182 
ON THE 16-NODAL QUARTIC SURFACE. 
[664 
we have at once 
f[ac] \/[a&] = {( a fd' e> + a'f'de) fbcb'c' + (be' + b'c) Vadea'd'e 
f[bd] v / [cc?] = ^_ 1 ^.^ 2 {(dfa'e -F d'f'ae) fbcb'c' 4- (6c' + 6'c) Vadea'd'e); 
the difference of these two expressions is 
= „|/y 2 — a'd) (/e' -/ e) fbcb'c, 
where substituting for a, d, e, f a', ... their values a — £, d — %, e— £, f—£, a — ... 
we have ad' - ad = (a- d)(£ - £'), /e' —/'e = (/■- e) (£- f'); also V&c&'c' = V[6] V[c]; and 
we have thus the required relation 
V[ac] f[ab] - V[&d] V[cd] — -(a — d)(e— f) \/[&] Vjc]. 
As regards the first mentioned relation, if for greater generality, 6 being arbitrary, 
we write [6] = 66', that is, —(6— %)(0 — £'), then it is easy to see that there exists 
a relation of the form 
V [6] = A [ab] + B [ac] + G (WJ + 1) [cd], 
where A+B + C + D = 0. The right-hand side is thus a linear function of the 
differences [ab] — [ac\ [ab] - [bd], [ab] — [cd]; and each of these, as the irrational 
terms disappear and the rational terms divide by (£ — £') 2 , is a mere linear function 
of 1, £+£', whence there is a relation of the form in question. I found 
without much difficulty the actual formula; viz. this is 
(a-d) (b - c) (e-f) 
1, e + f, ef [6] 
1, b+ c, be 
1, a + d, ad 
1, e, f, ef 
[ac] 
1, e, f ef 
[ab]- 
1, e, f, 
[cd] + 
1, e, f ef 
1, b, c, be 
1, c, b, be 
1, b, c, be 
1, c, b, be 
1, d, a, ad 
1, d, a, ad 
1, a, d, ad 
1, a, d, ad 
i, o, e, e* 
i, e, e, e 2 
1, e, e, e* 
1, e, 6, № 
where observe that on the right-hand side the last three determinants are obtained 
from the first one by interchanging b, c: or a, d: or b, c and a, d simultaneously: a 
single interchange gives the sign but for two interchanges the sign remains +.