704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
483 
and we thence have 
satisfying 
k 2 4- k' 2 — 1. 
41. Observe further that, substituting for a, b, c, f, g, h their values, we have 
2t 2 = c — b.b — d.c — d, = c—d.d—b.b — c, 
S3 - = c — a.c — d.a — d, = d—a.a—c.c — d, 
(£ 2 = a — b.a—d.b — d, = — a — b.b— d .d — a, 
3) 2 = c — b. c — a. a — b, — — b — c.c — a .a — b, 
where in the first set of values all the differences are positive, but in the second set 
of values, we take the triads of abed, in the cyclical order bed, eda, dab, abc. There 
is in this last form an apparent want of symmetry as to the signs (viz. the order 
which might have been expected is H 1—), but taking the order of the letters to 
be (£, 21, S3, il) and c, a, b, d, then the cyclical arrangement is 
(£2 = _ b—d.d—a.a — b, 
2i 2 = — d — c . c —b.b—d, 
S3 2 = — c — a.a — d.d — c, 
< £) 2 = — a — b.b—c.c — a, 
where the four outside signs are all —. Observe that the triads of abed, and abdc, are 
bed, eda, dab, abc, 
where in the first and second columns the terms of the same column correspond to 
each other with a reversal of sign, whereas in the third and fourth columns the lower 
term of either column corresponds to the upper term of the other column, but without 
a reversal of sign. 
The product-sets, u ±u': and vl indefinitely small, differential formidee. 
42. Coming now to the product-sets, these may be written 
k{C.A + A.C} = X,X;, 
,,{D.B+B.D}= Yj;, 
UC.a-a.c} = y,y;, 
„ {.D.B-B.D} = X / Y:, 
±{B.A + A.B}= (P + Q)(P' + Q'), %{B.A-A.B}= (P - Q)(P' - Q'), 
,,{D.G + C.D}= i(P - Q)(P' + Q'X D.C-G.D}= i(P + Q)(P' - Q'), 
61—2