292 On the Solution of cei^tain Systems of Equations. 



and the other may then, after a lineai* substitution, be denoted by 

 x^-\-bxy + c^y'^+dx + ey-\-fz=:Oi . . . {g,) 



let us now examine the relation (g) +X(/.). 

 (12.) First, let \=2c— 6, and we have 



{x^cyY-\-dx + ey-\-2ac—ah-\-f=0, . . (A.) 



Next, let Xsr — 2c— i, and we have 



(ar-cy)* + (^ + ey-2ac-«6+/=0. . . (f.) 



(13.) Now let 



a?H-cy4-/'t'=X 

 and 



then we may give {h.) the form 



X2 4-M + (^-2/i)^+(e-2c/i)y=0. . . (j.) 

 So, if we make 



x—cy-^-v^Y, 

 and 



-2flc-a64-/-v2=N, 



we may represent (i.) by 



Y2 + N+(</-2v)^+(e + 2cv)y=0. . . . {k.) 

 (14.) If we assume that 



e— 2c/i< e + 2cv 



d-2fi "" ^"" </— 2v' 

 and, consequently, that 



fjL={4ic)~\cd-{-e), v=t{4c)~\cd-e), 

 and 



d-2fi={2c)-\cd^e), d-2v=(2c)-\cd+e)', 



and if we also make 



W=^M.-(ScT\cd-^eY, N'=N-(8c2)-\c</+^)^ 



then (j.) and (A:.) may be put under the respective forms 



X2+(2c)-'(c(?-c)Y+M'=0 (/.) 



Y^+(2c)-'{cd-\-e)X-\-W=0 (m.) 



(15.) The equations (/.) and (m.) may be still further simpli- 

 fied as follows ; assume that 



X=etx, Y=^y, 



M'=a*m, and N'=/9*», 

 and determine a and ^ so as to satisfy the relations 



a-^^(2cy\cd''€):=l=^-'ot{2cy\cd'{-e), 



