316 PROCEEDINGS OF 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 £ v , and so the right side 

 must be, 



v = ^> 



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

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

 tion T of the rath degree. So, unless the function T(rj, £) in (28) is 

 composed of n equal factors of form 



bi + s (0?i ( 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], 'Q either becomes an E function or 

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

 (25) if we divide out the factor £<"»-i)»\ 



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 



& = £-<» 2 (0j (33) 



i) v — y] v — oj a (£) ) 

 applied to the surface 



iu 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 r x or r 2 in (25) and (26) 

 is zero. Then iu 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 



