462 The Rev. George Salmon on the Degree of the 



Let n denote the order of the surface, that is to say, the number of points 



in which an arbitrary line meets it. 

 n', the class of the surface, that is to say, the number of tangent planes 



which can be drawn through an arbitrary right line, 

 a, the order of the tangent cone, drawn from any point to the surface. 

 5, the number of its double edges. 

 A-, the number of its cuspidal edges. 



l\ the order of any double curve which may exist on the surface. 

 k, the number of its apparent double points ; that is to say, the number of 



lines which can be drawn from an arbitrary point, twice intersecting the 



double curve. 

 t, the number of triple points on the double curve, which are also triple 



points on the surface. 

 c, the order of any cuspidal curve which may exist on the surface. 

 h, the number of its apparent double points. 

 /3, the number of intersections of the double and cuspidal curve which are 



stationary points on the latter. 

 7, the number of intersections which are stationary points on the former. 

 /, the number of intersections which are singular points on neither. 

 p, the number of points where the double curve b is met by the curve of 



contact a. 

 <7, the number of points where the cuspidal curve c is met by a. 

 Let the same letters accented denote the corresponding singularities of the 



reciprocal surface. 



Having made this preliminary enumeration, I give in the first place the 

 theory of the reciprocal of a surface of the n"" degree, having no multiple line. 

 To know the nature of a section of the reciprocal surface, it is only necessary 

 to know the nature of a tangent cone to the original surface. Having the degree 

 of this cone, and the number of its double and cuspidal edges, we shall know 

 at once by M. Plucker's formula3, the characteristics of its reciprocal, namely, 

 the section of the reciprocal surface. 



I investigate the equation of the tangent cone by the method which M. Joa- 

 CHiMSTAL has given for plane curves ("Crelle," vol. xxxiv. p. 24). Let the 

 quadriplanar co-ordinates of two points be xyzw, x'y'z'w', then — 



