92 MR. C. J. HARGREAVE ON THE CALCULATION OP ATTRACTIONS, 



Consequently 



l\N« 



CMa 



{Ma , „, 3/„ l\Na /»,, „, Ma") 



3^ ( 1 + 2 s p2) - y (p^ - y ) -^ - 3- ( 1 ~ ^2) _ I , 



= 4 ^ f ^ { 1 + 2 S ^2 + (1 _ 3 ^2) (, _ ^) _ ^ (1 ^ ^2) j (by B.), 



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 — g sin^ /) is the 

 equation of the curve which generates the surface of equilibrium, where the value of 

 2 depends on m, or on the velocity of rotation : but as the coeflScients of higher 

 powers of sin / 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 -}- g>2 4. A' (/,'^), or r' = a' (l — g>2 _|_ (g'2 _ a') (/j'^^, where g' and A' 

 are functions of a' as % a! and B a'. Then 



<|) a' = 9 r' 4- Fr>2 -I- ^'^^lli^ ^ + r <j)' r r fji.'^ = ^r + Fr.^2 + Ur.[i\ suppose. 

 The (/2 + l)th term of V for an internal point now becomes 



By writing for r and R their values in terms of a and a, it is easily found that 



+ ( w-3 ) s^ "2 the same functions of a and g^. 



And that similar equations are true for the two remaining integrals. 

 1st. Let w = 0, and we have 



"^ij-i d{^' 



=2„W 



J^\' cj) a'da'-{-a^(l>a[ - g ^'2 + ^'4 (^1 s®- A)) + a^ (f)' a'^ 



+ iW-'^T^" «' . F a' rfa' + ^'2 «2 F a (~ g ^'2) + ^'^ f^ d Ua'da' 



. — same function of a and gj, 

 which gives 



4Te|/%'(pa'rfa'H-a2<p«(._j- + |(|-g2_.A))+a3<p'«^^_l^"F«'.«'.rfa^ 



e 1 /*a ■) 



— a2 F a y + YJ a! .U a' da! > — 4Tr^ (same function of aj g^ and Aj). 



