53] ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL. 423 



f *T7 M ( 7 I 1 l \ M f 1 ^ 



therefore 8 V = -- Ida- (-. = \dcr - 



47J-J \r' rj 4w ) >* 



M f , s ,d<r Ma'Sa' 

 = 4n) aSa r'->= ~4,r 



Now the volume of the cone whose base is So- and vertex and 

 radius unity is - Scr ; hence the volume of the corresponding cone enveloping 



the element at P 1 or P' is -'6VSo-; therefore if Sw be the solid angle 



o 



of the cone 



or 



j 5 17 / s / 



and we have o V = a oa 



a'VJ 6V 



Hence it follows that the attraction of shell E at P, in the direction 



f P P' 8V _K A l/ M a/8a/ 



1 ' L e ' P,P' ' ] 6V PjP' ~ ^77? ' P,P ' 



Now if x = a 1 l, y = b 1 m, z = c 1 n be the coordinates of P 1} those of P' will 

 be of I, b'm, c'n and the projections of P,P' on the axes will be I8a', TO 86', 

 nSc'. 



^ ^ 



Putting for Z the value , = K. a! and so for m and n, we get 



a a 



I8a' = . a'Sa', mW = . I'W, nSc' = . c'Sc'; 

 a'- o'- c- 



oc u % 

 but the direction cosines of the normal are as /2 : ^H ' ~n 



Hence PjP' is ultimately in the direction of the normal at P a . 



Hence attraction of shell E at P! which has been shewn to act in 



the direction of this normal = ,fr, , where p, is the perpendicular from 



((/ U C 



on the tangent plane at Pj. 



If we call p the density of shell E, the volume of the shell is 4irtabc, 

 and we have M^irptahc, 



therefore the attraction of the shell = V .t. p r 



