86 MR. C. J. HARGREAVE ON THE CALCULATION OF ATTRACTIONS, 



The first term is 



= 2 T^ («2 ^^ ~ ^ _ 7-2 V {e being the eccentricity). 

 The third term is 



= _ 3t^ (1-^2) ^2 (I - ^ + "^^^^^in-ie), for K^ = 9. 



4 ff p 7^ 'I* 

 The value of V for the sphere of radius r, calculated by the usual method, is — ^ — ; 



consequently, for the whole ellipsoid, the value of V is 



Differentiate tof, by means of the equations r^ :=. f^ -\- g^ -\- h^ and (ji, = —^ and we 

 have 



— -j> = attraction m x= -\- 2 t^/ ( 1 — -^ H —3 — sin- ^ e) ; 



so — -J— = attraction 



/ 1 v^ri — e*) \ 



ionin3/= + 2t^^\^1 — ^+ \s sin-^ej; 



so — j^ = attraction in 3/= +4cr^A(4'i2 ^a — sin- ^ e) ; 



which are the 

 common ex- 

 pressions 

 otherwise 

 found -|~. 



Also 



_ -^ = attraction to centre == +4T^|y+ |-r(y - ^*-^)(~}2 + ^^^^^ 



10. By a similar process, I have deduced the attraction of an oblate spheroid, on 

 a point within it ; the density varying inversely as the distance from the centre. The 

 corresponding expressions are 



rfV_ « 3 TTgff f 1 1 Al-eT, 2 1-en 14-e-) 



* See Pratt. Mec. Phil., § 172. 



t Ibid. § 158. 



