296 PROCEEDINGS OP THE AMERICAN ACADEMY. 



can have multiple values of | only for a finite number of values of ^, 

 these being the values for which the equations 



91 

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

 yj become infinite. 



Now we consider all such values of r; 



'^ = Cr, r = 1, 2, ly 



for which the equation, considered as an equation in |^ 



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

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

 greater than unity, i. e. some of the equations ^ = 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 rji or t,. 

 Surround these points by neighborhoods 



|^J<S, hil<8, \^\<h, 



i. e. 



|?-aj<5, i:^_c,|<8, ia<8» 



which are to be considered later. 



Now let »; = i be any value for which the equation 



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

 in ^ to the first degree, free from rji and t,, and thus the points of the 

 surface g = lying in the neighborhood of the point ^^ = 0, rji = 0, 

 ^ = 0, can be represented by a power series 



So, in this case, we have m developments 



^^ = P^{m,0^ o-=l, 2, m, 



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



i = Pa(v,0^ 0-= 1, 2, m. 



It is readily seen that the function 



i = pivuO 



