The collected mathematical papers of Arthur Cayley.

512

ON POLYZOMAL CURVES.

[414

Article Nos. 99 to 104. Further Properties in relation to the same Sets
(A, B, C, D) and (A1, B,, C0, DA).
99. It is to be shown that in virtue of these equations, and if moreover - + +- +- = 0,
then it is possible to find 1,, mi, n,, pi, such that we have identically
- lM + mn3 + n - pT + 1111 - vm,13 - nl + piS = 0.
This equation will in fact be identically true if only
-ff'l + gg'm + hh'n. - gh' - -g'hn^ = 0,
cc'zI + bb'n -ffp. + cb'm + bc'n, = 0,
gc'm - hb'n + ff'l + gb'mn - hc'ne = 0,
cg'm - bh'n. + ch'm1 + bg'n1 +ff'pl = 0.
From the first and second equations eliminating mnz or nt,, 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
bc'ff'l + c'g'afm + b h a'f'n + g'hff'p = 0,
b'cff'l + cga'f 'm + b'h'afn + ghff'p = O;
which equations may also be written
f' c'g a' f'g'
a ' l + -b-h n+ n + -+f p O,
ah bh af ab
Cf + C, af fg
a~T - z + Of', + n + a'-'P = 0;
a'h' b'h a af/ a/b
and it thus appears that the equations are equivalent to each other, and to the
assumed relation
-+ ++ =0.
a b c d
100. Similarly, from the third and fourth equations eliminating m or n, the other
of these quantities disappears of itself, and we find
cg'ff'1 - cga'f'm, + afc'g'n, - c'gff'p, = 0,
bh'ff'l - afb'h'mn + bhaf'n, - b'hff'p = 0,
equations which may be written
cf -, c af fk
/ - / m + -, nI - J-, p = 0,
a'c' c'g' a' g'
f'h' b'h' af' b'g
ah - - z+ b- p =,
ah bg af ab