AN INTERRUPTED MEDIUM. 523 



. cosar sin a r coskr sm 



(cos a r sin a r cos k r sin k r\ a > 

 / r=o J 

 = 



Thus, that portion of the value of c 3 , which we obtain by substituting integrals 

 in place of sums, is zero. This conclusion I arrived at previously, and it appears 

 to be, in every way, conformable with the nature of the function. 



Let us now recur to equation (d), and endeavour to obtain the approximate 

 form of c 2 by summation. We will suppose (as an approximation merely) that the 

 particles may be regarded as aggregated in spherical surfaces about the molecule 

 under consideration. Let e be the distance between two consecutive particles, 

 r=n e, where n is a number ; then 4 T n 2 is the number of particles in a spherical 

 surface, of which the radius is r. Putting, therefore, instead of 2 m, S 4 TT m n 8 , 

 where S refers to the number n : we get 



1 f sin ane 3 sin an 3 cos an 



c 2 =4:'!rmS ;-]- , 8 ,+ a y 



we 3 ane a* n 6 e 3 or tir * 



-( 



sin knc 3 sin Awe 3 cos k n 

 ~kn~e "--*-* 



4 TT m f 1 {sin (3 n 3 sin n 3 cos /3 n 

 ~~ ** 3 



1 /sin an 3 sin an 3 cos a : 

 ' a /a4 +' 



a\n 2 tfn* an 



where /3 is written for a e, and a for k e. 



_ 4'7rM_f / Q6? ^sin /3 w cos /3 w 1 ^ n d {sin a n cos a M 

 ~ e 3 



4 I 7T> y ( a d 1. _^_ sin /3n _d 1 ^ sin a w "1 

 ~" P d0pd~fi fin* a Ta ddt an* J 



rf /I rf /I sinaMxX . ? /I <? -vl s 



The value of this expression depends on the summation of the series S t 



from n=l, to w=oo . There is no known process, so far as I am aware, by which 

 this summation can be effected. We can, however, obtain that part of the series 

 which is required for our present purpose by the following process. 



sin a + sin 2 a + sin 3 a + Sec. = - cot -- 



2 *2t 



cos a cos 2 a cos 3 a , a 2 ^ 



/. -- j --- g --- 3 - &c. = log a - 2^ + &c. + C. 



VOL. XV. PART IV. 7 B 



