198 CHAPTER VIII. 



Hence the inverse of the following substitution, 



3 = <*12 Si + ^23^3 + ^24 4 H ----- 



->$< (*-!,.:., at; t 



will transform f into 



summed for $, j = 1, 3, 4, . . ., m\ i <j. Applying the substitution 



we obtain as the new coefficient of || the function a 22 A 2 + /5 n , which 

 may be made to vanish by determining A. 



We may therefore suppose that # u = in our original function /. 

 Since the ccij are not all zero, we may assume that a 12 =f=0. Apply- 

 ing to / the inverse of the substitution 



Si = a i2?s + %3^3 H ----- f- iwSw? SS = S (* = 1? 3, 4, . . ., m) 

 we obtain the function *=2... 



Replacing | t + y 22 | 2 + y 23 | 3 + . . - + y 2m | m by ^, we get 



Similarly, if m ^> 5, we can transform /"' into 



If m be odd, we reach ultimately the form 



Applying to it the substitution which replaces m by x~ 1/2 % m , we 

 obtain .F. 



If m be even, we reach ultimately the form 



If aSm-i + /J6w-iSm+ySJi ^ reducible in the ^jP[2], i.e., be the 

 product of two linear homogeneous functions of m _ i and m? an 

 evident substitution will reduce to F Q . In the contrary case, a, /?, y 

 are certainly distinct from zero, so that the substitution 



