104 ^ ■^^^'^ SOLU'TION 



wpon the two following more fimple equations of the quadratic 

 form : 



yr + 2Aab -\- Bb' =1^(3 +2A + B), 



Aa- + 2B^ib + ^Cb- = + (A -f 2B + 3C). 



For thefe two equations are no other than the two parts of 

 the firft of the preceding equations : and if we multiply the 

 firft of them by a, and the fecond by /», we fliall have the two 

 parts of the fecond of the preceding equations. 



To determine a and b, I now write M = 3 + 2A + B ; 

 N = A + 2B + 3C ; and ay ~ b : thus we have, 

 a-x{i + lAy + Bf) = + M, 

 a'- X (A + 2B/ + 30^0 rz +' N. 



Multiply the firft equation by N ; the fecond by M ; fub- 

 trad; the one from the other ; and divide by a"- ; and there will 

 refult this equation for )' ; 



(3N — AM) + 2 (AN — BM)y + (BN — 3CM) y' = 0. 



And in this equation there is no ambiguity of figns. 



If we fuppofe J =z i, the equation laft found is equivalent to 

 the identical equation MN — MN — o. One value of j is there- 

 fore unity. But this is precifely the cafe of a =: b, which we 

 have noticed above to be inapplicable to the prefent purpofe. 

 Nor is it to be wondered at that this value of y is of no ufe in 

 the prefent inquiry : for it is manifeft that it does not at all de- 

 pend on the- given quantities A, B, C, M and N, but merely on 

 the peculiar manner in which the equation is conftltuted. We 

 learn from hence, however, that there is always another value 

 of .V, whatever numbers A, B, C, M and N may denote; be- 

 caufe every quadratic mufl; have two roots or none at all. 

 ■ Quadratic equations, with one root — i, may be fuppofed 

 to be thus generated : Cy — i)x{my — n) = my'' — {rn -\- n)y + « = o, 

 i: ■ the 



