294 PROCEEDINGS OF THE AMERICAN ACADEMY. 



tion $ = 0, and thus represents the latter equation within the corre- 

 sponding limits, i. e., when 



or 



K|<S, hil<e, ^J<c, 



\i-aJ\<e\C\, h-^CKcUl, \t\<S. 



Next we consider points for which ^ = 0, but i, -q are not both zero. 

 For these we use the transformation 



^ = l>?, K=Crj. (8) 



Then, by the same method of treatment as above, putting t, for "rj and 

 7} for ^, and taking 13 = 0, we derive a set of surfaces 



ffA^r^V,t) = 0, T=l, 2, t<m, 



on which are mapped all points of the original neighborhood for which 



hl<8„ _^ 



< ei, 



and so all points for which 



hi < Si, K! <h\v\, \^-%v\ <h\v\' 



Here, we have a function corresponding to (f, >;) : 

 Now, for the infinite roots of 

 we put the equation into the form 



(i,a) 



-, 1, ^ ) - 0. 



\ rj 

 So the equation 



«^(l, = 



is such that its roots for | = are the same as the ratios of the infinite 

 roots of the equation 



and by 3, 3) these ratios are all finite. 



