AND IMAGINARY ROOTS OF EQUATIONS. 
615 
or of 
or of 
b c d e 
c d e i, 
abed 
c d e i. 
Under the first or third supposition F must contain four equal factors ; under the 
second becomes a^-\-irf‘; under the fourth or fifth it is readily seen that the form 
becomes 
respectively, so that the second, fourth, and fifth suppositions conduct alike to the form 
<y 5 +<p 5 , a particular case of the preceding one. 
It remains only to consider the sixth supposition, viz. that the first minors of 
are all zero. 
In this case if we write 
abed 
c d e i 
\/ ax-\-y/ cy=u , 
ax—y/ cy=v , 
A + B = 4: 
A— B= 
and if neither a nor c is zero, it will readily be seen that F(x, y) becomes Am 5 +Bw 5 by 
virtue of the relations 
If a=0 or c=0, the preceding transformation fails. 
But unless also i=0 or e=0 at the same time as a— 0 or c— 0, a legitimate transforma- 
tion similar to the above may be performed by interchanging «, c, X, y with i, a, y, x. 
If now 
«= 0, it will easily be seen that a, b , c, d or else a, c, e are each zero. 
Similarly, if 
i=0, it will easily be seen that i, e, d, c or else i, d, b are each zero. 
Again, if 
c=0, it will easily be seen that a, b, c, d or else c, e . are each zero ; 
’ and if 
^=0, it will easily be seen that c, d, e, i or else d, b are each zero. 
( 29 ) Thus -we see that the equation ax 5 ■\-hbx A -\-\Qacx 2 + lObcx^+oa^x + he 2 — 0 belongs to the class of soluble 
forms. 
