518 ME. J. W. L. GLAISHER ON THE THEORY OF ELLIPTIC FUNCTIONS. 
It is curious that all the formulae of the form 
<px+<p(x— j &)±<P(&+/*)+& c .= series of sines or cosines 
which can be obtained by definite integrals, and which possess any interest, should be 
in reality elliptic-function identities. Of course every result that can be derived from 
these identities by differentiation, by multiplication by a factor and integration, &c., 
can as a rule be obtained directly from an integral, which integral itself would arise 
from a similar treatment of the original integral. This is true of the identities in the 
Philosophical Magazine, ser. 4, vol. xlii. pp. 422 et seq. (December 1871); and, for 
example, such an integral as 
f e — eric, (a— b)-\-e 2a:b eric (a+b)} . . . (57) 
(where erfc,z:=J e-^dx) would give rise to identities which, however, could be deduced 
from (28) and (53) by a similar process to that by which (57) can be derived from 
fV* 2 cos2 bxdx=^e->\ 
•Jo 2 
