86] AN EQUATION OF THE FIFTH ORDER. 503 
(2bt 4- 3c) (at 2 + 2bt 4- c) 
— (2at + 46) (bt 2 + 2ct + d) 
+ a (ct 2 + 2dt 4- e) = -|(7, 
(formulae the first and third of which are readily deduced from an equation given in 
the Note on Hyperdeterminants above quoted). The connexion between the quantities 
A, B, G and a, /3, y, 8, is given by 
Aa — ZBb + Cc = — 65, 
Ab — 2 Be + Cd = — 6y, 
Ac — 2 Bd + Ce — — 6/3, 
Ad — 2 Be A Cf = — 6a. 
The theory of the stationary points being thus obtained, the next question is that of 
finding the equations of the edge of regression. We have for this to eliminate t from 
the three cubic equations, 
at 3 + 361 2 + 3ct + d = 0, 
bt 3 + 3 ct 2 + 3 dt + e — 0, 
ct 3 + 3dt 2 + 3et +f= 0 : 
treating the quantities t 3 , t 2 , t 1 , t° as if they were independent, we at once obtain 
/3t + a = 0, 8t + y = 0, yt 2 — a = 0, 8t 2 — /3 = 0; 
or as this system may be more conveniently written, 
/3t + cc = 0, yi + /3 = 0, 8t A y = 0. 
But the most simple forms are obtained from the identical equations, 
ft (at 3 + 3bt 2 + 3ct + d), 
— (3et + f) (bt 3 + 3ct 2 + 3dt + e), 
+ (2dt + e) (ct 3 + 3dt 2 4* 3et + f) = t 3 (Bt 4■ A) 
(bt 4- c) (at 3 + 3bt 2 4- 3ct 4- d), 
— (at 4- 36) (bt 3 4- 3ct 2 4 3dt + e), 
4- a (ct 3 + 3dt 2 4- 3et +f)= Ct + B ; 
equations which, combined with those which precede, give the complete system 
/3t + a = 0, yt 4~ /3 = 0, 8t + y = 0, Bt A A = 0, 
CtAB= 0: