902 
PROFESSOR A. CAYLEY OR THE SINGLE 
which by a preceding formula is 
or the function is altered at most in its sign. And again writing 2 z, 2iv for z, w we 
have 
In reference to the foregoing results we say that the theta-functions have the 
quarter-periods (1, 1), the half-periods (2, 2), and the whole periods (4, 4). 
The conjoint quarter quasi-periods. 
10. Taking x, y integers, we consider the effect of the change, u, v into 
u-\- \(ax-{-hy), v-\-—.(hx-{-by). 
It is convenient to start from the function 
’,8 vJ r~£! lxJ rty)^’ 
the argument of the exponential is here 
4(«, h, bXm+a—x, ?i+/3 —yf 
"l - TjTri\^ni -j- ol— x,uy -)- -(ax+hy). —|—.—(— / 3 — y.n-\- §-J- .(Jtx-\-by ) 
which is 
=\{a, h, bjjii- k a , n-\-/3Y-\-^TTi(m-\-a.u-{-y. +.w+yS.v+S) 
-j- other terms which are as follows : viz., they are 
— \{a, h, + n-\-(3Jx, y) a.ax-\-hy.n-\-fi.hx-\-by) 
+i(«, h, bjx, yf — 17 ri(x.u+y.+.y.v+h) 
—^(x . ax + by .+. y . hx -f by), 
where the terms of the right hand column are in fact 
= bXm -ra, 7i-\-(3X x > y) 
. —\Tri{x.U-\-y.-\-:y.v-\-h) 
~i(«> K 1>Xx, yf, 
