ME. J. W. L. GrLAISHER ON THE THEORY OE ELLIPTIC FUNCTIONS. 513 
whereas the true equation is 
0 'Y' Q/jr /oirj 0,2 S1T1 Az \ 
tanh^+tanh(^-^)+tanh(^+^) + &c.=^ + ^| gIs ^+ iI ^+&c. j. . (49) 
It is well known that if an infinite system of series be summed by rows and by 
columns, the results need not necessarily be the same ; but the above is a striking 
instance of such a disagreement. We should be prepared for some ambiguity from the 
observation that although the value of the left-hand side is liable to a change of a unit 
according as the number of terms retained is even or uneven, yet in the process of 
transformation no condition whatever with regard to the number of terms in the columns 
is, or can be, imposed ; but we should scarcely expect to obtain an absolutely erroneous 
result by an apparently definite process. 
If the same method be applied to the hyperbolic cotangent, we have 
and finally 
2x 
which is also erroneous, the term — being omitted on the right-hand side. 
The method of §15, however, yields correct results, for 
2tt 
— cosech2v sin 4 z 
whence 
+ 
MDCCCLXXV, 
3 T 
