324 PROCEEDINGS OP THE AMERICAN ACADEMY. 



reduced to a value q' ^ less than r — p^, in which case the sum of the 

 exponents of the three variables, 



p^-\- q\^- in — r, 



is less than m and reduction ensues. So it is only in the case in vphich 

 for every term 



p,>s, s = 2, 3, m, 



that we are not sure of reduction. Suppose the number of transforma- 

 tions after this point to be n. Then we get for the new exponent of t, 



qs + n{p, — s). 

 Now by taking n large enough we can make the quotient 



" {Ps - ■s) + qs 



s 



iy "■"" s 

 have the lowest value for the term in which — is lowest, while if 



this is the same for two or more terms, we can make the fraction above 



lowest for the one in which — is lowest. Accordingly, by a finite 



number of transformations of type (43) we secure the condition that 



p — y p . . . Q . 



^-^ and so — is lowest in the same term in which — is lowest. 



r r 



6. A succession of transformations of type 



^M = Wi^, (46) 



followed by a succession of type 



secures the surface with condition 5, 2) in the form 



^pi^.p, Vp> = ^p(fp. Vpi V)E{Ep, r)p, 



= [C+<'C''^('/p'OC' + --- + <'"^''"^('7p'0]^(^p.'7p,0 y ^4g^ 

 where for some particular term in Xp, the rth, 



Pr <r, qr< r. 



Consider the surface (42) with the condition 5, 2), the sth term being 



J 



V 



and suppose we apply to the surface n transformations of type (46), 

 dividing out each time the factor ^. The resulting term is 



