908 
PROFESSOR A. CAYLEY ON THE SINGLE 
X, Y, X', Y' which occur therein, we obtain expressions for the four products such as 
A— u ). One of these equations is 
C 2 O.C(m+w0C(w-m0 = C 5 ^C ss w / -D 2 mDV. 
Taking herein u' indefinitely small, we obtain 
CuC"u-(C'uy c"o m'oy 
C hi 
CO 
CO / G 2 u 
where the left hand side is in fact ; , log C u, or this second derived function of the 
theta-function Ou is given in terms of the quotient-function — : hence integrating 
\jVj 
twice, and taking the exponential of each side we obtain Cu as an exponential the 
TD hi, 
argument of which contains the double integral j — ( du ) 3 , of a squared quotient- 
function. This in fact corresponds to Jacobi’s equation 
®U- 
21. From the same equation C 3 0.C(Y + u) C (u — u') = C 2 uC 2 u' — D~uU~u', differen¬ 
tiating logarithmically in regard to u', and integrating in regard to u, we obtain an 
equation containing on the left hand side a term log ^ ~ + ■ ' )’ an< ^ 011 hand 
an integral in regard to u, and which in fact corresponds to Jacobi’s equation 
© (u—d) 
5 ©(« + «) 
©'a , , -. @(?t—a) . . 
u— +ilog=——7=n(«, a), 
©« 
k 2 sn a cn a cln a sn 2 udu 
i 1 — k 2 sn 3 a sn 2 u 
22. It may further be noticed that if in the equation in question, and in the three 
other equations of the system, we introduce into the integral the variable x in place 
of u, and the corresponding quantity £ in place of u', then the integral is that of an 
expression such as 
dx 
T*,/ a — x.b — x.c—x.d—x 
where T is =x — £, or is — any one of three forms such as 
1, as+£ xg . 
1, a +&, ab 
1, c-\-d, cci 
