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



When w = 2, the term is 



To calculate V for the sphere whose radius is r. 



Let the sphere be referred to polar coordinates, the centre 

 being the pole. 



LetOC = r,OP = ri,andPOC = 0; thenPC= ^/ {r'^ ^ r^^ 

 — 2rriCos0). 

 Mass of the element at V =^ ^r^ (pr^dr^^mBdd d a, and 



(r« + ri^ — 2rri cos 9) "^"^^J o J o ^/{r^ +r,* — 2rr,cos5) 



K being the value of r^'^ ^^ rj — 2 r^ (p„ rj + 2 (pn^ r^, when rj = 0. 

 The w^hole value of V then is 



4 cr^ |«<|)^a - <^^^a - -^e2fl2^a - (<^,,r - |-<;>,^, ^ + 7") ~ j^ ^^ (/^' ~ 1") ^' ^ "} ' 

 And the attraction toward the centre 



where 





13. To find the attraction of the same spheroid on a particle without it. 

 The series (Art. 3.) is 



/»R />i / r'* r" \ , 



Now 



^/*<|) r' . r"* + 1 . </ r' = /" + 1 <^^ r' - (n + 1 ) r"* </)^^ r' + w (w. + 1 ) r'" " ^ <p,,, /■' - &c. (A) ; 

 and the general (wth) term of V is 



^/!.\P„_i (r"* + ' ?^y - {n + I) r'"</),r' + w (,z + 1) r'""! (/),,r' - &c.) rf^' 



_^:LP /'^ p cduJ 



