138 
two of these quantities are changed from positive to nega- 
tive there will be no alteration in the biquadratic (24). 
Hence, 
are roots of (24) equally with (^/^, + + v' y)- 
It remains, then, to determine a, /3, r, so as to coincide 
with (2) ; for this purpose it is necessary that the following 
conditions should be satisfied : — 
a+(j + v= 
^ ^ - 4c 
ap + vv + pv=— Y0 — 
By the theory of equations a, /3, v, are the three roots of 
the auxiliary cubic — 
(25) 
q Ct a 
+ —z^ + 
2 
- 4c ¥ ^ 
The coefficient of (x) in (24J is positive when any one of 
the three terms or all of them negative. 
Hence ( — \/ a — -v//3 — \/ v) ;—\/ a + a/ j3 + y/v) ; + 
(\/a + - y/ v) are the roots oi x'^ + ax^ + hx + c = 0. 
It appears, therefore, that the introduction “of the am- 
biguities in sign” mentioned by Mr. Todhunter, page 120 of 
his Equations, is not necessary to establish Euler’s results. 
6. In Euler’s process there appears considerable latitude 
for the exercise of the inventive faculty in the arrangement 
and maojnitude of the roots. 
O 
The following occurred to me as being simple. Form the 
biquadratic as follows — 
I X- 
v/ a 
■ — Ih 
\/ a_ y/P ^ 
\/A 
VP 
. Vy 
\ 
m 
m 
m J\ 
m 
m 
m J 
\ m 
m 
m 
( 
( \/ a 
x + - 
-XP 
_\/y\ ^ 
0 .. 
(26) 
^ m 
m 
m J 
■■■! 
- 
m ) 
_ (^/^ - 
m 
2 
} {(^^ 
y/ a"' 
m J 
yVy 
Or. 
) 
0 
