512 
ON POLYZOMAL CURVES. 
[414 
Article Nos. 99 to 104. Further Properties in relation to the same Sets 
(A, B, G, D) and (A x , B x , G 1 , D x ). 
99. It is to be shown that in virtue of these equations, and if moreover - + ^ + -+ ^ = 0 
abed 
then it is possible to find l x , m x , n x , p x , such that we have identically 
— ¿21 + m23 + w(S - pT) + ¿j2( x — m£& x — n x ^. x + p& x = 0. 
This equation will in fact be identically true if only 
-ffl + gg'ni + hh'n . - gh'm x - g'hn x = 0, 
cc'm + hh'n —ffp . + cb'm 1 + bo'n x = 0, 
gc'm — hh'n + ff'l x + gh'm x — hc'n x = 0, 
eg'm — hh'n . -f ch'm x + hg'n x +ff'p x — 0. 
From the first and second equations eliminating m x or n x , the other of these quantities 
disappears of itself, and we thus obtain two equations which must be equivalent to 
a single one, viz., we have 
he'ffl 4- c'g'a fm + hh a'f'n + g'hff'p = 0, 
h'cff'l + cga'f'm + h'h'afn + gh'ff'p = 0 ; 
which equations may also be written 
cf i c 'g' a f f'g' « 
-4- l + -7T m + ^P.n + J ~- p = 0, 
ah hh aj ah 1 
sL i+%L m+ “f n+ S> 
ah' + b'h' + off o/l/ 1 ’ 
and it thus appears that the equations are equivalent to each other, and to the 
assumed relation 
l m n » 
-+ t- + - + T = °. 
abed 
100. Similarly, from the third and fourth equations eliminating m or n, the other 
of these quantities disappears of itself, and we find 
cgffli - cga'fm, + afc'g'n, - c'gff'p, = 0, 
hh'ffl! — afh'h'm^ + hhaf'n x — h'hffp x = 0, 
equations which may be written 
c ll_ c 9_ m .of fg 
a'c' c'g' + a'f 
f'h' 
7 b'h' a'f 
7 l —r - m + -f?r n— 7 
ah bg af ah 
ga 
Vf 
P = 0, 
p = 0,