180 Prof. M. J. M. Hill. Theory of the Locus of [Nov. 19, 



IV. " On the Locus of Singular Points and Lines which occur 

 in connexion with the Theory of the Locus of Ultimate 

 Intersections of a System of Surfaces." By M. J. M. HILL, 

 M.A., Sc.D., Professor of Mathematics at University College, 

 London. Communicated by Professor HENRICI, F.R.S. 

 Received October 5, 1891. 



(Abstract.) 

 Introduction. 



In a paper " On the c- and ^-Discriminants of Ordinary Integrable 

 Differential Equations of the First Order," published in vol. 19 of 

 the ' Proceedings of the London Mathematical Society,' the factors 

 which occur in the c-discriminant of an equation of the form 

 /(#, T/, c) = 0, where /(, y, c) is a rational integral function of 

 x, y, c, are determined analytically. 



It is shown* that if E = be the equation of the envelope locus of 

 the curves /(#,?/, c) = ; if IS" = be the equation of their node 

 locus ; if C = be the equation of their cusp locus ; then the factors 

 of the discriminant are E, N 2 , C 3 . 



The singularities considered are those whose forms depend on the 

 terms of the second degree only, when the origin of coordinates is at 

 the singular point. 



The object of this paper is to extend these results to surfaces. 



It is well known that if the equation of a system of surfaces con- 

 tain arbitrary parameters, and if a locus of ultimate intersections 

 exist, then there cannot be more than two independent parameters. 



Hence the investigation falls naturally into two parts : the first is 

 the case where there is only one independent parameter, and the 

 second is the case where there are two. 



The investigation given in this paper is limited to the case in 

 which the equation is rational and integral both as regards the 

 coordinates and the parameters. 



PART I. 



The Equation of the Surfaces is a Rational Integral Function of the 

 Coordinates and one Arbitrary Parameter. 



In the case in which there is only one arbitrary parameter each 

 surface of the system intersects the consecutive surface in a curve 



* The theorem was originally given by Professor Cayley, in the ' Messenger of 

 Mathematics,' vol. 2, 1872, pp. 6-12. 



