296 PROCEEDINGS OP THE AMERICAN ACADEMY. 



can have multiple values of $ only for a finite number of values of 77, 

 these being the values for which the equations 



* = 0, ^ = 



have common roots, and by the condition 3, 3) none of these values of 

 r) become infinite. 



Now we consider all such values of -q 



V = c r , r = 1, 2, /, 



for which the equation, considered as an equation in J,, 



*(?,}) = <> 



has multiple roots. Deal with each of these as in 5, c r taking the place 

 of /3 in (6) ; then, in equation (7), some of the // 's will, in general, be 

 greater than unity, i. e. some of the equations g ■=. will have for the 

 lowest terms in £ alone exponents greater than 1. For such as have 

 their /x = 1, there are regular points. The others will afford singular 

 points unless they have terms of the first degree in either ^ or £. 

 Surround these points by neighborhoods 



141 < 8, \m\<*, |CI<*» 



i. e. 



|?-aj<&, \V-C r \ < J, \t\< 8, 



which are to be considered later. 



Now let t] = b be any value for which the equation 



<£ (?, V) = o 

 has not equal roots. Then the equations g = of (7) each have a term 

 in £ to the first degree, free from -q x and £, and thus the points of the 

 surface g = lying in the neighborhood of the point $ a = 0, rj x = 0, 

 £ = 0, can be represented by a power series 



So, in this case, we have m developments 



$,=£rO&i©i <r=l, 2, m, 



and, by using the relations (4), we have 



£=p 9 (r), Q, <r=l,2, m. 



It is readily seen that the function 



