AND DOUBLE THETA-FUNCTIONS. 
991 
Then 
X Y Z W 
X 
Y Z W 
X' Y' 
Z' W' 
P = 
X' —Y' + Z' — W', 
p = a 
—/3 +y —S, 
<0 
if 
R 
To 
+y —8, 
Q = 
><l 
1 
k- 1 
+ 
N 
1 
k 
— y + £ — a, 
q'= s y 
-f /3 — a, 
it = 
X' + Y' -71 -w; 
r = a 
+£ — y —8, 
r' = a -f /3 
—y — 8, 
s = 
1 
if 
1 
N 
+ 
s = S 
+ 
i 
To 
1 
8 
4= S +y 
— /3 —a. 
Hence 
a (p -f r) — S (2 ■+ s) = 2 (a- - 
■S 3 )X, 
a ( p '-j-/)_ 
■8(</+s / ) =2 (a- — 
S 2 )x; 
8 „ —a „ =2 
W, 
s - 
^ 5 3 - " 3 3 
W, 
—P(p - r ) 4 -y(q~ s) = 2 (/3 2 —y 2 )Y, 
—y +/3 =2 ,, Z, 
—fi(p / —>' / )-\- , y{q , —s')=2 (ft 2 — y 2 ) Y 
—y » +/3 jj =2 ,, Z. 
and by means of these values 
4(a- — S~)~X'X — «'( j p+^')(_p , + /') + S'(</+s)(y +s )— a 8[(p+ r )(2 /- M )-f (p +f)(^+*')], 
4 „ W'W = S 2 „ -fa 2 „ -aS[ „ „ ]• 
4 (ft 2 - y 2 ) 2 YT = fi 2 (p —r)( p - r) +y% - s ) (q - s') ~ /3y[ (p - r) (q - s') + (p' - r')(q - s) ], 
4 „ ZZ =y 2 ,, +/f „ ~/3y[ j, >> ]• 
Hence 
4(a’-V)(X'X-W'W) = (p + r)(p'+r)-( q + s )(q'+s'), 
4(/3*-/)(Y'Y-Z' Z ) = ( p -r)(p'-~,■■)-(, 
and substituting in the expressions for P and It, 
4(*»-S»)(^-y»)P = 
(0--y-)[(p+r)(p : '+r') — (q+s)(q'+/)']—(a 
4 ,, Tt= 
2 §3\r 
o - >j (p — O—(?—«)(?'—*')]• 
[ 
i » 
Similarly 
