PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



607 



Let C be the centre of a sphere, A the attracted point. AP=r, AC=a, 

 CAP=<^, CP=p, and the radius of the sphere =R, f(r) the attraction of an unit 

 at the distance r. 



A c 



Tlien the area of an annulus is inri^mKpdr dtp, and its attraction on the 

 point 2 IT r' d r d (j) ain <p cos (f> f (r) . 



Hence the whole attraction is the following double integral, 



where 



and 



IT I dr.r'fir) I aia2 (pdcp 



'J a — B ^ a 



cos (^, : 



lar 



®"° 'P' = A~r:^ — 



Consequently, the whole attraction is 



-f: 



Or—R 



dr. r^/{r) sin'^ <p, 



= -rY / dr.{4a?r^-r^ + a^-'R^ )f{r). 



This result may be easily reduced to the same form as the first side of the 

 equation which constitutes our theorem, as follows. 



Let r=a—'R + 6 



therefore attraction = -^f^^ ^ / (a - R + 0) { 4 o^ (a - B + 0)^ 



-{a-n+d' + a^-E^y} de 

 =^J'^^de.f(a-B, + 6) { 4aa(a + R-2E-e)2 



-(a + R-2'R^I +a2-R=')2 } 



+ 4a' (2B,-ey-4:a^ {a + ny-4'^+R\2B,~e'' -2R^' 

 + 8«.(« + E)2(2R-e)-4a(a + E)(2R-0)^ + 4^+R.2R-^'} 

 VOL. XIV. PART II. 5 S 



