482 The Rev. Geokge Salmon on the Degree of the 



I have called the cuspidal points on the double line (see " Cambridge and Dub- 

 lin Mathematical Journal," vol. ii. p. 72), viz., the points at which the two tan- 

 gent planes to the surface coincide ; for these points are determined by the 

 condition B- = 4 J. C* It will be found, in like manner, that if the surface have 

 a triple right line, there will be on that right line 4 (n — 3) points, at which 

 two tangent planes coincide, and that the lines to these points are edges of the 

 cone of simple contact ; and, generally, that if the surface have a right line of 

 the degree ;? of multiplicity, there will be on that right line 2 (p - 1) {n - f) 

 points, the lines to which are edges of the cone of simple contact. 



We return now to the case of a surface having a double line. Any two 

 polar surfaces will then pass through that line, and the question is, in how 

 many points not on that line will they intersect the original surface. 



We give first the solution of the question. Three surfaces, whose degrees 

 are a, h, c, have a right line common, — in how many other points do they in- 

 tersect ? The intersection of the first two surfaces consists of that line and of 

 a curve of the degree ah - 1, which latter meets the third surface in c{ah - 1) 

 points. But a certain number of these points will lie on the right line in 

 question. In fact, let ^.r -f % = 0, Cx + Dy - 0, represent two surfaces 

 having a right line in common, and of the degrees a and b respectivel)^ then at 

 the a + b -2 points, where the right line xy meets the surface AD = BC, the 

 two surfaces will have the same tangent plane, and therefore (see "Cambridge 

 and Dublin Mathematical Journal," vol. v. p. 34) this right line will meet the 

 remaining {ab-l) curve of intersection. Subtracting this number (a + b~2) 

 from the number c{ab-l) previously found, we learn, that if three surfaces 

 have a right line common, this will replace a + b+ c -2 of the points of 

 intersection. 



Let us now apply this theory to the case with which we are concerned. 

 The two polar surfaces intersect in the double right line, and also in a curve of 

 decree (n — lf — l, which, according to the theory just explained, meets that 

 right line in 2n — 4 points ; namely, the cuspidal points on the double line. 

 Since at each of these points the two polar surfaces will touch the original sur- 



* It was in the manner given in the text that I was led in the year 1846 to the consideration 

 of these cuspidal points. It is obvious that this includes a theorem concerning discriminants 

 which has been since stated by M. Joachimstal. 



