SECTION III., 1886. INSSMI] TRANS. Roy. Soc. CANADA. 
IX.—Abels Forms of the Roots of the Solvable Equation of the Fifth Degree. 
By GEORGE Paxton YounG, University College, Toronto. 
(Read May 27, 1886.) 
I.—OBJECT OF THE PAPER. 
$ 1. Jerrard having shown, by a peculiar application of the method of Tschirnhäus, 
that every equation can be deprived of its second, third and fourth terms, the problem of 
the solution of equations of the fifth degree is reduced to that of the solution of the 
trinomial quintic 
D +p, c+ ps —0. (1) 
1 
When p, is zero, the roots of this equation are the five values of —p 5. Inthe “ American 
Journal of Mathematics,” (Vol. VII. pp. 170-177) the present writer has demonstrated, that, 
when p, is distinct from zero, the equation does not admit of algebraical solution, unless 
,__ 5 At (8=B) | 
te Be Reale © & 
5 AP @2+B) | i 
ts rer | 
This is the criterion of solvability. To solve the equation, assuming that p, and p, are 
related as in (2), find À from the quartic equation 
M—BM—6V+BA41=0. (3) 
Do) 1 
Era) | a 
nt eee | 

(16+ B)(A+1) A+)’ | 
Then the root of equation (1) is 
ae 2 ul = 
oS? 22a A? ce cea? MG (5) 
A general form of the roots of solvable equations of the fifth degree was found, though 
without deduction, among the papers of Abel after his death, and is given in ‘ Crelle’s 
Journal” (Vol. V. p.386.) Let 
B=p+qvVA+é)+vV fAdt+eythyv 1+ &)} 
6Z=p—q/d+e+v jsAdt+eay—svyd+ e)} | 
B=p+av(i+e—V fRA+E)+R VA +e 
B=p—qgvV(i+e)—V fhdte—hv (1 + e)} 
(6) 
| = 
