994 
PROFESSOR A. CAYLEY ON THE SINGLE 
a 0 6“^o 
(tyo) 2 
z 0 b\ 
(^o) 2 
0 /£) 9 /Cf 
—y~y 
i 
+ ' 2 
— w 2 iv' 2 — x~x 0 : 
1 9 0 
+^'V 
-fa: 3 
9 
-r 
+2 2 
— IV 2 . 
9 /o 0 /o 
x~y - -y-x ~ 
i 9 7 9 
+ -ur - 
9/9 9 9 
— w 3 z 0 3 
-r 
+x 3 
— ltd 
+z 3 , 
9 /9 9 /9 
arz ^ —y^w ~ 
I 9 /9 
+z~x~ 
9/9 9 9 
—w z y* = 
I 9 9 
-f z 3 
9 
—iv~ 
-fa: 3 
-2/ 2 , 
0/9 9 /9, 
art# * — y « 
1 9 /9 
+ ~T/ - 
— lO't'X * =—W*Xq 
9 9 
-A 
— IV 2 
-j-z 2 
-r 
-fa: 3 
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 
(Oo) 2 
C cr *0 
GO 2 
(z 0 4 -V) -(&»)*} = 
» {y&y -(ty) 2 } = 
„ {zb 2 z — (dz) 2 } = 
„ {wb 2 w—(bw) 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 ) + ( x 0 2 w 2 —z 0 2 y 2 ), 
— ( X o“y~ Z Q vJi ) ~( x o“ X ~ "t z o“ z ~ ) ~( X 0 2 W 2 —Z 0 2 y 2 ) —(—.1’ 0 2 2 2 + z 0 2 <« 2 ), 
( - x 2 z 2 + z 2 x 2 ) + ( x 2 w 2 -z 2 y 2 ) + ( - x 2 x 2 + z 0 2 * 2 ) + ( x 2 y 2 ~^ w ~)> 
-( x 0 2 w 2 -z 0 2 y 2 ) —(—x 0 2 z 2 +z 0 2 x 2 ) -( x 2 y 2 -z 2 w 2 ) - (- Xo 2 x 2 +z 2 z 2 ), 
where it will be recollected that x, y, z, w mean $ 4 , S- 7 , 9- 8 , S n ( u ) ; a: 0 is 3-± (0) that is 
c' 4 , and z Q is S- s (0) that is c 8 . But the formulae contain also 
A= (c"', c*, v'f, 
b% = (c s "', c 8 iy , c s v ,Xu', v'f, 
b !/o=i C 7> <Xu, V), 
H)=( c ll'> C n "Xll, v ). 
The formulae may be written 
c.b-c , 
1 
rA 9 - 
-(^) 2 } 
c 2 
..+ 
9 
c 
£ 2 
OV-Tbi 
4 
4 
4 } = 
(- 
-4 
4 
+ 8 
8 ) 
+ ( 
» 
7 
• \ — 
-( 
4 
7 
-8 
ID 
-( 
8 
8 
8 } = 
(- 
-4 
8 
+ 8 
4 ) 
+ ( 
55 
f ll 
11 
11 } = - 
-( 
4 
11 
-8 
7 ) 
+ ( 
(cq ) 2 
c 3 . i 2 cT r~ 
4 7-8 11) 
(-4 8 +8 4) 
Cg-9-Tg 
~V“ 
G C ll)" 
c 3 .d 2 c 2 . $ 2 
y~ 
c 2 . d 2 c 2 
+ (—4 8 +8 4) +( 4 11 -8 7), 
-( 4 11 -8 7) -(-4 8 +8 4), 
7) +(—4 4 +8 8)| + ( 4 7-8 11), 
-( 4 7 —8 11)|—(—4 4 +8 8), 
where b 3 c 4 , b 2 c s , bc~, bc n are written in place of b~x 0 , b~z 0 , by Q , bz 0 . There is of course a 
like system of equations for each of the Gopel tetrads. 
136. Observe that dividing the first equation by \ 2 (u), or say by 0/, the left hand 
side is a mere constant multiple of b~ log ; and the right hand side depends only on 
the quotient-functions d 7 -+ .9- 4 , d 8 -+ D n +- d 4 ,; each side is a quadric function of (>/, v'), 
and equating the terms in u'~, u'v, v' 2 respectively, we have 
<P_ 
dv 2 
log If, 
d 2 
du dv 
log T 4 , 
d? , n 
rW l0 §' 5 
4 
each of them expressed as a linear function of the squares of the quotient-functions 
d 7 +-d 4 , d s -+ d n ~ .9.,, The formula is thus a second-deviative formula serving for 
the expression of a double theta-function by means of three quotient-functions. 
