On the Attraction of Ellipsoids. 363 



Substituting these values for cos a , cos j3 , cos 70, in the expres- 

 sion for Q, and observing that 



COS a' COS a + COS |3' COS )3o + COS y' COS 70 = 0, 



we get 



Q = T. - : cos a' cos Q' cos y'= o. (9) 



r 4 pr sm 



PROPOSITION YI. 



The same things being supposed, to find the other Components 

 of the Attraction, namely It in the direction of the centre of gravity 

 MO, and P in the transverse direction TS. 



To find ^; , 



JR = X cos a + Y cos jS' + Z cos y' ; 



= +~(A + + C-3I). (10) 



To find P ; 



P = X cos ai + Y cos )3i + Z cos 71 ; 

 but, 



sin cos ai = cos a cos - cos a, 



sin0 cos]3i= cos|3'cos0 - cos/3, 



sin cos 71 = cos 7' cos - cos 7. 



Substituting for cos a, cos|3, 0087 their values from (8), we get 



cosai = -- r^- cos a', cos/3i = -- : cosjS', 

 pr sm ^>r sm 



c 2 -^ 2 



COS 71 = -- r COS 7 . 



jt?r sm 



Substituting these values of cosai, cosjS,, 0087!, and observing 



that 



cos a cos d! + cos ]3' cos ]3i + cos 7' cos 71 = 0, 



