704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
475 
are found in terms of the functions of (u, v) and of (u', v'): in any such expression 
taking u, v' each of them indefinitely small, but with their ratio arbitrary, we obtain 
the value of 
n u u u 
A.dB-B.dA, 
^viz. u here stands for the two arguments (u, v), and d denotes total differentiation 
dA = du ~ A (u, v) + dv A (u, v)J, 
as a quadric function of the functions of (u, v): or dividing by A 2 , the form is d -j equal 
A 
to a function of the quotient-functions & c -> that 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 dx, dy, du, dv, viz. putting for shortness 
and 
these are of the form 
X = ci — x. b — x. c — x. d — x. e — x .f— x, 
Y=a—y.b —y.c—y. cl —y.e —y./- y, 
dx dy xdx y dy 
vT _ V;P’ WT~ VF’ 
each of them equal to a linear function of du and dv: so that the quotient-functions 
are in fact the 15 hyperelliptic functions belonging to the integrals 
and 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, 
v = 0, and then u' =0, v' = 0, and using the results to eliminate the functions on the 
right-hand side, give expressions for 
u + u u-u' 
A . B, &c., 
that is, they give A (u + u!, v + v').B(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, 2, 3: 
60—2