374 M^a MURPHY'S THIRD MEMOIR ON THE 



42. Applications, when the law of force is the inverse square of the 

 distance. 



(1) Let AB be the axis of a very broad cylindrical plate, the 

 round side of which is covered with a fluid, attractive or repulsive, 

 and so distributed as to exert no action on any point in the axis. 



Put AB = 1, APo = a the radius of the base. 



Let ab be one of the very small annuli into which the edge is 

 divided, and put aPo = x. 



Then it is easy to prove that the action of the annulus a 5 on the 



11 



point A is proportional to -^ -jr, or ultimately to the differential 



1 ■ X . 



coefficient of -7— with respect to x, that is, to -t-t? ryj, which quan- 



Aa {o'' + arj» 



tity expanded is proportional to a; — f rj + &c. ; and as b is very great 



compared with x, we need only take the first term of this expansion. 



In this case we may therefore put ao = l, a, = 2, ai = 3, &c., 

 and therefore, M = sin0 + 2sin30 + 3sin50 + 4sin70 + &c. 



The calculus of S„ as indicated in the preceding article will be as 

 follows : 



So = pay Si = |OoCOS&, 



f ^ coefficient of sin2g in ZS^u ^31 

 \ ' "~ coefficient of sin0 in 2SoU ~ 2)' 



■ • Si = 2coseSi - x^So 



= Po Jcos20-^}; ■ • 



