272 Wilder''s Algebraic Solution. 



x^-i-z^-3pz+^^=c, (S). These three 



equations joined to x'' -\-{y-\-z)x-{-p—0, {D), are sufficient 



to determine y. 



For brevity maiie2=0, then 



6 

 (2) P=-i' 



and (2) and (3) x3=|±Vp+|^ =^(&c) 



for then (D) y= becomes 



y=— ' which is the rule of 



Cardan. 



So also the function r-i r~, ; ' -Ttt^, be- 



x^-\-i/x^-{-px+q (F)' 



comes, when (F) is a factor of (E) independently of sc, 



x^ — {y^ —•^p)x'^ -\-{j)^ —'2qy)x^ - q^ 



x^ -\-yx'^ -{-px-\-q 

 writing (2n?/4-22) for y, and {'2n'^y^ -\-Anzy-\-'2z^ —p) for 

 p, and it is changed to 



x^ - 2px^ +(4w ^y '^ + 1 6?z^2y + (24??^2^ — An^p)y^ + 



:r3 4-(2//3/ + 22)a:2 + 

 (IGnz^'— 8??.p2 — 4/ig)y + 42'' —4>:'Z'-\-p^ —4qz)x^—q~ 

 (2n2^M- 4n2; V + 22 ^ — p)x * 



dividing by An'^x^, the function (E)=0, and then comparing 

 with y*-\-ay^+by^-{-cy-{-d=o^ 



42 



we shall have — =«, (1) 



^(ez'^P)=b, (2) 

 1 



-(423-2p2-9)=c, (3) 

 1 ;r* px'^ »2 g^ 



These equations joined to 



x"" -{-{'2mj+'2z)x'^ -{-{2n^y^ -^4nzy+9z^ -p)x-\-q=0, (F), 

 are sufficient to determine?/. When 2=0 and w=l, we 

 have x^ -{-'ibx* -\-(b^ —4d)x'' — c^=0, which is the reduced 

 of Des Cartes. When 2=0 and w=|, we have 



