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 <l> for <£, ^ for 3> M , (^ = 1, 2, v) 



where the <i>'s are derived successively in the same way as the 3>'s. At 

 each stage the $ factor must contain all the terms of lowest degree in 

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

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

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

 required terms would not be present. 

 Now, by (19), 



5* _5$ 9£y_ J_5<5> 



Also 



(22) 



(23) 



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

 have the relation 



L($, v , 0* + M& V) || = R(v, + 0. (24) 



Then, substituting for £ and 77 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 



A(fe v„ 0** + m v ($ v , Vv , %r = ? l O* + <»i(QT*fi(v» 0, (25) 

 PAiv, v» Q*v + Qv&, vv, 0p- = M£ + ^(m^{U 0, (26) 



vrj v 

 where 



<M&, v^ = *v(£ v , vv, 0^i (&, v^ = **(& vv, 0&(€v, vv, £)■ (27) 



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

 press it in the form 



R(v, = n(v, On + (* 0n + i + 3 = ?S(v, 9, 



where 



S(v, 0) * o. 



If (17, £)n contains no term in 77, the first transformation of (14) will al- 

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

 and thus securing the form (25) at once with 



