This equation coincides with (2) when the following equa- 
tions are satisfied : 
l3 + V — a^ — a 'I 
a(v-/3) = ±6 V (13) 
(3v = c ) 
These conditions 
give 
2a 
2a 
(14) 
(15) 
And an auxiliary cubic 
(a^)^ + 2a{aJ + {a? - ic){a?) ~b^ = 0 (16) 
Now, if the three roots of (16) he represented by r, s, t 
a- sjr ) 
r + s + f= -2a > (17) 
rst = h^ j 
Substitute these values in (14), (15), 
^”4" 4 
4 4 
Then the form (B) becomes 
*’+ + + =0...(18) 
By solving these two quadratics. 
2x= ^/r + ^~s:^s/l or, - 
2x= ^ r+\/s'^Jt or, J 
The upper sign applies when (h) is positive, and the lower 
when (6) is negative. The auxiliary cubic is the same for 
+ ax^ + hx + c = 0 as for + ax^ - hx + c — 0. 
The omission of the double sign of (5) by Mr. Todhunter 
and other writers on this subject render their solutions un- 
necessarily restricted to the solution of x“^ + ax^ -^-hx + c = 0. 
It readily follows from (19) that when the biquadratic 
equation x^ + ax^zkhx + c = 0 has equal roots, the auxiliary cubic 
(16) has equal roots. (See Todhunter’s Theory of Equa- 
tions, page 115.) When the auxiliary cubic has equal roots 
it would not be safe to infer that the biquadratic has equal 
