92 
MR. C. J. HARGREAVE ON THE CALCULATION OF ATTRACTIONS, 
Consequently 
where G is the equatorial gravity and / the latitude. 
18. From this it appears that up to terms of the 1st order, R = a (1 — s sin 2 l) is the 
equation of the curve which generates the surface of equilibrium, where the value of 
e depends on m, or on the velocity of rotation : but as the coefficients of higher 
powers of sin l may be considerable, it will be useful to find the surface of equilibrium 
to a greater degree of exactness. For this purpose we must introduce the fourth 
power of sin /, whose coefficient will be of the second order. Let the equation of the 
strata be a' = r' (1 + s' p 2 + A' p 4 ), or r' — a! (T — s' -J- (s' 2 — A') p 4 ^), where s' and A' 
are functions of a 1 as % a’ and 0 a'. Then 
(f) a' = q> r + Fr' ;m 2 + r <p' r 0 r p 4 = <p r F r.[/ > 2 + II r.p 4 , suppose. 
The (n + 1 ) th term of V for an internal point now becomes 
2 » e V r. { J? ^ J? d f' + 1*' 4 d s } . 
By writing for r and R their values in terms of a and a, it is easily found that 
d r' equals da! + (J^)( ~ s (* 2 + ^ (* 2 ~ A') - p' 4 ^ s 2 ) 
( d/ a \ uJ^ 
_ 3 ) s 2 ^ the same functions of a and s 1 . 
And that similar equations are true for the two remaining integrals. 
1st. Let n = 0, and we have 
>■> 
J ^ a! <p a! d a! + a 2 <p a ( — s fd 2 + fd 4 (J- s 2 — A^ + n 3 
2 1C P f , d Uj pa pa 
SJ ~ 4 j + (jJ 2 J q a! .¥a! da! + fj] 2 a 2 ¥a{- s^' 2 ) + (d 4 J o a'Ua'da’ 
L — same function of a and s 1? 
which gives 
a ' V a ^ a ' $ a (“ 1} irCy £2 — A)) + a? <p' a ^ F a!. a!. da! 
£ 1 P a 1 
— « 2 F a y + — J ^ a! .11 a! da! > — 4 w § (same function of a x s 2 and A x ). 
