.4 1—
ay and a, are arbitrary, and w, and W, may be chosen in any way

such that {r] <1 . In the usual notation of elliptic functions,

this form of r is denoted by ¢° . The above product becomes

e 8

aq Az a ZAZ Bgd ¢3AZ voo
oy Lra0e . GO, \

The examples given in Fart II are sufficient to show that
the formulas derived in Fart I may be used tc expand into series
a large pumber of infinite products which have been treated by
special metheds, previded the produgt satisfies the conditions

of Theorem IV.