502 
ON THE DEVELOPABLE DERIVED FROM 
[86 
To effect the reduction of this expression, consider in the first place the equations 
which determine the stationary points of the edge of regression. Writing instead of 
| : 7] the single letter t, these equations are 
at 2 + 2 bt + c = 0, 
bt 2 + 2 ct + d = 0, 
ct 2 + 2 dt + e = 0, 
dt 2 + 2 et +f = 0, 
write for shortness 
A = 2 (bf— 4ce + 3d 2 ), 
B = af— 3 be + 2cd, 
C — 2 (ae — 4bd + 3c 2 ), 
and let a, 3/3, 37, 8 represent the terms of 
a, 
b, 
c, 
d 
b, 
c, 
d, 
e 
c, 
d, 
e, 
f 
viz. a = bdf — be 2 + 2 cde — c 2 f — d s , 
'3/3 — adf— ae 2 — bcf + bde + c 2 e — cd 
37 = acf — ade — b 2 f + bd 2 + bee — c 2 d, 
8 = ace — ad 2 - b 2 e + 2 bed — c 3 ; 
it is obvious at first sight that the result of the elimination of t from the four 
quadratic equations is the system (equivalent of course to three equations), 
a= 0, /3=0, 7 = 0, 8 = 0. 
The system in question may however be represented under the more simple form 
(which shows at once that the number of stationary points is, as it ought to be, eight), 
A=0, B = 0, (7=0; 
this appears from the identical equations, 
(2c£ + 3d) (bt 2 + 2ci + d) 
— (2 bt + 4c) (ct 2 + 2 dt + e ) 
+ b (dt 2 + 2et +/) = \A ; 
(2ci + 3d) (at 2 + 2bt + c) 
— c (bt 2 + 2c£ + d) 
— (2at + 38) (ct 2 + 2di + e) 
+ a (dt 2 + 2e£ +/) = i?;