x11 +α x2,

is an asymptote to the former.

If n=1, and a,-1,....a respectively = 0, the asymptote is a right line, and the corresponding infinite branch is of the hyperbolic kind: for all other values of n,, &c. a,, &c. the asymptote and curve are of the parabolic kind.

(5.) Methods of expanding y in a descending series in terms of x.

or

for

[1] Assume u= =y, then by Maclaurin's theorem

y x

*

[2] Assume y=aam, and after substituting this value y, equate the indices of any two terms: then if the other terms, when the value obtained for m has been substituted, are of less dimensions than these, we shall obtain the first term of a descending series. By assuming

y= ax + bx",

a second term may be obtained, and so on: this method is generally very tedious, if applied to an equation consisting of more than four terms.

A rectilinear asymptote, may frequently be conveniently obtained by finding the value of x- and y-xdy,

y

day'

when ∞; if either or both of these quantities be finite, there is an asymptote, the direction of which will be determined by the value of the quantity d, Y.

SINGULAR POINTS.

(6.) Let the equation of a curve, and its differential be p(x,y)=0, M+Ndy=0,

respectively

then the values of x and y which satisfy the equations

p(x,y) =0, M=0,

but do not satisfy N=0, give a maximum or minimum ordinate, according as day is negative or positive: and those values which satisfy N=0,

p(x,y) =0,

but not M=0, give a maximum or minimum abscissa.

If day=0, (x,y) is a point of inflexion, or contrary flexure, at which point the tangent cuts the curve: if however d3y=0 is satisfied by the same values of x and y, (x,y) is a point of undulation, at which point the tangent merely meets the curve.

Generally, if several successive differential coefficients, as far as the nth, vanish for the same values of x and

y, the curve has a contact of the nth order with its tangent, which intersects the curve or meets it, according as n is even or odd.

(7.) If the same values of x and y satisfy the equations

then dry=

p(x,y)=0, M=0,

N=0,

=, in which case the values of dy may be found by preceding methods. (Diff. Calc. 15.)

If the equation d,y=- has two or more unequal possible roots, then (x,y) is a multiple point, through which as many branches of the curve pass as the above equation has possible

this equation may be determined as above: if it has two or more unequal possible roots, (x,y) is a point of osculation ;