290 PROCEEDINGS OP THE AMERICAN ACADEMY, 



and 



*ffi* = 34* ???***- 



Now choose 8 so that r 8 < o x . 

 Then, when 



|?] = r x , |^| = r 1} K| = 8, r x >i, 



the absolute value of the general term of series <fr becomes 



< | A iJk | . iy + ->' . &+j+*-" 



Accordingly, the series 



is convergent when 



\l\<Tt |7l<Ti |C|<8,* 



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

 5. The family of lines tangent to the surface <J> (f, ??, £) = at the 

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

 $ (£, 77) = 0. If the line -q/t, = /?, £/£ = a, (a and /? being finite) is 

 one of this family, then the point £ = a, -q = ft, £ = of the surface 

 <1> (£, 77, £) = 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 <I> ($, rj, £) = 0. In fact, 

 cut the surface 



* (6 * = 



by the plane 



»7-/3£=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 



v -j3£=0) 



> cr = 1, 2, s < m. 



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



