1891.] Ultimate Intersections of a System of Surfaces. 183 



mate intersections is found by eliminating the parameters between 

 this equation and the two equations obtained by differentiating it 

 with regard to the parameters. These equations will in this part of 

 the investigation be called the fundamental equations. 



In general the locus of ultimate intersections is a surface, for the 

 coordinates of each point on it can be expressed as functions of the 

 two arbitrary parameters. The exceptional cases in which it is not 

 a surface are enumerated at the end of the paper. These include the 

 case where the equation of the system of surfaces is of the first degree 

 in the parameters. Hence it will be supposed that the degree of the 

 equation of the system of surfaces in the parameters is above the 

 first. 



In general also the locus of ultimate intersections possesses the 

 envelope property, and the equation of the envelope is determined by 

 equating the discriminant, or a factor of it, to zero. 



If factors of the discriminant exist which, when equated to zero, 

 give surfaces not possessing the envelope property, then, as in Part I, 

 it is shown that these surfaces are connected with loci of singular 

 points. 



Now the locus of singular points of a system of surfaces whose 

 equation contains two arbitrary parameters is in general a curve 

 (not a surface), whose equations can be obtained by eliminating the 

 two parameters from the equation of the system of surfaces and the 

 three equations obtained by differentiating it with regard to the 

 coordinates. Hence its equations cannot be determined by equating 

 to zero a factor of the discriminant. 



But if every surface of the system have a singular point, then in 

 general its coordinates may be expressed as functions of the two 

 parameters of the surface to which it belongs. Hence the locus of 

 the singular points is a surface. It will be proved that it is a part 

 of the locus of ultimate intersections. Hence its equations can be 

 obtained by equating to zero a factor of the discriminant. 



Let now E = be the equation of the envelope-locus, 

 C = the equation of the conic node locus, 

 B = the equation of the biplanar node locus, 

 U = the equation ,of the uniplanar node locus. 



Now at any point on the locus of ultimate intersections 



(I.) There may be one system of values of the parameters satisfying 

 the fundamental equations. 



In this case there may be envelope, conic node, or biplanar node 

 loci; and the results are given in the following table : 



o 2 



