42 AN ILLUSTRATION OF THE THEORY OF THE FUNCTIONS. [716 
Evidently is an even function : (— u) = ^su. Moreover, it is at once seen that 
we have 
^ (u + tt) = 'à- (it + i%) = — e%~ 2lu Sm, 
whence also 
(u + rmr + ni%), 
where m and n are any positive or negative integers, is the product of by an 
exponential factor, or say simply that it is a multiple of ^su. 
Writing u — — we have ^ (— ^ that is, 
*(№) = <>, 
and therefore also 
[mir + (n + |) = 0. 
The above properties are general, but if $ be real, then k, K, K', q being as in 
Jacobi (consequently k being real, positive, and less than 1, and K and K' real and 
t T K’ 
positive), and assuming § = ^ , or, what is the same thing, 
ttK' 
q(=e K ) = e-S, 
the function ^ is given in terms of Jacobi’s © by the equation = © 
what is the same thing, ®u = ‘J 
We hence at once obtain expressions of the elliptic functions sn u, cn u, dn u in 
terms of S-, viz. these are 
snu =7 eItiE " 2m>a (£ +ii% ) *'* (m) • 
cn “ : = \] (I) ’ 2 “ > * (S + ^ +a (S) ’ 
dn u = sjk' * +K) 
Consider now the integral 
i dx f dx 
JaV{(~ ) x ~ a.x — b.x — c.x — d) ’ ] a \J(X) su PP ose > 
where a, b, c, d are taken to be real, and in the order of increasing magnitude, viz. 
it is assumed that b — a, c — a, d — a, c — b, d — b, d — c are all positive; x considered 
as the variable under the integral sign is always real; when it is between a and b 
or between c and d, X is positive, and we assume that \/(X) denotes the positive 
value of the radical; but if x is between b and c, X is negative, and we assume