294 PROCEEDINGS OF THE AMERICAN ACADEMY. 



tion 4> = 0, and thus represents the latter equation within the corre- 

 sponding limits, i. e., when 



|£|<s, M<*> U< c > 



or 



|*-a,{|<e|C| f h-i»C|<«|t|. |f|<«. 



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

 For these we use the transformation 



t = €v, t = Zy- (8) 



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

 7) for £, and taking (3 = 0, we derive a set of surfaces 



9 v (£fV>l) = 0, t= 1, 2, <<m, 



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



£ 



M < 8i » i — < € i » 



I v 



and so all points for which 



Here, we have a function corresponding to <£ (J, 77) : 



<M?,f) = (|, i,1) m 



Now, for the infinite roots of 



*(£})r=0, 



we put the equation into the form 



( f -,.,4)=o. 



So the equation 



<M?,f) = o 



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

 roots of the equation 



$(lv) =o, 



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



