753] ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. 249
that is, since a = (a),
h — — (rj') + 2 (&)') a;
and we thence deduce
hw = — ([r]‘') w + 2 (&/) aw.
But from the equation 2aw- v = 0, we have 2 (&>') aw - (©') 77 = 0, and the equation
thus becomes hw = — (t)') w + (w') tj ; which, in virtue of 2hw — iri = 0, becomes
-i7n = — (rj') w + (w') rj, (second result).
From the equation above obtained, h = — (77') + 2 (w') a, we have
hw' = — (?/) 2 (a)') aw';
in virtue of 2hw' — 6 = 0, this becomes - 2 (7?') &>' 4- 4 (to) a&>' = 6; an equation which
may also be written — 2 ((?/) «') + 4 ((ft)') a<o') = (6), or, what is the same thing,
— 2 (co') 7] + 4 (<o') (a) &)' = (6); or since (a) = <2 and (6) — 6, this is
— 2 (co') 7] + 4 (ft)') a&)' = 6 :
and comparing with the original equation
— 2 (77') &) / + 4 (a)') a&)' = 6,
we obtain
(to') 77' — (77') co' = 0, (third result).
We have thus the three systems
(ft)) 77 — (77) ft) = 0 , Ip (p — 1) equations,
(to') 77 -(77') w =|7rt, p 2 „
(w') 77' - (77') w' = 0 , |p (p - 1)
in all p(2p —1) equations. As to these systems, observe that («0)77, (77) w, etc., are
all of them matrices of p 2 terms; each of the three systems denotes therefore in the
first instance p 2 equations, viz. the equations obtained by equating to zero the several
terms of such a matrix: but in the first system each diagonal term so equated to
zero gives the identity 0 = 0; and equating to zero the terms which are symmetrical
in regard to the diagonal we obtain twice over, in the forms P — 0, and —P = 0>
one and the same equation; the number of equations is thus diminished from p 2 to
|p(p-l); and similarly in the third system the number of equations is = fp (p -1):
but for the second system the number of equations is really = p 2 . It is hardly
necessary to remark that in this second system |7ri is as before regarded as a matrix.
The foregoing three systems of equations are in fact the equations (6) p. 4 of
Dr Schottky’s work.
Cambridge, 12 Jidy, 1880.
C. XT.
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