1891.] Ultimate Intersections of a System of Surfaces. 181 



whose equations are the equation of the surface and the equation 

 obtained by differentiating it with regard to the parameter. These 

 equations will be called the fundamental equations in this part. Hence 

 each surface touches the envelope along a curve, which is called a 

 characteristic. It is known that the equation of the envelope may 

 be obtained by eliminating the parameter from the fundamental 

 equations and equating a factor of the result to zero. But it frequently 

 happens that there are other factors of the result (or discriminant as 

 it will in future be called) which when equated to zero do not give 

 the equation of the envelope. 



Following out the same line of argument as that used in reference 

 to a system of plane curves, it will be shown that these factors are 

 connected with loci of singular points. Now if each surface have 

 one singular point, then its coordinates may in general be expressed 

 as functions of the parameter of the surface to which it belongs. 

 Hence the locus of all the singular points of the surfaces of the 

 system is a curve. Its equations, therefore, cannot be found by 

 equating a factor of the discriminant to zero. But if each surface 

 of the system have upon it a nodal line, then the locus of the nodal 

 lines of all the surfaces is a surface, and it will be shown that its 

 equation may be found by equating to zero a factor of the discrimi- 

 nant. 



The singular points in space, the form of which depends only on 

 the terms of the second order, when the origin of coordinates is 

 taken at the singular point, are : 



(i.) The conic node, where all the tangent lines to the surface lie 

 on a cone of the second order. 



(ii.) The biplanar node or binode. This is the particular case of 

 the preceding, in which the tangent cone to the surface breaks up 

 into two non-coincident planes. These planes are called the biplanes, 

 and their intersection is called the edge of the binode. 



(iii.) The uniplanar node or unode. This is the particular case of 

 the conic node, in which the tangent cone breaks up into two co- 

 incident planes. The plane with which these planes coincide is 

 called the uniplane. 



It is shown that a surface cannot have upon it a curve at 

 every point of which there is a conic node. Hence there are two 

 varieties of nodal lines to be considered ; the first, being such that 

 every point is a binode, may be called a binodal line ; and the second, 

 being such that every point on it is a unode, may be called a unodal 

 line. 



It will be proved that if E = be the equation of the envelope 

 locus, B = the equation of the locus of binodal lines, TJ = the 

 equation of the locus of unodal lines, then the factors of the discrimi- 

 nant are in general E, B 2 , U 3 . 



VOL. L. 0. 



