284 PROCEEDINGS OP THE AMERICAN ACADEMY. 



Next, as / {ii, V, w) can be supposed to be irreducible, we have a 



relation of the form 



9 

 L {u, V, w)f{u, V, w) + M{u, V, w) — [/(m, V, w)'\ = X 0', w) 



= (v, w)a + (y, w)a+i + + (v, w)„ ^ 0. (j) 



Now it is shown that the first member of equation (j) becomes divisible 

 by ^^•'"-Di'+i* after the substitutions (i), and the establishment of an 

 upper limit for the power of Cr which can then he taken out as a factor 

 of the function resulting from x (^» ^j? will secure a corresponding limit 

 for r, as is needed to finish the proof. 



B. — Critique of Kobe's Analysis. 



We now show in what respects Kobb's method and proof are at fault. 

 Some of these errors are noted in a memoir " Sulla riduzione delle singo- 

 larita puntuali delle superficie algebriche dello spazio ordinario per tras- 

 formazioiii quadratiche," by Beppo Levi.* 



4. Kobb overlooks in his succession of transformations of type (g) 

 the occurrence of transformations which arise from 1, 3), c. These are 

 equivalent to 



V = 



\ 

 ^ ... 



and here the number corresponding to yg' of (h) is zero; so that the 

 proof, even if correct in other respects, would fail to cover all the cases 

 involved.! 



5. Without specific discussion of several unwarranted assumptions 

 of Kobb.t we show by an example the failure of his proof for the 

 upper limit of the exponent of the power of ^,. to be taken out as a 

 factor of X ('^i "') i^ (j) under the substitution (i). Let the given sur- 

 face be 



f= )<2 — 2uw — v'-^ + 2inv + uvw — vw'^ — tiw' + w^ = 0. (k) 



Here, 



X (v, w) — - (4 + w"^) (w — vy. 

 The curve 



(/) (u, v) = u^ — 2u — v"- + 2v = 



* Annali di matematica, Series 2, Vol. XXVI. (1897), p. 219. 

 t Cf. Levi, 1. c, p. 224. t Cf. Levi, I. c, pp. 225-6. 



