WITH AN APPENDIX BY PROFESSOR A. LODGK. 97 



Thus, when 9 is small, and the series tends to be divergent, we get from (17) 



= ._ 



" 27T . 2c sin 



(23); 



and this is the correct value, seeing that 2c sin (^6} represents the distance between 

 the source and the point of observation, and that on account of the sphere the value 

 of \jf is twice as great in the neighbourhood of the source as it would be were the 

 source situated in the open. 



When = 180, i.e., at the point on the sphere immediately opposite to the source, 

 the series converges, since P takes alternately the values + 1 and 1. It will be 

 convenient to re-tabulate continuously these values from 7; = 18 onwards without 

 regard to sign and to exhibit the differences. 



In summing the infinite series, we have to add together the terms as they actually 

 occur up to a certain point and then estimate the value of the remainder. The 

 simple addition is carried as far as n = 21 inclusive, and the result is for the even 

 values of n 



18-3939 + 9-3506 i, 



and for the odd values 



- 19-4734 + 9-1721 i, 



or, with signs reversed to correspond with Po B+ i (180) = 1, 



+ 19-4734 - 9-1721 i. 



The complete sum up to n = 21 inclusive is thus 



+ 1-0795 + -1785t . 



(24). 



The remainder is to be found by the methods of Finite Differences. The formula 

 applicable to series of this kind may be written 

 VOL. com. A. o 



