MR. J. W. L. GLAISHER ON THE THEORY OF ELLIPTIC FUNCTIONS. 499 
A similar course of procedure shows that 
coth x sin nx dx— nr coth — =7r< 1 + - K? 
hA 
'- 1 r 
from which (22) may be derived. 
In his ‘ Nouvelles Tables,’ T. 265, Prof. De Haan assigns definite values to the inde- 
1 tanh x sin nx dx and 1 coth x sin nx dx ; 
Jo J# 
and it is noticeable that, if these values be used, they lead to the same results as those 
just investigated. The reason is that the integrals in De Haan are in effect evaluated 
on the assumption that cos oo = 0; and if in (30) we had, in place of the first two 
terms, viz. 
written 
+-W+I 
0+-+0+-, 
n 1 n' 
it is clear that the final result would have been the same. 
It may be remarked that the identities (19) and (22) may be somewhat generalized 
by means of the integrals 
sinh ax . tt 
c - 5 SS sm **<**=6 
. , nx . ax 
smh 2b sm ¥b 
. nx ax’ 
cosh — + cos -r- 
b b 
cosh ax . x 
sinh bx sm nx dx= 2, b 
sinh 
, nx , ax 
cosh -£- + cos ~r 
b b 
while other identities may be derived from 
C ” cosh ax -w x 
. cSihS cos “ & =4 
, nx ax 
cosh —j cos -y 
2b 2b 
, nx ax' 
cosh ~r + cos -y- 
b b 
5 sinh ax 7r 
stah Tx C0snxdx =Yb 
, nx , ax 
cosh -^ + 008 — 
b a 
in which, of course, a is to be supposed less than b. 
§ 9. The well-known reciprocity oif and <p in the formulae 
f( n )=\ / (ir) • J o $( X ) C0S dx ’ /(») = \/ (') 'J o <P( X ) sin nX dx 
leads to a corresponding reciprocity in the formulae (24) to (27). Thus from the first 
of the integrals we deduce that, <p and /'being both even functions, if 
3 u 2 
