556 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704 
4 
By means of these values, we have 
4 (a 2 — 8 2 ) 2 X'X = cl- (p + r) {p' + r) + 8 2 (q + s) (q 4- s') — a8 [(p + r) (q + s') + (p' + r') {q + s)], 
4 „ W'W=8 2 „ + a 2 „ -aS[ „ „ ], 
4 (0 s - y 2 ) 2 Y'Y = ¡3 2 (p - r) (p' - r) + y 2 (q - s) (q' - s') - ¡3^ [(p - r) (q - s') + (p' -r')(q- s)], 
4 „ Z'Z = y 2 „ + ft 2 „ - £y [ » „ ]. 
Hence 
4 (a 2 — 8 2 ) (X'X — W' W) = (p + r) (p' 4- r) -(q-\-s) (q' + s'), 
4 (ft 2 — 7 2 ) (F'F — Z' Z) = (p — r)(p — r) — (q — s) (q — s'), 
and substituting in the expressions for P and R, 
4 (a 2 — 8 2 ) (£ 2 -7 2 )P = 
(ft 2 - 7 2 ) C(p> + r) (p' 4- r) — (q + s) (q' + s')] - (a 2 - S 2 ) [(p - r) (p' - r) - (q-s) (q - s')], 
4 „ R = 
>> [ »» >> ] j> [ » >j ]• 
Similarly 
4 (a 2 - S 2 ) 2 W'X = 
a8 [(p + r) (p + r') + (q + s) (q 4- s')] - a 2 (p + r) (q 4- s') - 8 2 (q + s) (p + r), 
4 „ X'W = 
» [ >3 )) ] ^ 33 ® 33 3 
4 (j3 2 — y 2 ) 2 Z'Y = 
fty [(p - r) (p' - r) + (q — s) (q' - s')] - ft 2 (p - r) (q' - s') ~r(q- s) (p - r'), 
4 „ YZ = 
33 [ >> » ] y 33 ft 33 3 
4 (a 2 — 8 2 ) ( W'X — X’ W) — — [(p 4- r) (q' 4- s') — (p + r) (q + s)], 
4 (ft 2 ~ 7 2 ) (Z' Y - Y'Z) = -[(p-r)(q'-s')-(p'-r)(q-s)l 
and substituting in the expressions for Q and S 
4 (a 2 — 8 2 ) (f3 2 — rf)Q = 
- (ft 2 ~ 7 2 ) [(P + r) (q + s') - (p + r) (q + s)] + (a 2 - 8 2 ) [(p - r) (q' - s') - (p - r') (q - s)],