MR. J. W. L. GrLAISHER ON THE THEORY OF ELLIPTIC FUNCTIONS. 515 
nearly equal to unity ; but as probably the theta functions (or their transformations as 
in (28)) would always afford the best means of actually calculating the elliptic functions, 
I have not investigated whether (10) to (17) would present any advantages over the for- 
mulae which result directly from the change of modulus from k to k ', as ex.gr. the formula 
at the beginning of § 4, viz. 
cos am {jTfFJ- TT~ r ( e ‘+ e ") + 
§ 17. There are two well-marked classes of identities that are derived from the theory 
of elliptic functions, viz. pure algebraical identities, in which only one single letter is 
involved, as ex. gr. 
(1_2^+2^-&c.) 4 +(2^ + 2^+&c.) 4 =(1+2^+2^+&c.) 4 , 
and what may for the sake of distinction be called transcendental identities, viz. in which 
7T 2 
a function of (ju is equated to a function of — . To this latter class belong the chief 
identities discussed in this memoir ; and if special values be assigned to x such that the 
left-hand member of the equation is of the same function of p that the right-hand 
member is of — , or, in other words, if the identity is of the form where gjv=ir 2 , 
such a result is usually very interesting. The best known identity of this class is 
\Zlog^(i+2+2 4 +2 9 +&c.)=^/logi(|+r-l-f 4 +r 9 -f«&c.) ; . . (51) 
but there is another elegant formula of the same kind to which Abel has drawn 
attention (CEuvres, t. i. p. 307), viz. 
.■^(l+!)(l+2 3 )(l+S s )---=^.(l+»-)(l +«")(! +»")•••. • • • (52) 
the relations between q and r being of course 
log . log T — TT 2 . 
It seems probable that all the transcendental formulae of this latter class can be 
deduced from the trigonometrical identities in § 11 and at the beginning of § 12 by 
elementary methods, without the introduction of elliptic-function formulae ; and it is of 
some interest to verify (52) in this way. 
Starting from the formula (23), which may be written 
sin 3# 
— x cosec 
s+v__ c -(*+i0 
-j- &c. 
we have, on differentiation with regard to x , 
x cos x cos x 3 cos 2>x „ tt 2 ( e z + e~ z 
4 sin 2 # - * - 1 l+e 3,x ’ 4 ~ C 2ju, 2 \{e z — e _z ) 2 
3 t 2 
e8 - y + e - (z _ y) e s +y + e -(* + y) 
