AN INTERRUPTED MEDIUM. 519 



&c .... , .. .... 23 



2.4...(2 + 2) ( 2.4...(2 + 4) " 1.2.3 



- &c.j 

 Hence the part of the required co-efficient which lies within brackets is evi- 



2n + l f m _ a. 



dently the same as that of a in J n_2 a + a 2 ) 5 rf/>- 



, - 2 t l-a 2 



* p 2 a* ~2a* 



.-. The above co-efficient is the same thing as the co-efficient of a 2 " hi the 

 expansion of 



m fl-a l-a 2 VT+a? l-a 2 



- 



r -2n+3 2a * - 2a 2 (l-a) " 2a 2 2 a 2 



, ,, 



S 



But this co-efficient is evidently zero. Hence, every term in the expression of 

 ' "- -- is 



We have consequently proved, that a particle of a system exerting forces 

 which vary inversely as the square of the distance, will not tend to move at all 

 by the action of the other particles of the system on it when out of its position of 

 equilibrium. 



Let us next proceed to find the values of c 2 , M, &c. For the first of them, we 

 will take the sum throughout an unform medium, whereby its value will become 



c* = lm((pr + ^ d y 2 ) 2 sm 2 k S 2 X (1) 



where 5 #, 8 y, d z are the co-ordinates of any particle m, measured from that 

 under consideration, r the distance between the two, and r r the law of force. 



Now, 2 sin 2 ^ = 1 -cos k 8 x = 1 - (1 - ( -^|^+ &c.) 



.-. c* = ?m(4>r+ ^dy^-Zmtyr+^di, 2 ) (1 - ^^ + &c.) 



This expression can be reduced by means of the Theorems just obtained. 

 Equation A (1) gives im 8z* V"- 2 = \ - r 2 Sx- n ^ 



T ~ H JL T 



which, by applying equation (B), is reduced to 



- m - > - , 



r (2n i) (2 B + i) r 



VOL XV. PART IV. 7 A 



