322 TRANSACTIONS OF SECTION A. 



2. Relations connecting the Branch Points and the Double Points of an 

 Algebraic Curve. By Professor J. C. Fields. 



This paper concerns itself with an algebraic curve. 



F(a?, y) = y + F„- 1 2/"~' + F„ = (1) 



which presents no singularities at infinity and whose finite point singularities 

 consist of nodes and ordinary cusps. On representing an arbitrary polynomial 

 in (x, y) of degree n-s by the notation. 



G{*, y) = 2 d r ,x<y s (2) 



a simple proof is given of the formula 



j^^nvb" te (3) 



where the summation is extended to all points (a , b ) which are ordinary branch 

 points, nodes or cusps of the curve, the coefficient c A having as value 1, 2, or 3, 

 according as the corresponding point (a,, 6^) is an ordinary branch point, a node 



or a cusp. If in particular for G(a;, y) we take the polynomial F' (x, y) for- 

 mula (3) reduces to one of Pluecker's formulae. The proof of the formula is 



effected through representation of the function ~- by the aid of partial frac- 

 tions, on taking account of the fact that the degree of the reduced form of a 

 rational function of (z, y) cannot be positive if the function possesses no infinity 

 at infinity. 



It may be noted that formula (3) was given by the author of the present 

 paper for a more restricted case at the Ithaca meeting of the American Mathe- 

 matical Society in 1901. In the paper here in question the proof of the formula 

 has been greatly facilitated and abbreviated by the utilisation of the properties 

 of the function R (r, u) which presents itself in the writer's general theory of the 

 algebraic functions. 1 



3. The Infinitesimal Transformation of an Electromagnetic Field into Itself. 



By H. Bateman, M.A. 



4. Report on the further Tabulation of Bessel and other Functions. 



See Reports, p. 67. 



Department of General Physics. 



The following Papers were read :— 



1. On the Radiation producing Aurora Borealis. By L. Vegard. 



Starting from the view advocated by Birkeland, that aurorae are caused by 

 electric solar radiations, the author treats the problem of determining their phy- 

 sical properties. 



From the form and structure of the luminosity a method is found of examining 

 the way in which the solar rays are absorbed by matter, and he arrives at the 

 conclusion that the law of absorption of the solar rays is essentially the same 

 as that of o-rays. This coincidence with regard to law of absorption has led 



1 "Theory of the Algebraic Functions of the Complex Variable." Berlin : 

 Mayer and Miiller, 1906. 



