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