282 MESSRS. PROUDMAN, DOODSON, AND KENNEDY ON THE DIFFRACTION OF 
where 
and 
a = £ (-1), a>+i s,MC M f (m) , 
n = i n (n + 1) E„ ( K a) \ 
A? _ v / i> 2n+l bs n 2 (kci) p? / \ 
“ ' n{n+ 1) |E„M| 2 n ^ ) ’ 
c = i (~ 1 )»( 2 »+ 1 ) S "h ffl) , c d' c , Qt) PM, 
n = i . 1 1 t n I 
D= £ (-l^+U-rff/^P.M. 
« = i IArt) I 
A'= 2 (-1 S ' , i^l C \I** a) F - W’ 
n = i n(n+l) ! E' n («a) 1 2 
R'— 4 / i Vi 2».+ l SV (tea) p, / \ 
13 ~ id i} n(n+ 1) |E' n M| 2 ,,W ’ 
c'= £ (-i)»( 2 „+i) s '»y, c \ir ) p.w, 
7! == 1 I -tL* n \KCi) | 
D'= 2 (— 1)” (2n+ l) |yA g ) P,( M ). 
( 6 ) 
(7) 
Further, we easily obtain 
8Yj _ /xY, + 
S/X 
Mi 
+ 
C/JL 
1 2 
1 —/X 
/xZj + Y x 
h 2 
L —,(x 
SC' 
sy 2 
/X Y 2 + Zo 
SD' 1 
S/x ’ 
d/u 
1 
h-■* 
1 
to 
9 /x ’ 
sc 
SZ 3 
/xZ 2 + Y 3 
SD 
S M ’ 
CfJL 
1 —fjC 
Sm ’ „ 
• ( 8 ) 
while if we denote SC/S/x, SD/9/x, SC'/S/x, SD'/S^ by c, d, c', d' respectively, we have 
c = 
2 (-1)*. (2» +1) WW P'„ W. 
! f„ i “ 
?i = 1 
d = 
7* C= 1 
I E n («a) I 
2 (- 1)“ (2»+ 1) M, 
! F „ (zca) 
• (9) 
7? = 1 
d! = 
2 (-1)’'[2,1+1) p'„(m). 
71 = 1 
E'.MI 
In summing the series in (6), (7), (9), supplementing values of 0 may be considered 
at the same time if the odd and even terms of these series are taken separately. We 
