MR. J. W. L. GLAISHER ON THE THEORY OF ELLIPTIC FUNCTIONS. 
507 
and in order to obtain the correct result we must replace the indeterminate series, 
1 — 1 + 1 — 1 + 1 — &c., by Cases in which the method gives results absolutely erro- 
neous will be noticed in § 16. 
It will have been seen that the process of § 11 consists in replacing each term 
of the original series by n terms (n infinite), and therefore the original expression itself 
by n 2 terms. Each series of n terms formed by adding the vertical columns is trans- 
formed into another series of n terms, so that we thus replace the first scheme of n 2 terms 
by a second scheme of n 2 terms, which latter system, being such that the columns admit 
of being summed as ordinary geometrical progressions, gives the second side of the 
identity to be proved. 
§ 13. A question that naturally arises is to inquire what are the results which we 
should obtain if, instead of using (39) and the similar formulae for the conversion of 
one series into another, we were to replace at once these series by their finite summa- 
tions, i.e. instead of (39) to take 
x^+a* (x — fx^ + a 2 (# + ju.) 2 -|-a 2 
+ &c. =^jjcosec ^ ( x—ai ) — cosec ^ (x -f- ai) j 
„ nx . %ai 
2 cos — - sin — 
V- V- 
' 2txi . a ttx . o Teai 
sin 2 l-sm 2 — 
ix fx 
«. . „ irx . , ana 
sin 2 [- smh 2 — 
ix fx 
We thus find 
sech x — sech (x— ft) — sech (,r-}- ft)-j-&c. 
sinh — 
2]* 
27 r nx 
=— cos — ■ 
fX fX I . „1tx • v 3 
r 1 sin 2 — ■+- sinh 2 — 
IX 2 (X 
• i 3 tT 2 
smh — 
2fx 
, nX 
>3^ 
2 fx 
+ &C. >, 
. . (40) 
while the left-hand side also 
2% ( , 7T 2 HX , 37T 2 Znx D \ 
— sech cos — bsech-.r-cos — 4-&c. . . . . 
fx\ 2(X fx 2ft fx 1 ) 
(41) 
from (18). Although (40) is the identity which we have absolutely proved, we may 
regard the fresh identity as being that which follows from (40) and (41), viz. (writing 
for the moment x in place of and ft in place of 
cos x cos Sx _ f sinh x sinh „ ) 
cosh %ft' cosh %[x' C C0S sin 2 x-\- sinh 2 j^x sin 2 x + sinh 2 %[x ' C ‘j ' 
(42) 
This result follows immediately from another form of the series for the cosine 
3x2 
