286 PROCEEDINGS OP THE AMERICAN ACADEMY. 



use in 2 only the set of points determined in 1. We are not warranted, 

 either, in assuming that, when a reduction of singularity arises from the 

 appearance of a term of lower degree in a different'variable from that 

 with reference to which the first development is derived, the resulting 

 development will hold throughout the same region as the first develop- 

 ment. As an example consider the surface 



r 2 + cr£ - £ = 0. 



Regarded as a development for t, its coefficients converge for all finite 

 values of o- and £ ; but when we develop for £, 



*> — , 5 



1 — O" 



and the resulting series converges only when 



| o- 1 < 1. 



7. From geometrical considerations we should not expect the quad- 

 ratic transformation used to resolve the singularity in all cases. In 

 ordinary space the transformation 



£ = t£, rj — cr£, 



will transform in a one-to-one manner, without change of the £ coordinate, 

 all points except those in the £ = plane. Now in the surface from (m), 



t 2 - a' + t(t£ = 0, 



all points in the £-axis are singular, and whatever the reduction that 

 may be secured for the origin, there will be within the neighborhood of 

 the origin points whose singularity is not reduced. The same consider- 

 ations would be seen to apply if we had any space curve as a singular 

 line. 



Levi, in the article previously mentioned, does not attempt a proof 

 of the entire proposition, but directs his work toward establishing by 

 geometrical considerations the reduction of the singularity, making ex- 

 ception, however, of certain cases,* which are closely related to the one 

 considered in 7. 



Having thus considered the failure of Kobb to establish the proposi- 

 tion even for the general case of an algebraic surface, we shall, in the 



im Grossen, the limit to the number of points taken being determined by finding 

 the extent of tbe domain of each ; while the developments about the later points 

 giye relations im Kleinen, as far as the first point is concerned. 



* Cf. Levi, 1. c. p. 227. Cf. also a second paper by Levi, Atti R. Ace. Sci. Torino, 

 Vol. XXXIIL, 5 Dec, 1897. 



