18 G Ultimate Intersections of a System of Surfaces. [Nov. 19, 



It \viil be noticed that when the equation of the system of surfaces 

 is of the second degree in the parameters, and the . condition holds 

 which expresses that the fundamental equations are satisfied by two 

 coinciding values of the parameters, there is a reduction in the 

 number of factors of the discriminant corresponding to the singular- 

 point loci, the factors C 3 , B 4 , U 6 becoming C 2 , B 3 , U 4 respectively. 



The explanation is as follows : 



The discriminant is formed by solving the second and third funda- 

 mental equations for the parameters, substituting each pair of values 

 in the left-hand side of the first fundamental equation, multiplying 

 the results together, and also multiplying by a rationalising factor. 

 Now in the case where the degree of the equation in the parameters 

 is the second, there is only one system of- roots corresponding to 

 the loci under consideration, whereas there are two when the degree 

 in the parameters is above the second. Hence this accounts for a 

 diminution in the number of factors when the degree in the para- 

 meters is the second. 



But this diminution is partly counterbalanced by an increase due 

 to the fact that the rationalising factor vanishes at every point on 

 the locus of ultimate intersections, and consequently increases the 

 number of factors corresponding to the singular point loci. The 

 result of the two causes is what has been stated above. 



