480 The Eev. George Salmon on the Degree of the 



for we have 



a = 2(/.-l), 6 = ^^~y^^^~^l ^ = 3(m-2), 5 = 2(^- 2) (^-3). 



IV. — Theory of Higher Multiple Lines. 



In order to complete the subject, I give another independent method of 

 investigating the theory of reciprocal surfaces, which is that which I first em- 

 ployed (" Cambridge and Dublin Mathematical Journal," vol. ii. p. 66). The 

 degree of the reciprocal surface, being measured by the number of points in 

 which an arbitrary line meets that surface, is equal to the number of tangent 

 planes which can be drawn through an arbitrary line to the original surface. 

 Now the points of contact of such planes lie on the polar surface of any point 

 on the arbitrary right line. Take then the polar surfaces of any two points on 

 the arbitrary line ; then the intersection of these two surfaces of the (n — 1)" 

 degree with the given surface determines n {n — 1 )^ points. The degree of the 

 reciprocal of a surface of the n"* degree is therefore n{n — \y. Should the 

 surface have a double point, this being an ordinary point on each of the two 

 polar surfaces, will count for two intersections of the three surfaces. A double 

 point, therefore, diminishes by two the degree of the reciprocal of a surface. 



Ex. 1. The surface of the third degree. 



has four double points, namely, the four points where three of the planes 

 x, y, z, w intersect : and the reciprocal, whose equation is of the form 



is reduced by the four double points to the fourth degree from the twelfth, 

 which it otherwise would have been. 



Ex.2. A surface of the fourth degree (for example, Fresnel's wave surface) 

 may have sixteen double points, and in this case the degree of its reciprocal will 

 be reduced from the thirty-sixth to the fourth. 



'^m- 



