284 PROCEEDINGS OP THE AMERICAN ACADEMY. 



Next, as f (u, 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) «-[/(«, v, w)~\ — x (v, w) 



C/U 



= (v, w) K + (v, w) K+ i + + (v, w) n =j= 0. (j) 



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

 by £ r (m-1) (r+1) after the substitutions (i), and the establishment of an 

 upper limit for the power of £ r which can then be taken out as a factor 

 of the function resulting from ^ (v, w), will secure a corresponding limit 

 for r, as is needed to finish the proof. 



B. — Critiqde of Kobb'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 siuso- 

 larita puntuali delle superficie algebriche dello spazio ordinario per tras- 

 formazioni 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 



£ = t\ v 

 v = 





and here the number corresponding to y 3 ' of (h) is zero; so that the 

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

 involved, f 



5. Without specific discussion of several unwarranted assumptions 

 of Kobb.J 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 ( v -> w ) m (j) under the substitution (i). Let the given sur- 

 face be 



f =: u 2 — 2uw — v 2 + 2vw + uvw — vw 2 — uw 2 + w z = 0. (k) 



Here, 



X (v, w) = (4 + w 2 ) (w — v) 2 . 

 The curve 



</> (u, v) = u 2 — 2u — v* + -2v = 



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

 t Cf. Levi, 1. c, p. 224. \ Cf. Levi, 1. c, pp. 225-G. 



