WITH AN APPENDIX BY PKOFESSOR A. LODGE. 93 



It will be seen that the imaginary part of 10" +1 x/, (10) tends to zero, as n increases. 

 It is true that if we continue the calculation, having used throughout, say, 5 figures, 

 we find that the terms begin to increase again. This, however, is but an imperfection 

 of calculation, due to the increasing value of y^ (2u +1) in the formula and 

 consequent loss of accuracy, as each term is deduced from the preceding pair. Any 

 doubt that may linger will be removed by reference to (4), according to which the 

 imaginary term in question has the expression 



,1+1 ( d \" sm r 

 \ r drj r 



Now, if we expand r -1 sinr and perform the differentiations, the various terms 

 disappear in order. For example, after the 25th operation we have 



25 sin r _ 5_0^_48 . . . 4 . 2 __ 52 . 50 . . ._ 6 . 4 ^ 54^ . .6 )A &c 

 r 51 ! 53 ! 55 ! 



the first term being in every case positive and the subsequent terms alternately 

 negative and positive. The series is convergent, since the numerical values of the 

 terms continually diminish, the ratio of consecutive terms being (when r 10) 



100 100 100 



- ) ) , <XC. 



2. 53 4. 55 I! . 57 



Accordingly the first term gives a limit to the sum of the series. On introduction of 

 the factor 10'' +1 , this becomes 



10 2fi 



1.3.5 ... 49. 51 ' 



i.e., approximately 10~ 8 X 3'0. A fortiori, when n is greater than 25, the imaginary 

 part of 10" +1 x,, (10) is wholly negligible. 



We can now form the coefficients of P under the sign of summation in (15), i.e., 

 the values of 



i H (2n+ I)!)," 1 (19)- 



For a reason that will presently appear, it is convenient to separate the odd and 

 even values of n. 



