486 The Rev. George Salmon on the Degree of the 



and for twice the reduction in the number of its double edges, 



2/ii5(/^-l)7r-/x(p-l) (14p-8)n + Mp(p-l) (8^.-2) -;r(p-l)V^ 



+p{p-l) (4p-6)/>. 



As a verification of this formula, let the surface n consist of p surfaces of 

 the »i"' degree, all having the same curve fx for their complete intersection, then 

 jx = iri^, n —jym, p = 2m- (in — I), and the formula for the reduction in the 

 degree of the reciprocal becomes 



m'\p{p-l) {3p + l)m-2p(p'-l)-2p'{p-l) (m-1); 



— Pip^— 1 ) ™' - 2p(p-l) nr. 



But this is the difference between mp{mp - 1)- and mp(p - 1)1 



I have endeavoured to apply this theory also to the class of ruled surfaces 

 which I have considered ("Cambridge and Dublin Mathematical Journal," 

 vol. viii. p. 45) generated by a right line resting on three directrices ; and I 

 have succeeded in verifying the theory in the case where two of the directrices 

 are right lines. In this case, if the degree of the third directrix be /u, the sur- 

 face is of the degree 2|U, and each of the right lines are multiples of the degree 

 fx. Now it is easy to see that the effect of two non-intersecting multiple lines, 

 in diminishing the degree of the reciprocal, is the sum of their separate effects, 

 and therefore, putting p = fi,n — 2/i, in the formula already obtained, the degree 

 of the reciprocal is reduced by 8/x(;u-l), but this is exactly the difference 

 between 2/i(2/x — 1)- and 2^. It is to be observed, however, that the ruled 

 surface in question has, as I have proved in the memoir referred to, not only 

 the two directrices for multiple lines, but has likewise a certain number of 

 generatrices which are double lines, and it is necessary to show that these have 

 no effect in depressing the degree of the reciprocal. Let there be \ such lines ; 

 now it is evident that the degree of the tangent cone is less than it otherwise 

 could have been by 2\, while I shall show that the number of cuspidal edges 

 of this cone is less by 6\ than it otherwise would have been. For we have 

 proved that the number of cuspidal edges is diminished by three times the 

 number of points where each double line meets the second polar surface, whose 

 degree is 2/^ — 2 : but since the directrices are multiple lines on that surface of 

 the degree fx — 2, subtracting twice this number from 2fJt — 2, there remain but 



