181 
664] ON THE 16-NODAL QUARTIC SURFACE, 
are proportional to constant multiples of the square-roots of these expressions; viz. 
the correspondence 
is 
$2 = ^13, 
R\ —^3> R —^04» 
Q=%, 
Q. — ^03) 
i\/a V[a], 
i\/b\/[b], 
i\/c y/[c], i </d V [d], 
i\/e V[e\, 
¿v / /V'[/l; 
Ql = ^2, 
Px = * 34, 
P = * 01 , S = -* u , 
P 2 = % 3 , 
P,=%, 
\/ab \/[ah], 
\/ac \/[ac], 
\/ad\/[ad], \/ae \/[ae], 
Vbc V[bc], 
Vbd s/[bd\; 
S 3 = ^23j 
Q3 ==< ^ r 0> R 3 = &4, 
Rs= ^03 > 
\/be y/[be], 
\/cd \/[cd], \/ceV[ce], 
\/ de \/[de]; 
where, under the signs ¿/, a signifies bcdef, that is, be . bd .be .bf. cd . ce. cf. de. df. ef, 
and ab signifies abf.cde, that is, ab.af.bf.cd.ce.de, in which expressions be, bd, ..., 
ab, af ... signify the differences b — c, b — d, ..., a — b, a —f, ... But in what follows, 
we are not concerned with the values of these constant multipliers. 
Prof. Borchardt’s coordinates x, y, z, w are 
X — ^0 = P j y = ^23 = S» Z — 14 — S', W — 'Aj — P3 j 
viz. P, S, P 3 , S s are a set connected by Gopel’s relation of the fourth order—and 
this relation can be found (according to Gopel’s method) by showing that Q 2 and R 2 
are each of them a linear function of the four squares P\ P ;i 2 , S' 2 , S 3 2 , and further 
that QR is a linear function of PS and P 3 S 3 ; for then, squaring the expression of 
QR, and for Q 1 and R 2 substituting their values, we have the required relation of 
the fourth order between P, S, P 3 , S 3 . 
Now we have P, S, P 3 , S 3 , Q, R = constant multiples of Vjoc], f\ab\ f\cd\ 
f[bd), V[6], V[c] respectively: and it of course follows that we must have the like 
relations between these six quantities; viz. we must have [6], [c] each of them a 
linear function of [ac\ [ab], [cd], [6d]; and moreover f [b] f[c] a linear function of 
\/[ac] V[a6] and \/[bd] V[cd]. 
As regards this last relation, starting from the formulae 
f [ac] = fZf' {^ ac fi'd'e + a cf'bde], 
V[bd] = g Kbdfae'e' + \/b'df'ace}, 
f[ab] = jZf {^ a kfdd'e' + Va'bf'cde}, 
f[cd] = gZf' {'dcdfa b'e 1 + Vc'df'abe],