112 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
the addition theorem, the elliptic functions of {m (A)y} -y, in terms 
of those of 6,, and in the same manner those wf {n (¢.n}-n in 
terms of those of ¢g, Since we know ¢ in terms of ¢,, we can elimi- 
nate g, and obtain a relation between @, and ¢, which would be a 
new transformation. However, we need not expect to discover in 
this manner any substitution sufficiently simple for practical use. 
The substitutions given above may of course be applied also to 
the complementary integral, and, since interesting relations will 
be thus discovered, I place the different series of forms together for 
comparison. 
i Se Cl HS Oe aa = Te fi”) 
(= Tal tan # (_] + of) 
= 94, (Pn = Fa (er a ei on |= lim 2 
ar 
== On=— Dy == Wa) 
\ (=5,¢- (52y) 
= 9p 9B == (PDB =, (Par = ve ee moe = (ve) 
(= : Ztan + (_] + ga) 
=9, (e-)6 = rare iA a eee = ay lim 
( aoe 
aly 1 ppd. eel aan 
(= (52p) 
