20 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
RESOLUTION OF ALGEBRAICAL EQUATIONS. 
Proof of the impossibility of representing in finite algebraical functions, in the most 
general case, the roots of algebraical equations of degrees higher than the fourth; 
with methods for finding the roots of equations of the 5th, 6th, Tih, &c., degrees, im 
those cases where the coefficients in the given equations involve a general or variable 
quantity, but where, in consequence of relations subsisting between the coefficients, 
the roots of the equations happen to admit of being exactly represented im finite 
algebraical functions. 
BY THE REY. GEORGE PAXTON YOUNG, M.A., 
PROFESSOR OF LOGIC AND METAPHYSICS IN KNOX’S COLLEGE, TORONTO. 
Read before the Canadian Institute, 19th February, 1859. 
DEFINITIONS. 
Def. 1. In the functions which are to be considered, a variable is 
involved; and, when quantities are spoken of as rational or irrational, 
the meaning always is, rational or irrational with respect to the vari- 
able. ‘Thus, ¢ being constant, and p variable, the former of the ex- 
pressions, ¢ + ./p, ./¢ + p, is surd or irrational; and the latter, 
rational. 
Def. 2. Surds may be distinguished as of different orders. The 
n* root of a rational expression, ” being a prime number, distinct 
from unity, is a surd of the first order. But the x™ root of a rational 
expression, when n=2,Nns...n,, each of the numbers, 7, m., &c., 
being a prime number distinct from unity, is a surd of the s order. 
Again, the x root of an expression involving surds of the s order, 
but of no higher order, when m=m,n,ns...m,, each of the numbers, 
MyNq, &c., being a prime number distinct from unity, is a surd of the 
(s+¢)" order, and soon. Thus, the first of the expressions, 
as jbo 
2 Ts ie { 6. 2 a \3 = 
(+p) » ete) tpfi,e+ [ (c+p) +P +o] , 
is asurd of the third order; the second, of the fourth order; and 
the third, of the seventh order. 
Def. 3. Every surd of a certain order is formed by the extraction 
of some root, (as the x), of an expression involving only surds of 
the order immediately inferior, 2 being a prime number, When we 
oi Be 
6 
