662] WITH A 16-NODAL QUARTIC SURFACE. 165 
We have therefore 
G — D = (c — d) [xx'abef ]; 
and in like manner we obtain the equations 
B — G = (b — c) [ccx'adef], A — JD = (a — d) [cox'beef], 
G — A = (c — a) [xx'bdef], B — D = {b — d) [xxcaef], 
A — B =(a — b) [xx'cdef ], G — D = (c — d) [xx'abef]. 
It is now easy to form the system of formulae 
E 
F 
A 
B 
G 
D 
ae. af. bed 
—be. bf. eda 
+ ce . cf. dab 
- de . df. abc 
= 0 
ad. bf. cf 
— ad.be .ce 
+ p f 
- <f 
= 0 
bd .cf. af 
— bd. ce .ae 
+ e / 
~ <f 
= 0 
cd . af. bf 
— cd. ae. be 
+ «/ 
~ *f 
= 0 
be . af. elf 
— be. ae. de 
+ f 
~ef 
= 0 
ca . bf. df 
— ca.be .de 
- <f 
+ f 
= 0 
ab .cf. df 
— ab .ce.de 
+ e/ 
~ <f 
-0 
— af. bed 
+ be .cd 
+ ce. db 
+ de .be 
= 0 
— bf. eda 
+ ae. cd 
+ ce .da 
+ de. ac 
= 0 
— cf. dab 
+ ae .bd 
+ be. da 
+ de. ab 
= 0 
— df. abc 
+ ae .be 
+ be. ca 
+ ce. ab 
= 0 
— ae . bed 
+ bf. cd 
+ cf. db 
+ df. be 
= 0 
— be . eda 
+ af. cd 
- 
+ cf. da 
+ df. ac 
= 0 
— ce . dab 
+ af. bd 
+ bf. da 
+ df. ab 
-0 
— df. abc 
+ af. be 
+ bf. ca 
+ cf. ab 
-o, 
where for shortness ab, ac, etc., are written to denote a — b, a — c, etc. ; also abc, etc., 
to denote (6 - c) (c - a) (a - b), etc.: the equations contain all of them only the 
differences of A, B, G, D; thus the first equation is equivalent to 
ae. af. bed {A — D) — be . bf. ede (B — D) + ce. cf. dab (C — D) = 0, 
and so in other cases. 
Cambridge, 14 March, 1877.