AND DOUBLE THETA-FUNCTIONS. 
919 
set of values, we take the triads of abed, in the cyclical order bed, eda, 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 -\ -1-), but taking the order of the letters to be 
r 3 , 13,0 and c,a,b,d, then the cyclical arrangement is 
(T~— —.b — d.d — a.a — b. 
3 2 = —. d — c.c — b .b — d. 
13 : = —.c — a.a — d.d — c. 
0 '= — -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, 
and bde, dca, cab, abd, 
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, n±u': and u f indefinitely small, differential formulae. 
42. Coming now to the product-sets, these may be written 
HC.A + A.C}=XX/, 
„{D.B + RD)=YX/, 
i(B.A + A.B} = (P + QKPRQ'), 
„(D.C + C. D} = i(P - Q )(P + Q'), 
i(D.A + A.D} = (P-iQ^P/H-iQ/), 
„{B.C + C.B} = -i(P-tQ / )(P/+tQ / / ), 
i(C.A - A.C} = YY;, 
,,{D.B - B.D} = X,Y/, 
i(B.A - A.B} = (P —- Q )(P' -r- Q'), 
„{D.C - C.D}= i{P -f Q )(P' — Q'), 
i{L».A - A.D} = (P.+iQ^PZ-iQ/), 
„{B.C - C.B} = -f(P- i Q / )(P;-fQ/). 
^3. We can from each set form two fractions (each of them a function of uf-vf and 
u—u), which are equal to one and the same function of u only: for instance, from the 
Y' . . 
first set we have two fractions, each ~ : putting in such equation u= 0 , we obtain a 
new expression for the function of u involving only the theta-functions A u’, &c., 
