906 
PROFESSOR A. CAYLEY OR THE SINGLE 
0.0 
1.0 
0.1 
1.1 
where X, Y denote ®^j(2 u), ®^j( 2a) respectively, and X', Y' the same functions 
of 2 u respectively. In the other equations we have on the left hand the product of 
different theta-functions of u-\-u', u —u respectively, and on the right hand expressions 
involving other functions, X l5 Y 1} X 1 / , Yf, &c., of 2 u and 2 u respectively. 
16. By writing u — 0, we have on the left hand, squares or products of theta- 
functions of u, and on the right hand expressions containing functions of 2 u : in 
particular the above equations show that the squares of the four theta-functions are 
equal to linear functions of X, Y; that is, there exist between the squared functions 
two linear relations : or again, introducing a variable argument x, the four squared 
functions may be taken to be proportional to linear functions 
&(a—x), 33(6—x), (B(c—x), W(d—x) 
where 23, Qt, 23, a, b, c, d, are constants. This suggests a new notation for the 
four functions, viz.: we write 
^Q( w )> ^(o)H sQiu), 
= A u, B u, Cu, Dm; 
and the result just mentioned then is 
Ahi : B ~u : C % : D hi 
='8L(a—x) : 23(6— x) : Qt(c—x) : 3&(d—x), 
which expresses that the four functions are the coordinates of a point on a quadri- 
quadric curve in ordinary space. 
17. The remaining 12 of the 16 equations then contain on the left hand products 
such as A (a-4- u ')-B (u — u') ; and by suitably combining them we obtain equations 
such as 
dQ ( M + A).dQ(w-A) = 
d 1 3- 1 — 
0 ” 0 ” _ 
d ° 1° 
i ” * i ” — 
n 1 n 1 
T Q. 
XX' + YY 7 , 
YX'+XY 7 , 
XX'-YY', 
— YX'+XY 7 ; 
