Young — Algebraical Equations of the Third Degree, ^-c. 31 



and, therefore, those of the original equation are 



^Azr + Ao, 2Azr + Az 





^3 





the sum of which roots is 







2r-- 



2^ 3^ + ^.2 



A, 



A, 



~ a: 



as we know it ought to be. In the case of the incomplete cubic, con- 

 sidered above, in which Az- 1, and ^3 = 0, this sum is, of course, zero. 

 (5,) It was sufficiently shown in my former Paper (Article 

 33), that, if any cubic equation P=0, be written in tbe form, 



P = {ax^ + 5a; + c){x\p) - 0, 



in which a, I, c, and p, have any real values ^vhatever, and that the 

 quadratic equation Q^ - 3FF' = 0, be 



Jx^ + Bx+ C=0, 



the following relations necessarily have place ; namely, 



If ^2 < 4A C, then ¥ > 4:ac ; 

 ,, B^>4.AC, ,, V- Kiac; 



and conversely ; from which relations it follows that 



If B' = AA C, then b"" = 4ac; 

 „ b' = iac, „ B^- = 4:AG. 



The determination of the indicating quadratic, 



Q^-ZPP' =Ax''^-£x+ C=Q, 



in any individual instance, involves but very little calculation ; because, 

 since we know that the first two terms of Q- are always the same as 

 the first two terms of 3PP , we need not take the trouble to compute 

 them ; it is sufiicient that we preserve, in the multiplications, only the 

 terms beyond the first two terms of each product. 



In order to illustrate this by an example, let the proposed equation be 



F = x^- 4x^ + 3x^2=0 

 .-. Q=3x'^~Sx+3 



P'=3X-4: 



