AND DOUBLE THETA-FUNCTIONS. 993 
( r o~“Po 2 )P — —PaiVP — qqfrr — ss) + r 0 (pv +p'/• — qs — qs), 
,, lt= r 0 ( ,, ) J9 0 ( ), 
» Q= p Q {pq'—p'q+rs—rs)—r Q ( „ ), 
” ^— M ” )d - Po( >> )• 
On writing in the equations u — 0, then P, Q, E, S, p, q, r, s' become = p, q, r, s, 
p 0 , 0, r 0 , 0 ; and the equations are (as they should be) true identically. The equations 
may be written 
U + X’J V.—vJ U V u U U v! U U U V, U U U vj Vj vJ 
(c 4 c 4 ) 9- . 9 c 2 (9~. 9~ 9 2 .9 2 9 2 .9 2 9 2 .9 2 ) c 2 (9 2 .9 2 9 2 .9 2 9°. 9° 9 2 .9 2 ) 
(8-4) 4 4 = -4(4.4 - 7.7 +8.8 -11.11) +8(4 8 + 8.4-7.11-11.7), 
( „ ) 8 8 = + 8 ( „ ) — 4 ( ,, ), 
( „ ) 7 7 = +4(4.7-7.4+8.11-11.8) -8(4.11-11.4 + 8.7 - 7.8), 
( „ ) 11 11 = -8( „ ) +4( „ ); 
and there is of course such a system for each of the 60 Gopel tetrads. 
Differential relations connecting the theta-functions with the quotient-functions. 
134. Imagine p, q, r, s, &c., changed into x~, y 2 , z 2 , w 3 ; that is, let x, y, z, iv represent 
the theta-functions 4, 7, 8, 11 of u, v ; and similarly x, f, z, w those of u', v', and 
x 0 , 0, z 0 , 0 those of 0, 0. Let u, v be each of them indefinitely small; and take 
,d ,d 
b, =+—+P—, as the symbol of total differentiation in regard to u, v, the infinitesimals 
u and v being arbitrary : then we have in general 
9(u-\-u, v J cv)—9(ii > p) + 6d(«, p) + |6+(p, v), 
and hence 
P= {x-\-bx-\-fb 2 x)(x — bx-\-^b 2 x), = xr + (xb 2 x — (bx)~), 
and similarly for Q, E, S. Moreover, observing that x, z are even functions, y', z' odd 
functions of (u', v), we have 
x, f, z, to —x Q -\-\b 2 x Q , by Q , z 0 f}b%, bw Q , 
where b 2 x 0 , by 0 , &c. are what b 2 x, by, &c., become on writing therein u— 0, v= 0 ; 
by 0 , bw 0 are of course linear functions, b 2 x 0 , b 2 z 0 quadric functions of u and v'. The 
values of x'~, y 2 , z' 2 , w 2 are thus cc 0 2 +x 0 b+ 0 , ( by 0 ) 2 , z 0 2 -\-z 0 b 2 z 0 , (bw Q ) 2 ; and we have 
MDCCCLXXX. 6 M 
