AND DOUBLE THETA-FUNCTIONS. 
92.3 
V=x— x' Vi = 
1 , X-\-X, XX 
, V 3 = 
1 , X-f.fi, 5C.7j' 
, V 3 = 
1 , CK+afi, Xxd 
1, a-\-d, ad 
1, 6+<i, bd 
1 , c + c?, cd 
1 , 5+c, be 
1, c+c?, cd 
1 , a-f&, a& 
50. In fact, from the equations A 2 u=^i(a—x), Adu =d&(a—x) 
V=^(BWu'-C?uWW), =^(C%AV-AW«'), 
=^(A.Wu'-WuAV), =^(B%W-D^W), - 
we have 
(AVBV-BhiAV), 
(C*uW-~DWu), 
where it wil] he recollected that f^J®=a3$(?T, = —h(5® = c 
Moreover 
5? 
1 
<r 
li 
1 
b—x.b—x, c—x.c—x' 
, (ci—d)v 1 = 
a—x.a—x, cl—x.d—x' 
b — ci.b — d, c — a.c — d 
a—b.a — c, d—b.d—c 
I 
II 
1 
0 
c — x.c—x, a — x.a—x 
, (b—d)\/,= 
b—x.b—x, cl—x.cl—x L 
c — b.c—d, a—b.a—d 
b — c.b — ci, d—c.b—a 
(u —6 )v 3 = — 
a — x.a—x , b — x.b—x 
, (c-d)v 3 = 
c — x.c — x , cl — x.cl—x 
a—c.a—cl, b — c.b — d 
c—a.c — cl, d—a.d—b 
or as these may be written 
Vi= - {bh. b—x. b—x. + cg.c—x.c —x], =~{gh.a— x.a — x'. +bc .d—x.d—x'}, 
8/ i 
Vo= — -f cf .c—x.c—x. 4 -ah .a—x.a—x'}, =-(hf.6 — x.b—x'. 4 - ca .d — x.d—x'}, 
h ^ J CF ^ 
J C5 
V3— --{a g.a—x.a—x. +bf .b — x.b — x},— ^{fg .c — x.c—x. fi-ab.cZ — x.d—x}, 
C II 
that i 
is 
or finally 
V^-ijg^BV+gcWu'}, = X {|4 2 «AV+L.D ! «DV}, 
V;=4{|c%CV+Aa%A%'}, =I|~B 2 hB%'+^D%D%'}, 
V S = ~l { | A 2 t( W+Zm BV }. = j { |c=« CV+pD%D%'} , 
6 c 2 
