1891.] Ultimate Intersections of a System of Surfaces. 185 



The case (/3) always falls under the next case. 



(IV.) The values of the parameters satisfying the fundamental equa- 

 tions may become indeterminate. 



If the equation of the system of surfaces be of the second degree 

 in the parameters, and the analytical condition hold which expresses 

 that the fundamental equations are satisfied by two coinciding 

 systems of values, then this condition requires to be specially inter- 

 preted. For now the second and third fundamental equations are of 

 the first degree in the parameters, so that if they are satisfied by two 

 coinciding systems of values, they must be indeterminate. 



It is, however, possible to determine a single system of values of 

 the parameters satisfying them. In this case the three surfaces repre- 

 sented by the fundamental equations intersect in a common curve 

 (which is fixed for fixed values of the parameters) lying on the locus 

 of ultimate intersections ; whereas in the previous cases they intersect 

 in a finite number of points lying on the locus of ultimate inter- 

 sections. 



The surface of the system, corresponding to the fixed values of 

 the parameters, .touches the locus of ultimate intersections along the 

 above-mentioned curve. 



In general there are two conic nodes of the system at every point of 

 the locus of ultimate intersections. The parameters of the surfaces 

 having the conic nodes are determined by two quadratic equations, 

 called the parametric quadratics ; and in general the roots of each 

 parametric quadratic are unequal. If the roots of loth parametric 

 quadratics are equal, the two surfaces having conic nodes are replaced 

 by one surface having a biplanar or uniplanar node. 



If the parameters of one of the surfaces having a conic node become 

 infinite, this surface may be considered to disappear, and there is but 

 one conic node at each point of the locus of ultimate intersections. 



If the parameters of loth surfaces having conic nodes become 

 infinite, both these surfaces may be considered to disappear, and the 

 locus of ultimate intersections is an envelope locus (touching each 

 surface of the system along a curve). 



If the parameters of loth surfaces having conic nodes become 

 indeterminate, then there are at each point an infinite number of 

 biplanar nodes, and each surface of the system has a binodal line 

 lying on the locus of ultimate intersections. 



The results are given in the following table : 



