802 
MR. A. R. FORSYTH OH THE THETA-EUHCTIOHS, 
ratios can be expressed in terms of two new variables. Be-arranging now the first 
parts of (34), (36), (37), and substituting for the c’s in terms of the /c’s, we have 
_ K 1 djg 2 j d 9 " ^ KgKg d 1Q 8 
g /C p/Cg/Cg r9*i2^ ^ $p2^ 
V.,d| 
/Cg/Cp /Cp/C 2^3 ^12^ /Cp/C $ 12 ^ 
is Q 2 ts' is 0- 2 ■R'Ta Q. 2 
_ k 3 "*13 j K \ K 2^3 "*9 ^ I XV 1 JV 2 " *0 
^p/^2 $ 12 ^ /Cp/Cg/C g ^ 12 ^ K^KyK g »S'p2^ 
agreeing with Bosenhaijst (95). Assuming a^, a? 3 as the new variables, put 
a 2 
AH— A ,y ™ 
Q O- 
^13 
^ 5 =B(l-* 1 )(l-a: 3 ) 
^13 
then if 
|^=0(i-«A 1 )(i-«iX) 
r^po 
§^=D(1 -k 3 3 *-i)(1 -k 2 3 * 2 ) 
^=E(1- Ks X)(1-«s%)- 
the equations (51) are satisfied if 
A=- 
K \HH 
B: 
D=- 
K.,/C 
3^1 
K 1 K % K S 
c= 
/Co /Co 
K 1 K 2^ 3 
<K 3 K 3 
E= 
/Ct /Co 
^KoK! 
d a \3 
16. The other ratios involve aq, a? 3 irrationally; thus to find (~-J , Boseniiain uses 
equations corresponding to (xxvii), (38), (39), and eliminates 3- lly d 14 between them, 
a\2 
giving a quadratic in . Having obtained this, the expressions for the other 
functions follow by substituting for the ratios already found in the equations (31)-(39); 
and the complete system of expressions for the fifteen ratios is as follows, the + 
sign being usually taken throughout (see Cayley, * Crelle/ t. 88, p. 81): 
