314 PROCEEDINGS OF THE AMERICAN ACADEMY. 



similar result is true for all of the succeeding transformations. So in the 



first part of (14) we could write ^ for <I>, ^^ for ^^, (/x = 1,2, v), 



where the *'s are derived successively in the same way as the $'s. At 

 each stage the <> factor must contain alL the terms of lowest degree in 

 the corresponding <J> (except for a constant multiple), and no lower 

 terms ; for, otherwise, either there would be lower terms in the product 

 by the corresponding £J factor on account of its constant term, or the 

 required terms would not be present. 

 Now, by (19), 



Also 



$=i^'«>'$,; (22) 



and, combining with (21), we have 



|| = r-||". (23) 



But as $ has no multiple factors vanishing at (0, 0, 0) (see § 1,3), we 

 have the relation 



L(^, rj, 0* + M{^, >?, fl = Ri-n, ^ 0. (24) 



Then, substituting for f and t] from (19) on the left side of equation (24) 

 and using the relations (22) and (23), we have the required relation (18). 

 5. If V is taken large enough the transformations (14) will lead to the 

 relations 



^(^., vr, 0^" + M.{^., v., ^ = ^ Iv^ + -h(OJ^J^(v^, 0, (25) 



p.{i., v., 0*^ + <?.(^v, ^., op- = ^[^'^ + -(o]^=^(^., o> (26) 



where 



^viL, V^, = ^.(f., V^; 0^1 i^r, ^., = ^^(^"^ v^, 0J^2(i., v^, 0- (27) 

 We consider the effect of the transformations (14) on R in (18). Ex- 

 press it in the form 



Riri, = ^'[{V, On + (V, On+l + ] = i'^iV, 0. 



where 



S(v, 0) ^ 0. 



If (rj, C)n contains no term in rj, the first transformation of (14) will al- 

 low the factor C to be taken out of S, leaving behind a constant term, 

 and thus securing the form (25) at once with 



