Wilder"^ s Algebraic Solution. 275 



making the absolute term equal to '^g'.y'^ —2^!/" —qy+P^ 



1 



— 4x* 



reducing a;'2_j-(8?/* —Spy^ — Sqy ~ 2p")x^ -\-{p^ _4y^o- 



— 4q'-'p)x'^—q^=^0, but Sy* — Spi/^ —8qi/~S"^g ; hence, 

 our reduced is 



_ —2 



which is the product of the factors (a)=0 and (6)=0, re- 

 sulting from the transposition of — 4V^^. 



From the foregoing examples, one would be led to think 

 the method pursued here was applicable to all rational al- 

 gebraic equations ; but let us, before we attempt to follow 

 the analogy, recall, and demonstrate the following proposi- 

 tions. 



First, let i:~n Ji^rr"! n-i , — ;;:r, 



X ^ -f-yx ^-\-px ^-^qx * 



. . . -I-S(-„_2)m^™+S(„_i)„, (A') 



. . . -^tx-\-u (B') 



be a function in which x, ?/, p, etc. are independent functions 

 of any number of other quantities whatever, then I say that 

 Sm, Sgm, Sgwi, etc. can always be determined in functions of 

 y,p, q., etc. independently of :r, so that (B') shall be a factor 

 of (A'); for, continue the operation indicated till the index of 

 X in the remainder is n — 2, and then make the remainder 

 equal to zero independently of x, which can always be done, 

 since the whole number of unknowns, S„j, Sgm, 83^, etc., 

 and the whole number of equations is n— 1 ; and it is evident 

 that they are of the first degree, relative to S„i, Sgw 183^, etc. 

 It is plain that the function (A') may be decomposed into 

 (w - 1 ) factors, (x^-f a™) (5:'"+^'") (a:™-f 7™) (^'"+5™) etc. ; and 

 if we put a+/3+7-l-^, etc. =3/ 



a^^ay-j-a^-f/Sy etc. =p 

 a^'y-\-a.^S-\-l3yS-\-a'yS etc. =q 

 ..... etc. 

 afSyS etc. =u ; 

 then adopting the notation and formula of Lacroix Comple- 

 ments des Elemens d'Algebre, we have 



