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



This is the general theorem, but it is assumed in the course of the 

 investigation, when the discriminant is being formed, that the fun- 

 damental equations are satisfied by only one value of the parameter at 

 each point on the envelope locus or on a locus of binodal or unodal 

 lines. 



The investigation is accordingly carried a step further, and it is 

 shown that if the fundamental equations are satisfied by two equal 

 values of the parameter at points on an envelope locus, or on a 

 locus of binodal or unodal lines, the factors of the discriminant are 

 E 2 ; B 3 , U 4 . 



The geometrical meaning of the condition that the fundamental 

 equations are satisfied by two equal values of the parameter in the case 

 of the envelope is that the line of contact of the envelope with each 

 surface of the system counts three times over as a curve of intersec- 

 tion, instead of twice as in the ordinary case, or that two consecutive 

 characteristics coincide. The meaning of the condition in the case 

 of the loci of singular lines is that each of these loci is also an 

 envelope. 



The results are given in the following table : 



PAET II. 



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

 of the Coordinates and two Arbitrary Parameters. 



In the case in which there are two arbitrary parameters in the 

 equation of the system of surfaces, the equation of the locus of ulti- 



