142 Proceedings of the Royal Society of Edinburgh. [Sess. 
than the first in the small quantities £ and t], it is easily found that (9) 
takes the form 
IK 
+?!,£? 1 + +++) + +? + +) + 1 
lai ^^2ai{2n—\) j 4a 2 ('2/4 — l^ 2 *-*2i(2 
4a 2 (2/4 - l) 2 
i(2na + a) 
+ 2/2(2wa + a) 2 2 o,v. 
+ 00 
1 
2(2/ia + a) 
i(2n- la + a ) 
“ 4(2/4- la + a) 2 
4K 
27T 
r +°o 
2 m 
2/(/4a + a) 
+ (+n ++?++> + « ■ - n + + - v> 
^ 4a 2 (2»i- l) 2 ^ 
4(2/4 - la + a) 2 
+ 00 
+ y P? 
9/Qoo/y _i_ „ \2 
( 10 ) 
2(2 na + a) 5 
since the first two terms vanish and the fourth and sixth combine together 
c> 
into one simple series. 
In the same way the contribution to —u' + iv' due to (a) is 
+ « l 
li< 
27r ^{£' + ^( -a + a + r{)} - {£ + /(4/4 + 1 a + OL-rj)} 
+ etc. . . (11) 
there being four such terms identical with (9), except that the first member 
in the denominator in each case is the expression for the point Q. 
Abandoning once more terms of higher order than the first in f ^ + 
on expanding (11) becomes 
+ccr 1 (^ -•£) + +/-/? ) + l 
.24(2/4+ la - a) 4(2/4 + la -a) 2 2i(2na-a) 
W 1 J+ 1 (£ '-£) + i( v ' + v ) 
4a 2 (2/4+l) 2 
%K 
+ 
• +oo p 
4K 
2(2ua - a) 2 2i(2n+l)a inai 
1 (£' ~ i) +i(y +y) _ (£' - i) + i(y - y) 
2 i(na - a) 4a 2 (2/4 + 1 ) 2 
2(2«<l-a) 2 ] ^ 
4(2/4 + la - a) 2 
The infinite series in (10) and (12) are all summable as follows: 
2?za + a 2^a + a 2 
1 1 
1 — 
na — a a 
+ co 
^na - a 
1 00 1 
+ 2 “2 
7r an r 
2„2 _ „2 = „ C °t — (13) 
n L a * — a* a 
by a well-known expansion. 
+ 0° 1 +c° 1 _2 
(2 n - l) 2 “+(2/4+l) 2 = T ' 
1 
71 5 
= ++'sec' 
2( 2 „-la + a) 2 ^{2n + la-af 4« 2 
+ 00 1 +00 -1 2 
J- ^ 
^(2/4a + a) 2 2na-aS l 4a 2COS 
2a 
7T 
cosec 2 
. (14) 
. (15) 
• (16) 
* Omitting the term n = 0. 
m 
