Wilder'^s Algebraic Solution. 



281 



It would be useless to repeat the calculation by wiiich we 

 have determined the functions, 85,8,^,8,0; it is sufficient 

 to say that (A') will be, 



— dqi/jy- 

 — 5py^ I —5rp^ 



'-\-5py 

 ■bry 

 bpq 



-bq^y^p y x'^'^ 



+p' 



— bprq^ 

 -\-br-yq" 



— bp^r~q 



fx' 



+P' 



— bqry^ | —br^q \ 



— bq^y — bpr^y J 

 + bp^-y- 



— lOr^p 

 + bq'r J 



and B', x*-\-yx^-{-j)x^ -{-qx-\-?\ 



Let us now suppose that such a function ^(xpqr) is 

 written for ?/, as renders x^'>{y^ — bpy^ -{-bqy^ -\-op~y — bry 

 — bpq)x^ ' +(p^ —brp^ -^bq^p^ — 10r-p + bq^r)x' '>+{q^ — 

 bprq^ -\-bp^r^q — br^q)x^ -\-r^ =^0^ we may then compare it 

 with y^ -^-hy^-^-cy^ -|-f^y-f.e=0, after dividing the first equa- 

 tion by a; ■ ^ ; this done we have, 



5p=-&, (1), 



bq=c, (2), 



bp^-br—d, (3), 

 x^ —bpq-\-{p^ —brp^-\-bq^p^ — 10r-p — bq^r)-{- 



{q^ — bprq^-\-bp^r^q — bp^q) r^ 



or better, 

 x^ '>—{e-\-bpq)x^ ^-\.{p'^ —brp^-^bq^p^ — lOr'^p — bq^r)x^ "+ 



(q^ —bprq^-{-bp^r^q — br^q)x^-{-q^=0. 



These four equations, joined to y-{-{x*-\-px^-{-qx-\-r)=0^ 



are sufficient to determine y. 



It is very evident, that it is a matter of indifference, which 

 of the letters ^/, p, </, or r, we treat as the unknown ; for they 

 are all of five dimensions in the function before us. 



This remark is applicable to other cases ; as, for example, 



. (C) ,.. . x^ + {y^-3py)x^+p^ , 



the function yr^t which is ^-1 -; — ; here, ma- 



(D) x"-}-yx-{-p ' 



Vol. XVI.— No. 2. 9 



