558 
. Mr. Airy on the 
if the ordinates be measured from the centre, and a? be 
the axis of revolution, and T the time of revolution, the 
only force besides the attraction is the centrifugal force ; 
the resolved part of which in the direction of x is o, that 
. a 
in the direction of y is ^jT*y, that in the direction of z is 
z. This contributes to <JU the terms ML (y <5y -}- zSz) > 
and to U the terms ML (y 2 -j- z 2 ) = ML . AT r 2 . \ — {*». 
1 3 2 1 
And if £M be any attracting particle whose co-ordinates 
are x', y', zf, its attractions in those directions are 
(/— x) S M (y'—y) j'M 
(#' — a:! 2 + / — j/] 3 + z'— z | 2 )~? (x ‘ — a:l 2 + y'—yf + z' — z] 2 )i ’ 
— - r -M - — - — •• and consequently it contributes to 
[x'—xl +y—y\ + Z — l\)2 
iU the term iU (» , -^> ^ + , (/- . *) + and y h 
(TUTf + + 7-TP)# 
term //==- — — ==- v The expression then which 
V[x — a?) +y — y) + z'—z\) 
the attraction of the whole adds to U is the sum of all the 
quantities ^ ==y*v or the sum of the pro- 
V{x'—x] +y—y J +3 — 2 ]; 
ducts of each particle by the reciprocal of its distance from 
the attracted point ; that is, V. The whole value of U then 
is V + ML . ML . r 2 . IT- [x 3 ; and the equation to a sur- 
face of equal density is C = \— t — + { {r (a) — r(a)} 
+ (%* + r* {*(«) - *(-)}) • 
2 T 3 
1 - (- 
( 14 .) If then for r we put its value in terms of the semi- 
axis of the spheroid which terminates it, and the angle which 
it makes with the equator, we shall have an equation which, 
so far as it depends on that angle, must be identically true. 
