Professor Mac CuLLAon's Lectures on the Attraction of Ellipsoids. 389 

 hence, by substitution, we have 



J2 _ (jS c^ _ ^2 



cos oo = : — r cos Q' cos 7', COS Ri, = : — - cos y cos a, 



pr sm (f) pr sm 



a" -b' 



cos 7o = ■■ — - cos a cos /3 . 



pr sm 



Substituting these values for cosoo, cos|3o, cos 70? in the expression forQ, 

 and observing that 



cos a cos Oo + cos (3' cos Po + cos 7' cos 70 = 0, 

 we get 



^ ZMa" (b- - c') + P (c' - a') + & (a' - b') , ^ , ^ ... 



Q = —^—^ ^—^ > — ■ ^^ — > ^ cosa'cos^'cos7'= 0. (9 



r* pr sin r- « \ / 



Pkoposition VI. 



The same things being supposed, to find the components of the attraction, namely, 

 R in the direction of the centre of gravity MO, and P in the transverse direction TS. 



To find R ; 



R = X cos a + Y cos p' + Z cos 7', 



••• i? = ^ + 2^J^ + i? + C-5/) + ^, 



To find P ; 

 but, 



^ = ^. + 273(^+^ + ^-3^^ (10) 



P = X cos a, + 5^ COS j3, + Z cos 7, ; 

 sin cos o, — cos a! cos (/> — cos a, 



sin cos |8| = cos ^' cos <p — cos /3, 

 sin cos 7i = cos 7' cos — cos 7. 

 Substituting for cos a, cos/3, cos 7, their vahies from (8), we get 



-y2 „2 



¥-■ 



a -P , „ 0- —p- ^1 



cos O, = r-^ — cos a, COS fl, = r^ COS H , 



pr sin /jr sin 



C^ — 7)^ 



cos 7, = 7^-— cos 7'. 



pr smtf) 



3 E 2 



