PEOFESSOE CAYLEY ON PEEPOTENTI ALS . 
767 
viz. the discriminant taken negatively is 
I *+/»... ,-af 
fj -|— 7& 2 , — ch 
— af ... — ch —(a 2 . . . +c 2 +e 2 +l 2 )+t 
which is 
— t -\~f " . . . . t-\-Jc 2 ^ t — (l 2 . . . — C 2 — 6 2 — h 
+ 
a 9 P 
t+p" 
cVi 2 e 2 F \ 
't + hZ + l + k*)’ 
-t . ( t+f 2 . . . t+h 2 . t+k 2 ) 
=7+A...7-|-C.7+E.7+L, 
and consequently - A . . . — C, — E, — L are the roots of the equation 
t+p ••• Z-M 9 Z+F Z“ ■ 
148. The roots are all real ; moreover there is one and only one positive root. Hence 
taking — L to be the positive root, we have A . . . C, E, — L all positive ; and therefore 
a fortiori A— L, . . . C — L, E— L all positive, which agrees with a foregoing provisional 
assumption. Or, writing for greater convenience 6 to denote the positive quantity — L, 
that is taking 6 to be the positive root of the equation 
have 
a 9 c 2 e 2 Z 2 „ 
i ~6+P ’ ' ' — $ + It 1 ~ f+¥~ 0 — U ’ 
f ',„_ dS 
J +(c — hzf-\-(e — kwf + l^\ 2S 
= 38*0 - 
\J t • t+P- •■t+h?.t+k' i ( 1 — f+p • • • “ 
Z + A 2 t + & t 
y 
or, what is the same thing, we have 
AH 
dx ... dz 
f • • -h J + w {(a— xf . . . +(c — ^) 2 + (e + /cw ) 2 + Z 2 } 5 
* ( 1_ ^ • • ■ -rh-fi-i) A* • t+r . . . t +v. <+m 
where on the left-hand side w now denotes 
. . a? 2 ^ 2 , 
tion is j : 2 . . . 
149. Suppose 7=0, then if 
aA f a • • • 
and the limiting e^ua- 
