834 
MR. A. R. FORSYTH OH THE THETA-FUNCTIOXS, 
V ^ a + W + W + W = V V + &<? V+ V V+ V V s 
08 a V + V V+W + V V= V V +W +<5? V+W 
^i a v+W+^ ! V+V V =<w +<w + w + w 
w+ 0i' v +w + v v=v v+v v + v vm ( 
V+W+<V V +V§*=^11 S V +VV-W V +W 
^i a V+W + 0 3 a V + V V=<w + tf 4 %» + V a, *+V v . 
W + W+W +W=V V+VV+ViiN-VV h " ' 
V V+W+V V + w =W + W+V V -K a V 
0i a V+V V-K 2 V +V V = W + w + V V + V v 
V V+W + V V+W = w + V V+W +V V 
<W + W + W +W = w+W+V V-K*V 
V V +^3 a V +W +V V=V -V + V V +0/ V+W j 
The equations necessary to reduce the forms first given for such, expressions as 
c 2 c 3 © 1 ©' 0 will be found at the end of the sixteen sets, each of sixteen equations. 
40. First Set, with ®' 0 . 
C 0 C 0 ®0 ® Q = ++ + ++ + + + + + +0+0+0+0+ +S+3+3+3 
^2 C 3 ®1 ® 0— ++ + + + + + + + +0+1+ + +4+S+2+3 
CjC 3 © 3 © 0 =++ + +—+ + + + + + +1+ +0 +3+5+2+4 
C 1 C 2 ®3 ® o~ ++ + ++ + + + + + + +1+ +0++3+4+2+5 
C 8 C 12©4 ® 0=++2+ + ++0+4+ + + +3+ + +1+5+ + 
C 1 C 4 ®5 ® 0 =++ + + + + + + + ++0+5+1+4++4+1+0+5 
C 9 C 15®6 ® , 0 = ++5+ ++ + + + +4“+3+1+ + — + + +0+2 
<q°6 ®? ® 0 = ++ + +++ + + + + +0+3+2+1+ +1+2+0+3 
^4^12©8 ® 0 = ++2+ +++ +3+ + + +4+ +0 + +5+ +1 
CgCi 5 ©g © o = ++5+ + +4+ + + ++ +1+ +3 + +0+2+ 
C 2 C 8 <+o©'o=++ +0+ + +0+ + + ++3+ + +5 + + +5+ +3 
C 3 C 8 +11® 0=++ +1+++1+ + + ++ +4+ +3 + + +3+ +4 
C 4 C 8 ®12® / 0 = ++ +2+“+ +3+ + ++ +0+ +4“ + +1+ +5 
C l C 12®13© / 0 = ++2+3+ + +3+ ^ +a“+ +0+ +1 — + +1+ +0 
C 2 C 12®14® 0 = ++2+4+++4+ + +2 + +1+ + + + + +1 
C 6 C 9 ®15®0 = ++ +5+ + + +0+2+ ++ +1+ +3++ + + +4 . 
Konigsberger, in his paper already referred to, gives a set of sixteen similar equa¬ 
tions expressing the sixteen functions © r © / 0 in terms of products to which the above 
are similar ; but the constants on the left-hand side are for the most part different 
from his, with the result of making the combinations on the right-hand side different. 
