206 Mr. Arry on the Figure of a Fluid Mass . 
hence the sum for the pyramid is J,Ap* = 40 ; which for the 
whole pyramid is aii 
(5.) To find the length of p and the value of 4, suppose c to be 
the ordinate parallel to the axis of the solid: suppose the solid 
divided into wedges by planes passing through the line c; let two 
of these planes make, with the plane of xz, the angles ¢ and ¢+6¢: 
suppose the included wedge divided into pyramids by lines on 
these planes, drawn from the attracted point; let two of them 
make with c the angles @ and 9+60. Then 4 =sin. 0.80.5¢. 
Also x = a—p sin. 0 cos. ?; y =p sin. 0. sin. P: = =c—p cos. 0; x, y, 2, 
being the co-ordinates of the point at which p meets the surface 
again. Substituting these values in the equation to the surface, 
the value of p will be had in terms of @ and ¢. 
(6.) The disturbing forces being small, we will neglect the 
squares and higher powers of the disturbing forces and quan- 
tities dependent on them. And as the body, without this dis- 
turbance, would be a sphere, we will assume for its equation 
r+y+2=r'+y(z), where x(z) is a function of z involving 
a small multiplier. Substituting for x, y, z, the values found above, 
a’ +c°—2p (c cos. 0+ a Cos. p. sin. 0) + p* =7* + x (C—p COS. 4). 
But @ and c are co-ordinates of a point in the surface: hence 
(ie tte =r +x (c). 
Subtracting, p’—2p, (c cos. +a cos. ¢ sin, 6) = x (c—p Cos. 8) — x (€) ; 
“. p= 2(ccos. 0+ a cos. p sin. 0) + pase DLL Delia ON 
An approximate value of p is 2(c cos.@¢+acos.¢ sin. 6): let this 
be v; substituting it in the small term, 
x(e-—v cos. 0)-x (c) 
p=vut And & =5 +x (c-v cos. #)—x (0). 
«By f,Ap is meant what is usually written {4 pdp, the quantity whose differential 
coefficient, taken with respect to p, is Ap. 
