910 
PROFESSOR A. CAYLEY ON THE SINGLE 
(where the arguments, written above, are used to denote the two arguments, viz. : 
u-\-u' to denote (u-\-u, v-\-v) and u—u to denote (u — u, v—v'); and where the 
letters A, B denote each or either of them a single or double letter) in terms of tlie 
functions of (u, v ) and of (u', v ) : and in any such expression taking u' , v' each of them 
indefinitely small, but with their ratio arbitrary, we obtain the value of 
u u u u 
A.dB— B.oA, 
(viz., u here stands for the two arguments (u, v), and b denotes total differentiation 
d d 
bA=du—A(u, v)-\-dv—A{u, v)) as a quadric function of the functions of (u, v) : or 
B B 
dividing by A 3 , the form is b—=a function of the quotient-functions —, &c., that is, we 
have the differentials of the quotient-functions in terms of the quotient-functions 
themselves. Substituting for the quotient-functions their values in terms of x, y, we 
should obtain the differential relations between clx, dy, du, dv, viz., putting for 
shortness ~X.=a—x.b—x.c — x.d—x.e—x.f —x, and Y = a—y.b—y.c—y.d — y.e—y.f—y, 
these are of the form 
dx dy xdx ydy 
each of them equal to a linear function of du and dv : so that the quotient-functions are 
f dx f xdx 
in fact the 15 hyper-elliptic functions belonging to the integrals J anc ^ there 
is thus an addition-theorem for them, in accordance with the theory of these integrals. 
26. The first 16 equations of the product-theorem, putting therein first u — 0, r=0, 
and then u' = 0, v'—Q, and using the results to eliminate the functions on the right 
hand side, give expressions for 
u+vJ u—u' 
A . B, &c. 
(that is, A{u-\-u, v-\-v')X(u — u, v—v) &c.) in terms of the functions of (u, v) and 
('u , v') : and we have thus an addition-with-subtraction theorem for the double theta- 
functions. And we have thence also consequences analogous to those which present 
themselves in the theory of the single functions. 
Remark as to notation. 
27. I remark as regards the single theta-functions that the characteristics 
might for shortness be represented by a series of current numbers 
0 , 
1 , 
3 
