316 PROCEEDINGS OP THE AMERICAN ACADEMY. 



in 4, the succession of transformations (14) which leaves the degree of 

 T unchanged will secure for equation (31) a form 



The left side of the equation is divisible by ^^ and so the right side 

 must be, 



V < X, 



and we have an upper limit for y, the number of transformations which 

 leave the factor N^. of the second degree, and as a result leave the func- 

 tion T of the nth degree. So, unless the function T{rj, C) in (28) is 

 composed of ?^ equal factors of form 



Lv + ^ (OT, (32) 



the transformation of (14) will finally reduce its degree. Then, by ap- 

 plying the same reasoning to the resulting function, we see that finally 

 the function corresponding to S(r}, ^) either becomes an J^ function or 

 has besides the JiJ factor a factor of form (32), thus securing the form 

 (25) if we divide out the factor ^("•-D", 



The condition (26) is secured by using on the second equation in 

 (20) the same kind of reasoning as applied in 4 and 5. Then we take 

 for V the larger of the two values required to secure conditions (25) 

 and (26). 



C. — Further Transformations. 

 6. A transformation 



^V = iw — Wg (0 ) .ggs 



Vv = rj„ — 0)1 (^) ) 

 applied to the surface 



^.(^., /i., = 



in 5 will secure a form in which the singularity will be reduced by 

 either 



1 ) a further succession of transformations as in 3, 



2) the method of the Lemma, § 2. 



Let us consider here the case in which either rj or r, in (25) and (26) 

 is zero. Then in one of the equations a further succession of trans- 

 formations of type (14) will not change the power of ^ as a factor on 

 the right ; and if there are /x such further transformations, the reasoning 



