Wilder'' s Algebraic Solution. 279 



A=3p, (1), 

 B=3q+Sp^~a, (2), 

 C=6pq+p^ -3ap,{3), 

 3aq+3p^a-a= -\-3pb=:Sq^ +3p'q, (4), 

 6apq-{-p''a — 3pa^—2ab-{-3bq + 3p''b = 3pq^, (5) 

 x^ -{-q'' =6pqb — 3pab—b'' , (6). 

 If we multiply (4) by p, and subtract it from (5), we shall 

 have {3q-2a)p^ + {3q~2a)ap-\-{3q-2a)b=0. This equa- 

 tion is satisfied by making 3q=2a. 

 2 

 Writing ^ a for ^, in (4) and (6,) and we have, 



a^ 

 «p2_^36p — — =0; and 



x^^bp^-i-abp—b^ — ^', from whence 



we have?/; for when y^-^ay-\-b=0, we ha.ye y^-jrpy+q 

 — a; 3=0 ; and we have already p, q and x, in functions a and 

 6. This is the rule of Tschirnaus. We may still vary the 

 calculation, by assuming the function, 



(a3-^a2^^a.^py^ly^q^x){8^+l3^-{-{a-\-p)^+b-Jrq-\-x)x 



y^-{-ay-\-b 



for, making the coefficients of x and x^, equal to nothing, 

 and ehminating the symmetrical function a-J-/3-f-7, a2_|_|S 2 

 +7^, etc. by means of a and 6, we obtain, 

 3q~2a, 



a^ 

 ap2-|-36p— Y=0, and 



x^=bp^-{-abp — b'^ —-^iz- ' 



now, if we make y^-\-ay-{-b=0, we also have y^+py+q 

 =0, from whence y is known, y standing for one of the let- 

 ters a, /3, or 7. 



Let us next recall the function, 

 x ''^—(y^—4py^+4py + 2p^)x^^{p*-4qyp''+Aq^p-^ 



x^+yx^+px-\-q 

 2q^y^)x^-{-q^ (G) 

 ' (H)' 



