PROFESSOR KELLAND ON GENERAL DIFFERENTIATION, 



607 



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

 CAP = 0, CP=?, and the radius of the sphere =R, /(r) the attraction of an unit 

 at the distance r. 



A c 



Then the area of an annulus is 2'n-r^sm<f}dr dcp, and its attraction on the 

 point 2'7rr^drd(psm(pcos(pf(r). 



Hence the whole attraction is the following double integral, 



where 



and 



/a + R ftp, 



d r . i^f (r) I &m2(p d(f> 

 „ — R •-' o 



r^ + a^-W 



o^R 



COS (p, = 



2ar 



sin^ 0^ = ^ L 



Consequently, the whole attraction is 



rjT fa + B, 



fa + B, 

 ^Try dr.r^f(r) ain^cp, 



TT Pa + B, . 2 



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

 equation which constitutes our theorem, as follows. 

 Let 



therefore attraction 



=^J^^ d6.f{a-^+6) { 4«2(a + R-2R-0)2 



=r- 2 X 



-(a + R-2R-^| +a2_R2)2 | 



TT /"2R ^ ^ , 



^■T^J ^^/(«-I^ + ^) {4a2(a + R)2_8fl2(a + R).2R_a 



+ 4 a^ (2 R - ^)2 _ 4 a2 (« + R)2 _ 4 ^:^^ , 2R3;g' - 2R:I^' 



+ 8a(« + R)2(2R-d)-4a(« + R)(2R-a)^ + 4^+R.2R:ra'} 

 VOL. XIV. PART II. 5 S 



