558 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
[704 
and hence 
P = (x+dx + ^d 2 x) (x — dx -f £d 2 x), = x- + [xd-x — (?te) 2 }, 
and similarly for Q, R, S. Moreover, observing that x' and z' are even functions, 
y' and w' are odd functions, of u', v, we have 
x', y, z, iv =x, + ^d 2 xo, 0y o , z 0 + %d%, dw 0 , 
where d 2 x 0 , dy 0 , &c., are what dfx, dy, &c., become on writing therein u = 0, v = 0; 
dy 0 , dw 0 are of course linear functions, 3 2 ^ 0 , d 2 z, quadric functions of u and v. The 
values of x'-, y'-, z 2 , iv' 2 are thus x, 2 + x 0 d 2 x 0 , (0y o ) 2 , zf + z 0 d 2 z 0 , (dw 0 ) 2 ; and we have 
œ 0 d 2 æ 0 
%«) 2 
z,d-z 0 
(0W O ) 2 
x 2 x' 2 
- y-y' 2 
+ z 2 z' 2 
— w 2 w' 2 = xfx 2 
+ Z 2 Z 0 2 
+ X 2 
-y 2 
+ z 2 
— vf, 
x 2 y' 2 
-y 2 x' 2 
+ z 2 w' 2 
— w 2 z' 2 = —y 2 x 0 2 
- w‘% 2 
-y- 
+ X 2 
— vf 
fz 2 , 
x 2 z' 2 
— y 2 w' 2 
+ z 2 x’ 2 
— W 2 y 2 = Z 2 xf 
+ x 2 zf 
+ z 2 
— vf 
+ X 2 
-y-> 
x 2 w’ 2 
- y 2 z' 2 
4- z 2 y 2 
— W 2 x 2 = — W 2 X 0 2 
-yW 
— w 2 
+ z 2 
~y~ 
+ of. 
135. On substituting these values, the constant terms (or terms independent of 
u, v) disappear of themselves; and the equations, transposing the second and third 
of them, become 
x 0 d 2 x 0 (dy 0 ) 2 z 0 d% (dw 0 ) 2 
(z 0 4 -x 0 4 ){xd 2 x ~(dx) 2 }= (—x 0 2 x 2 + z 0 2 z 2 ) +( x 0 2 y 2 —z 0 2 w 2 ) +(—x 0 2 z 2 +z 0 2 x 2 ) + ( xfvf-zfy 2 ), 
» [yd-y -(3y) 2 }= -( x ti y-z 0 2 w 2 ) -(-x 0 2 of+z 0 2 z 2 ) -( x 0 2 vf z 2 y 2 ) -(-x 0 2 z 2 +z 0 2 x 2 ), 
„ {zd 2 z ~(dz) 2 }= (—x 0 2 z 2 +z 0 2 x 2 ) +( x 2 vf z 2 y 2 ) +(-x 0 2 x 2 +z 0 2 z 2 ) +( x 0 2 y 2 -z 0 2 w-), 
„ {wd 2 w-(dw) 2 } = -( XqW 2 z 2 y 2 ) -(-x 2 z 2 + z 2 xr) -( x 0 2 y 2 -z 0 2 w 2 ) -(-x 0 2 of +z 0 2 z 2 ), 
where it will be recollected that x, y, z, w mean S- 4 , %, S- 8 , ^- n (u); x 0 is S- 4 (0), 
that is, c 4 , and z 0 is S- 8 (0), that is, c s . But the formulae contain also 
d 2 x 0 = (<?;", c 4 iv , c/$V, vf, dy, = (Cy, c" \u, v), 
d 2 z 0 =(c 8 '", c 8 iv , c 8 v $V, v') 2 , dw 0 = (Cn, c u "$V, v). 
The formulae may be written 
0$% 
(9c 7 ) 2 
c 8 0 2 c 8 
(0Cn) 2 
U U It 
{■ ^.0%-(0^) 2 } c 2 .^ 2 c 2 .^ 2 
c 2 . ^ ' c 2 . + 2 
c 2 . ' c 2 . ^ 2 
C 2 .^- 2 (f. + 2 
(c 8 4 -c 4 4 ){ 4 4 4 }= (-4 4 +8 8) 
+(4 7-8 11) 
+(—4 8 +8 4) 
+( 4 11 -8 7> 
„{ 7 7 7 ] = -( 4 7 -8 11) 
-(-4 4 +8 8) 
-( 4 11 -8 7) 
-(-4 8 +8 4), 
„{ 8 8 8 }= (-4 8 +8 4) 
+( 4 11 -8 7) 
+(—4 4 +8 8) 
+( 4 7-8 11), 
„ {11 11 11 }=-( 4 11 -8 7) 
—(—4 8 +8 4) 
-(4 7-8 11) 
—(—4 4 +8 8), 
where 0 2 c 4 , d 2 c 8 , dc 7f 0c u are written in place of 0 2 ir o , d%, dy 0 , dw 0 . There is of course 
a like system of equations for each of the Gopel tetrads.