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Y. On the Locus of Singular Points and Lines which occur in connection 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,—Read November 19, 1891. 
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 f(x, y, c) — 0, where f{x, y, c) is a rational integral function of x, y, c, 
are determined analytically. 
It is shown* that if E = 0 be the equation of the envelope locus of the curves 
f(x, y, c) — 0 ; if N = 0 be the equation of their node-locus ; if C = 0 be the equation 
of their cusp-locus, then the factors of the discriminant are E, Ny 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 contain 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.! 
* The theorem was originally given by Professor Cayley in the ‘Messenger of Mathematics,’ vol. 2, 
1872, pp. 6-12. 
f An abstract of the contents of this paper has been printed in the Proceedings, vol. 50, pp. 180-186. 
A table of contents will be found below, pp. 274-278. 
21,6,92. 
