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By means of these eliminating p and q, we find

x2 + y2 + x2 = a2

as the singular solution.

(22) Let the equation be

(px − qy)3 q + 4mx3 (x − px) = 0.

d U

dp

d U

dq

=

=

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These two equations agree with the original one, if we

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and the singular solution found by eliminating p and q is

2 - 4m x3 y = 0.

Legendre, Mémoires de l'Académie, 1790, p. 238.

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dU

dp

dU

(≈ - px - qy)" = Apa qb.

= − mx (≈ − px − qy)TM-1 — a Apa1 qb = 0,

·➖ ➖ my (≈ - px -qy)-1-bAp"q-10. dq

Dividing the first of these by the second we find

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By means of these values, eliminating p and q, we find the singular solution to be

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If the partial differential equation be of the second order

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the conditions which must be satisfied in order that an equation should be the singular solution of the first order of the equation U= 0 are

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If the function is to be a singular solution belonging to the final integral, it must in addition satisfy the equations

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1 + x

+

(P

y = 0.

1+x

=0 is the only condition, and we have

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is the singular solution required. The integral of this is

≈ = (1 + x) † (y).

Poisson, Jour. de l'Ecole Polyt. Cah. XIII. p. 113.

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it is therefore a singular solution corresponding to the final

integral.

CHAPTER IX.

QUADRATURE OF AREAS AND SURFACES, RECTIFICATION
OF CURVES AND CUBATURE OF SOLIDS.

SECT. 1. Quadrature of Plane Areas.

WHEN an area is referred to rectangular co-ordinates a and y, the double integral ffddy taken between the proper limits gives the value of the area. One of the integrations may always be performed, so that we have either

fyda + C or fady + C,

and these integrals are to be taken between the limits of y. or x, which form the boundaries of the area. If we take the first of these expressions, the limiting values of y must either be constants or functions of a given by the equation to the bounding curve: therefore on substituting these values we obtain a function of a alone, which is to be integrated, and taken between the limits of that variable which are required by the problem. If after the first integration we suppose C = 0, the integral A = fyda expresses the area included between the axis of x, the curve, and two ordinates corresponding to the limits of v.

In taking the integral fyda between the final limits of a, it is necessary that the interval should not contain a value of a which causes y to vanish or become infinite, as in that case we might be led to an erroneous conclusion. Thus if we suppose a curve to be symmetrically situate in the first and third quadrants, and to intersect the axis at the origin; and if we were to integrate from x = a to x = a we should obtain zero as our result, instead of finding the area to be double of that from a 0 to a = a, or that from = 0 to ∞ = - a. Therefore when any interval from a to b contains a value c of a which makes y vanish or become infinite, we must break it up into two intervals, one from b to c and the other from c to a, and add the integrals corresponding

=

to these.

In like manner if the interval contain several· values of a which make y vanish or become infinite, we must split it up into as many smaller intervals, each having one of these values of r as a limit, and add them all together.

If the co-ordinates be not rectangular, and a be the angle between them, we must multiply the integral by sin a to obtain the value of the area.

Ex. (1) If we take the general equation to a parabola of any order

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(2) The general equation to hyperbolas referred to their asymptotes is

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This formula fails when m=n, in which case

A = C + a2 log x.

(3) When the common hyperbola is referred to its axes, the sectorial area ACP (fig. 53) is easily found.

For

Now

and fy dx

ACP = NCP – ANP = 1xy – fy dx.

=

b

2a

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{x (x2 - a2)1} - a2 log {x + (x2 − a2)}} + C.

Determining the constant by the condition that the area vanishes when a = a, we have

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