Surface reciprocal to a given one. 463 



Xx + fix', Xy + fxy', Xz+fiz', \w + ixw', 



- are the co-ordinates of any point on the line joining them. If these values be 

 substituted for the current co-ordinates in the equation of any surface, the 

 resulting equation solved for X: fi gives the co-ordinates of the points where 

 this line meets the surface. If then we form the condition (f>U=0, that this 

 equation in X: fj. should have equal roots, and in it consider xyzw as variable, 

 it will represent the locus of all points, such that the line joining them to x'y'z'vo' 

 touches the given surface ; or, in other words, it will be the equation of the 

 tangent cone whose vertex is x'y'z'w'. 



If the equation of the surface be U= 0, the result of this substitution will be 



[U] = X°t^+ \"-V A U+ ^p|- /^'U+ &c. = ; 



where A denotes the symbol 



, d , d , d , d 

 dx dy " dz dio' 



The result of elimination then between ,^ and , ^ is the equation of the 



dX dfx 



tangent cone. It will obviously be of the n(n— 1)" degree. 



Cuspidal edges on the tangent cone arise when any edge of the cone meets 

 the surface in three coincident points. \? xyzw be the co-ordinates of the point 

 of contact, the equation [ fJ] = must in this case be divisible by /x'. The 

 point must, therefore, be one of the intersections of the three surfaces U— 0, 

 A U— 0, A^ U=: ; and since these are of the degrees w, n — 1, w — 2, the num- 

 ber of such points is n(ra— 1) (w — 2). 



Double edges on the cone arise when any side touches the surface in two 

 distinct points. The equation [?7] will in this case have two values of ^ = 0, 

 and two of the remaining values of jx equal. The co-ordinates then of any point 

 of contact of a double tangent must satisfy the equations U=0^ AU = 0, and 

 ^U=0, where ■\jrU is the condition that the equation, 



_L x"-' A' U+ -l—^ X"-V ^'U+ &c. = 0, 



should have equal roots. ^Uis evidently of the degree (w - 2) (w — 3) in xyzw. 



3 p2 



