THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 187 
Results, corresponding to a certain extent with those 
above given, would have been obtained if, instead of con- 
fining ourselves to five, we had dealt with any number of 
symbols. 
21. We now proceed to show that, when the resolvent 
product 7,(7) vanishes, the general equation of the fifth 
degree admits of finite algebraic solution. In establishing 
the analagous proposition for the lower equations (Art. 
6, 7, 8), we supposed the product 7,_,(”) to vanish, with- 
out distinguishing its factors. But by assuming the 
evanescence of a factor we should of course be conducted 
to substantially the same results, and in dealing with the 
higher equations it will be found that the latter method 
possesses peculiar advantages over the former. In order 
to abridge the calculations, we shall suppose the given 
equation to be deprived of its second term; 7.e. that 
—@=24,+4,+43,+%,4+27;=0, 
a supposition that will not affect any of the values of f(z”). 
Next, if we make 
fli) =a, +i 4+? 2,48 v.47 25=0, 
f@=HtPxttati t+? 2=5A, 
f@®P =2,4 Pati attest? 27,=5f, 
and f=a2, +t ar,+8 a+? ati #,=5Ps, 
we shall have 
®=P,+P.+Ps, 2=t Bi +t B,+? Bs, 
M=t B+? P,+tPs, =P B, +0 Bo+t Bs, 
=i B,+t B,+? Bs, and b= 32, #,=— 58; Py. 
Whence we find 
a°=158, 83+ 1563 8s, 
Yxt=2063 8, + 808? B3 + 208, 63, 
and = 3a°=5Pi+ 1506; A, Aj + 100P, 62 83+ 563+ 523. 
But by the method of the limiting equation, or otherwise, 
we obtain 
Se=- 8c, Fxt=2h-Ad, and Fa°=5(be-e); 
and by comparison with the above, 
VOL, XV. cc 
