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



(II.) There may be more than one system of distinct values of the 

 parameters satisfying the fundamental equations. 



In this case the effect of the distinct values is additive. Thus if 

 there be p systems of values at a point on the envelope locus, the 

 factor E would occur to the p ih power. 



(III.) Two or more systems of values of the parameters satisfying the 

 fundamental equations may coincide. 



The results must be stated differently in the cases (a) where the 

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

 greater than two ; (/3) where the degree in the parameters of the 

 equation of the system of surfaces is two. 



In the case () it will be shown that there may be envelope-loci, in 

 which the envelope has stationary contact with each surface of the 

 system ; conic-node loci, which are also envelopes ; biplanar node 

 loci, in which the edge always touches the biplanar node locus; and 

 uniplanar node loci. The results are given in the following table : 



