AN INTERRUPTED MEDIUM. 517 



. j 



n (-])... (-r+l) r(r-l) .. . (r-j + 1) 



,,\ 2 ' 1-2 ____ / '1.2 ____ 8 



-A) = 2tl(2l ,_ lM2<| _ 2r + 1) 2r<2r-l).(2r-2 

 1 . 2 ....... 2 r 1.2...2 



fv . 



THEOREM 2. To find the relation between imf,r.(xf'f n and Sm/r/- 2 " 



This amounts to the determination of the* value of P in the above expression. 



Let 6 be the angle between r and 8 

 then 6 = r 8 cos 6 



j 2n 2 oS 2 n 



and e = r cos 



2n 2n a 

 P = 2mr COS C7/.* 1 



Now, the area of a spherical surface in the mass is //r 2 sin 6 dd d<j> 

 Hence for such a surface 



P = /7V 2 " +2 cos 2 *0 sin 6d6 dtj>f,r 



= -/- 



= /' 



2n+l 



or 2w-'- 



This proposition might have been proved with little difficulty without having re- 

 course to integrals, but the result is so obvious, that I do not think it necessary 

 to add such a proof. 



THEOREM 3. A system of particles act on one another by forces which vary 

 inversely as the square of the distance between them. One of the particles 

 is removed from its position of equilibrium, to find the force put in play 

 on it. 



Let the co-ordinates of this particle be measured in such directions that the 

 axis of x may be the line of motion. 



