A PLANE ELECTROMAGNETIC WAVE BY A PERFECTLY CONDUCTING SPHERE. 283 
shall use the suffixes 1, 2, to denote respectively the sums of odd terms and sums of 
even terms. Then for supplementary values of 9, the two values of each of 
K A'„ B'l C 2 , D 2 , C' 2 , D'„ Ci, di, c'i, d\. 
are equal, while the two values of each of 
A 2 , B 2 , A' a> B' 3> C l5 D l5 C'„ D\, c 2 , d 2 , c\, d',, 
2 ) 
are equal and opposite. For this reason in computing Y 1; for example, we evaluate 
Aj + jwA^—C r 2 , A 2 -p /x A! i — C , 1 
separately for O<0< |-7r. 
For 9 = 0 we have 
2P' n f) = n (n+1) P n (/x), 
so that 
C x = 2A l5 I \ = 2B 1} C\ = 2A\, T>\ = 2B\, 
C 2 = 2A 2 , D 2 = 2B 2 , C' a = 2A' 2 , D' a = 2B' a . 
Numerical Summation of Series. 
(By A. T. Doodson and C4. Kennedy.) 
3. The first step was to construct tables of 
log (2 n + 1) F„ fa) and log F n fa), 
where F„ fa) takes each of the forms 
| S„ fa) C„ fa) 1 
S„ 2 fa) 
I S'. M cr. Ml 
S'.* M 
|E*MI 2 5 
|E. Ml” 
1E', M | 2 ’ 
1 E', M | 2 
This was done by means of the Brit. Assoc, tables* mentioned in § 1, using seven- 
figure logarithms and afterwards reducing to five figures. To each of these was 
added log | P n (/x) | or log | P' n (f j for the same value of n, using five-figure logarithms. 
* In these tables the values of JE n (9)| 2 for n = 13, 14 are misprinted; the decimal point requires 
moving one place to the left in each case. Care should be taken in using the logarithms, as some of the 
negative characteristics are printed without the sign placed over them. 
