Surface reciprocal to a given one. 481 



If the tangent cone at the double point break up into two planes, then, such 

 a double point diminishes the degree of the reciprocal by three ; since the two 

 polar surfaces both touch the line of intersection of the two planes, which passes 

 through three consecutive points of the given surface. Should the two planes 

 coincide, the degree of the reciprocal will be diminished by six. 



It remains to trace by this method the effect of a double or other multiple 

 line in depressing the degree of the reciprocal. In this case each of the polar 

 surfaces will pass through the line in question, and we are led to the problem, 

 " Three surfaces have common a certain line, — in how many other points do 

 they intersect ?" It will be convenient to commence with the case when the 

 multiple line is a right line. 



Before we discuss this problem, however, it is useful to examine carefully 

 the nature of the intersection of the curve of simple contact with the double 

 line. If a surface have a double line, the tangent cone to it from any point con- 

 sists of the plane containing the point and the double line (reckoned twice), 

 and of the cone of simple contact whose degree is n- — n — 1. If now we con- 

 sider the intersection of this latter cone by the plane in question, it is evident 

 that (?i — 2) {n— 3) of the lines of intersection are the tangents from the point 

 to the curve (of degree n — 2), in which the plane cuts the given surface ; and 

 before investigation it was natural to think that the remaining 4 (w — 2) lines 

 must be the lines (reckoned four times) to the (n — 2) points, where the double 

 line meets this curve. Let, however, the equation of a surface containing a 

 double line be 



Ax- + Bxy + Cy- + Dy^ + &c. = 



(where A,B, C are functions of the co-ordinates of the degree n — 2), then the 

 discriminant of this equation, with regard to ?/, may represent any tangent cone 

 to the surface, since the plane y is arbitrary. This discriminant will contain x^ 

 as a factor, and if we divide by x^, and then make x = 0, the remainder will 

 be (B^ — 4AC) C^<f), where is the discriminant of the equation 



C+Dy + &c.^ 0. 



This proves that the section of the simple tangent cone by the plane a consists 

 of the lines which touch the plane section, of the lines (reckoned twice) to the 

 points where this section is met by the double line, and besides of lines to what 



