290 PROCEEDINGS OF THE AMERICAN ACADEMY. 



and 



Now choose S so that V S ^ hi. 

 Then, when 



the absolute value of the general term of series <I> becomes 



■ ^ 

 < ^ 



Accordingly, the series 

 is convergent when 



and it represents an analytic function for these values of the arguments. 

 5. The family of lines tangent to the surface ^ {$, t^, 4) = at the 

 point (0, 0, 0) forms a cone that cuts the plane ^ = 1 in the curve 

 <^ (f, 7]) = 0. If the line rj/^ = /S, $/C = a. (a and (3 being finite) is 

 one of this family, then the point $ ^^ a, t] =^ ^, I = of the surface 

 <I> (f, r], t,) = 0, (3) is at most a singular point of order m of that sur- 

 face, and its neighborhood corresponds to a portion of the neighborhood 

 of the singular point of the original surface <J> {i, rj, = 0. In fact, 

 cut the surface 



^ a, v,o = o 



by the plane 



rj-l3C=0. 



Then the curve of intersection C will have a multiple point at (0, 0, 0) 

 and the equations of the tangents to C at (0, 0, 0) will be 



C cr=l,2, s<m. 



* Cf. Stolz, Allgemeine Arithmetik, Vol. I. p. 293. 



