'274 Wilder'' s Algebraic Solution. 



2p=-a, (1) 



2q=b, (2) 



.r*+3p2=2c, (3) 



2pq=d, (4) 



px^+p^ -2cf = -2e, (5) 



qx^+p-q=~% (6) 



IG 8 '^ "^16 4 16a:4~^' vJ > 

 or better, .f '^ —<2p2xs_^pi_^2p_ i6^)x* -^4=0: 

 eliminating .r* from (3), (5) and (7) by the process indicated, 

 and we have 



p(3)_(5) . p^+q^^=pc-}-e,{S) 



9(3) -(6) iy'q=qc+M9) 



p(6)-q{5) q'=qe-f. (10). 



The equations (8), (9) and (10) are satisfied by makingp^ =c 

 cind q^=e, f being equal to nothing. This changes the 

 given equation, by writing —2p for «, and —2^ for —6, into 

 y"" —^jy^ —'2qij^ -\-p^y'^ -\-2pqij'^ -{-q^y^ -\-g—0, or better, 



which shows that the reduced is nothing but the rule of 

 Des Cartes, applied to the above equation. 



Since the given is parted into two factors, 

 {y*—py^ ~qy+^g){y'^ —py^ -qy — '^g)=0, the rule ap- 

 plied to y* —py^ - qij-\.*y g=LO^ gives for the reduced 



{a\ x^-\-1px'^-]^p'^x^ ~p^=.- W g ; changing the 

 signs of the second and fourth term of the first number, 

 we have (6), x° - 2px'^-\-p^x'' -\-q^ = —A\^g\ These two 

 equations give x^ ^ — 2p^x'^ -^{p* -i-4pq^)x^ —q'^ = l6g, 

 which is (7). It is easily seen that we have another equa- 

 tion, {x'^ —p^)y'^ —py^ —qy = (x^ —p^)'^g, which is the 

 same as the given equation. 



We might have treated the "function jtt\ otherwise by 



writing it thus, 



^'2_|.(8?/4— 8p?/2 —Sqy-'2.p^)x^ 



x^-\-2yx^-j-{2y^ —p)x-{-q "*" 

 {Ay'-4p y^-4qy^p' —Apq^-)x '-q' G. , , , , ^ 



we have by transposition, and extracting the square root, and 



