790 
MR. A. R„ FORSYTH ON THE THETA-FUNCTIONS, 
Actual. 
Quasi. 
X . 
! 
Hebmite’s 
Notation. 
y> 
Heemite’s 
Notation. 
X . 
Heemite’s 
Notation. 
y- 
Heemite’s 
Notation. 
4K 
Xl 0 
0 
To 
fi h ZP 
o 3 
4A i 
—r log r 
0 
4A 
*1 
4K 
—r log r 
m ° 
4A. 
- lo S2 
t 4 
This table exhibits the four pairs of conjugate periods in x and y ; one relation 
between them is immediately deduced, viz. : 
o 0 t 3 —n 3 T 0 +n x T„— n 2 Tj =o 
an equation which Hermite makes fundamental in his transformation theory (Comptes 
Rendus, t. xl.) The following equations, giving the relations for quarter and half 
period increase of the variables, are easily obtained 
$ 
{C0 a,+K ’ y 
*{£?)* y+ , 
*{(£ 3* *{£»)**' 
*{fc3* + ^ logM+ -; log? ’} 
3> {C^) a5+ S logp, ” +2 ^ log 
V 
X, y 
=p 2K <i> 
X p 
1 
<& {C;0* + S iogn2 ' + ^ iog? } 
®{C \i)*+^ lo % r >y+Vi lo zz} 
_ x jst , r (\ p \ 
=1 ‘ e “*iu»+ih y 
=q _1 e ~— ty<I> ? 
fl,V 
x,y 
( 8 ) 
(9) 
( 10 ) 
(11) 
