464 The Rev. George Salmon on the Degree of the 



The number, therefore, of points of contact of double tangents is 7i (n-l) 

 (n-2) (n — S); and the number of double tangents is of course half this. We 

 have proved, then, that the degree of the tangent cone is n {n — l) ; that it has 



.-, 1 , ^ n(n-l) (n-2) (n-3) , ,, , 

 »(n-l) (n-2) cuspidal edges, and ^-^ ^^ double edges. 



The degree of its reciprocal, then, is, by M. Pluckee's formula, 



n(n-l)]n(n-l)-l\-Sn(n-l) (n-2)-n(n-l) (n-2)(n-3) 



= n(n-iy-. 



And since the degree of the reciprocal surface is the same as the degree of a 

 plane section of it, we have in general n' =n (n— 1)^ 



Let us now proceed to the case where the given surface has multiple lines. 

 It appears by the same reasoning as for plane curves, that every line joining the 

 point x'y'z'w' to any point of a multiple line must be regarded as, in one sense, 

 a tangent line to the surface : and that tlie cone determined by the equation 

 <(>U= includes doubly the cone standing on the double curve b, and trebly 

 the cone standing on the cuspidal curve c If then a be the degree of the tan- 

 gent cone proper, we have 



n(n-l) = a + 2b + 3c. 



To find the multiple edges of the tangent cone, we have, as before, to exa- 

 mine the points where the line of contact meets the two surfaces A^ U, and -f U. 

 But the line of contact now consists of the complex line a+2b + 3c, and the 

 points where b and c meet A'U and -ft/" are plainly irrelevant to the question. 

 Neither shall we have cuspidal or double edges answering to all the points where 

 a meets these surfaces : since, if for example, any side of the cone a be also a 

 side of the cone b, this must be considered as a double edge of the complex 

 cone, although not a double line either on a or b. And any line passing through 

 an intersection of the curves a and c must be considered as a cuspidal edge of 

 the complex cone, although not so on either of the cones considered sepa- 

 rately. 



The following formulaj will be found to contain an analysis of the intersec- 

 tions of each of the curves a, b, c with the surface A'U. The signification of 

 the letters employed has been already explained : — 



