662] 
WITH A 16-NODAL QUARTIC SURFACE. 
163 
Vx, OX', X' being the same function of x' that X is of x. We may instead of the 
preceding form take X to be the general quintic function, or what is better take it 
to be the sextic function a — x.b — x.c — x.d — x.e — x.f— x\ and we thus obtain a 
remarkable algebraical theorem: viz. I say that the 16 squares, each divided by a 
proper constant factor, are proportional to six functions of the form 
and ten functions of the form 
a — x. a — x', 
(x - x'J 
[0 a — x.b — x.c — x.d — x' .e — x'. f — x' — 0 a — x .b — x'. c — x.d — x.e — x ,f— x\ 
and consequently that these 16 algebraical functions of x, oc are linearly connected in 
the manner of the 16 squares; viz. there exist 16 sixes such that, in each six, the 
remaining three functions can be linearly expressed in terms of any three of them. 
To further develop the theory, I remark that the six functions may be represented 
by A, B, C, D, E, F respectively: any one of the ten functions would be properly 
represented by ABC. DEF, but isolating one letter F, and writing BE to denote DEF, 
this function ABC. DEF may be represented simply as BE; and the ten functions 
thus are AB, AG, AD, AE, BC, BB, BE, CD, CE, DE. 
Writing for shortness a, b, c, d, e, f to denote a — x, b — x, etc., and similarly 
a , b', c', d!, e', f, to denote a — x, b — x, etc., we thus have 
(13) 
A = aa', 
(9) 
B = bb', 
(7) 
Q 
II 
O 
(8) 
D = dd', 
(6) 
E = ee , 
(= E), 
(1) 
h 
II 
(= n 
(3) 
DE = (FX x j abcd'e'f - \/ab'c'defY, 
(= D), 
(4) 
CE = —~7t 2 {Vabdc'ef — \/a'b'd'cef} 2 , 
\.0C OC j 
(= E), 
(2) 
CI) - (x _ x y abec'd'f - Oa'b'e'cdf}*, 
(14) BE = ^ ^ {Vacdb'ef - Va'c'd'bef } 2 , (= B), 
(16) BD = -—1 - ,,- 2 {Vaceb'df' — Va'c'e'bd/Y, 
(15) BG = Xadeb'cf — Vad/e'bcf}' 2 , 
(10) AE = j—^—7- {a/bcda'ef — \Zb'c'd'aef}' 2 , (= Ä), 
yOC QC )