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tion of all elements similar to PQ from 1 to K, that is, from y=0 to y = MK; that is, the aggregate of the elements is the band LPK; and as the area of the band will be expressed in terms of X, and is therefore the general value of all similar bands, the effect of a subsequent x-integration will be, to sum all such elemental bands of which the surface is composed, and the limits of this latter integration must be x = 0, and x = 01. If therefore MK, as determined by the equation to the surface, = Y, and OA = a, then

dzı2 dzı2) }, the area =

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If the x-integration is effected first, the effect will be to determine the band GPRI, and the limits of integration will be HI = x and 0; and the subsequent y-integration, with the limits OB=b and 0, will sum all such bands contained between parallel planes, and will give the area of the surface. In this latter case,

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The above is an outline of the general method of finding the area of such surfaces : the limits of integration will of course vary according to the conditions of each problem.

237.] Examples illustrative of the preceding formulæ. Ex. 1. The surface of the eighth part of a sphere.

Let the surface delineated in fig. 36 be that of the octant of a sphere: then, o being the centre, x2 + y2 +z = aạ.

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Ex. 2. A sphere is pierced by a right circular cylinder whose surface passes through the centre of the sphere, and the diameter of whose generating circle is equal to the radius of the sphere; it is required to find (1) the area of the surface of the sphere intercepted by the cylinder; (2) the area of the surface of the cylinder intercepted by the sphere.

Let the cylinder be perpendicular to the plane of (x, y): then

the equations to the cylinder and to the sphere are y2 = ax — xo, and x2 + y2 + 2 = a respectively. (1) If y= (ax — x2), then, from (34),

py a dy dx the area of the sphere =

Jo Jo (a_ *? y2)

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(2) Eliminating y, we have z = (a? — ax)} = 2, say; and the length-element of the trace of the cylinder on the plane of (x, y)



ja; therefore

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(a? - ax)t'
the area of the cylinder = 2/1 2

Jo J-2(a? – AX)* Ex. 3. On the double ordinates of a circle, and in planes perpendicular to the plane of the circle, isosceles triangles, whose vertical angle = 2 a, are described ; prove that the equation to the surface thus generated is x2 + (y +z tan a)2 = a?, and that the whole convex area = 2a{cot a ta (cosec a)2}.

238.] For another application of the preceding theory of quadrature let us consider that of the ellipsoid ; for although the integrals which determine this area become elliptic transcendents, and consequently do not admit of integration, yet by the introduction of new variables, and transformation according to the principles of the preceding chapter, they assume forms deserving notice on account of the geometrical interpretations which they yield. Let the equation to the ellipsoid be

xc2 22 22
az + 32 + 2 = 1,

(36) where a > b > c. And let a denote the whole surface; then, if r . (a? – x2)}, by the last of (29),

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which does not admit of further integration; let us however introduce new variables; and in the first place let the equation (36) be expressed in terms of subsidiary angles a and ß as follows:

x = a sin a cos B,
y = b sin a sin B,

(38) z = c cos a; for these equations satisfy (36).

Now we may either substitute these values in (37); or may apply to them immediately equation (30), Art. 236. In either case, if da denotes the infinitesimal surface-element, da= abcfc 2. (sin a)" (cos 3)2 , (sin a)2 (sin B)2 , (cosa) . +!

- sina dadß. (39) ? a? But if p = the length of the perpendicular from the centre of the ellipsoid on the tangent plane,

62 cm (sin a)? (cos B)2 + c^ ao (sin a)? (sin B)2 + ao b2 (cosa)2

ao b2c2 and therefore da = abc sin a da dB

(40) and if a = the whole surface of the ellipsoid,

ppsin a da dB A = abc

Todo P Hereby we may prove the following theorem. If da = the surface-element of an ellipsoid, and s is the area of the central section of the surface which is parallel to the plane of da, 1 4A = 4, when the ranges of integration include the whole ellipsoid.

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239.] Again, let us introduce two other subsidiary angles n and y, such that sin n cos y, sin n sin y, cos n are respectively the direction-cosines of the normal of the ellipsoid at the point (x, y, z);

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which is the equation to an elliptical cone whose vertex is at the centre of the ellipsoid ; and as n is the 2-direction-angle of the



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normal of the ellipsoid, the axis of z is the axis of the cone, and the ratio of the semi-axes of any plane elliptical section of it perpendicular to its axis is that of a? : 62. Now the y-integration, which has already been effected, between the limits 7 and — T, gives an annulus on the surface of the ellipsoid, the breadth of the annulus being due to the variation of n. Imagine therefore two cones, represented by equation (51), to be described corresponding to n and to n+dn; the lines of intersection of these cones with the ellipsoid will be two curves, infinitesimally near to each other, which contain between them the band of the ellipsoidal surface expressed by sin n (m+ ) dn.


mon3 and the sum of all these bands between the limits 7 and 0 will be the whole surface of the ellipsoid.

The projection on the plane of (x, y) of the intersection of the given ellipsoid with the cone (51) is an ellipse, whose equation determined by the elimination of z from (36) and (51) is

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Thus if the band which is expressed by (52) is projected on the plane of (x, y), one of its bounding lines is the ellipse (53), and the other is the ellipse corresponding to n+dn.

240.] The element-function of (37) also admits of another interpretation deserving notice. Let

n; so that dy dx = ab dndt; also the equation which determines the limits of integration is &? +m2 = 1; let

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l-a2 &2 – B2
= 6;

(54) 1-62 – 12 whereby we have a= abs L" sdn ;

(55) Now let us suppose $, n, Š, to be rectangular coordinates; then dn dẾ is an area-element of the plane ($, n), and as & is the coordinate parallel to the 6-axis, 5 dn df expresses the volume of a rectangular parallelepipedon whose base is dn , and whose height is Š ; Ś being related to & and n by the equation (54), which may be put into the form,

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