26 Proceedings of the Royal Irish Academy. 



lY. — On some geneeat, Formula for the Soltjtion of Algebraical 

 y/ Equations op the Third Degree, &c. By J, R, Young, for- 



merly Professor of Mathematics in Belfast College. 



[Eead May 25, 1874.] 



A general formula for the solution of any equation of the second de- 

 gree is readily obtained by means of the well-known expedient of 

 completing the square. I am not aware that any algebraist, as yet, has 

 investigated a like general formula for the solution of an equation of 

 the third degree ; that is, by the similar preliminary expedient of 

 completing the cube. It is the main purpose of the present Paper to 

 establish such a general formula ; previously, however, to which, it 

 will be necessary to dispose of the two following special cases, that 

 is to say, to prove that — -■' 



( 1 ) Whenever in an equation of the third degree, 



Asx^ + A^x^ + A^x -\- Aq = 0, , 



either the first triad of the coefficients, ^3, Ao, A^, or the second triad, 

 A2, Ai, Aq, furnishes the relation of equality 



3^3 A, = An% or 3A, A^ = A,K... [1] 



the first member of the equation can easily be converted into a complete V 

 cube, and thence a general expression for the root x be deduced.* . 



1. Let it be the first of the conditions [1] which has place) ^nd let 

 a quantity k be so determined that, by the addition of ^ to J 0, the 

 second condition also may be satisfied ; namely, the condition _^ 



3^3(^0 + ^) = ^.'; 



in order to which, the value of k must evidently be 



■■•>. 



A 2 

 ^=3-X-^- 



so that this quantity being added, there results the equation 



A ^ 

 A.x^ + A.x"- + AyX + — -r = ^ ; 

 3^2 



and, consequently, dividing by A^, and taking account of the stipulated 

 condition, we shall have 



