Surface reciprocal to a given one. ill 



It is also possible to determine a priori the number of apparent double points 

 h' belonging to the curve c'. For we can determine the rank of that system ; 

 or, in other words, the degree of the reciprocal developable. Two consecutive 

 planes which touch along the parabolic curve intersect in the line which meets 

 the surface along three consecutive points. 



Now suppose it were required to determine the degree of the surface gene- 

 rated by the lines which can be drawn to meet U in three consecutive points 

 along any curve UV, where V is of the js"" degree ; this is done by eliminating 

 between 



and the result is of the degree np(3n — 4). 



But in the present case |> = 4 (n — 2) ; and since the two lines which meet 

 in three consecutive points coincide along every point of UH, this result must 

 be a perfect square. Hence 



r'^2n(n-2) (3n - 4), 

 and 



2h' = c" -c' -r' -Z^ = n{n - 2) { 16w* - 64?i' + SOn^ - 108n + 156;. 



These are the only singularities of the reciprocal surface which I have been 

 able to determine a priori, except the number of cusps and double lines on the 

 tangent curve proper to the reciprocal surface ; these follow immediately from 

 the consideration that this line is the reciprocal to a plane section of the origi- 

 nal surface, supposed to be of the n"" degree, and having no multiple points. 



Hence 



K' = Zn{n-2); 28' = n(n- 2) (w^ - 9). 



For the sake of convenience, we assemble into a table the results already 



obtained — 



n' = n{n — 1)-. 



a' = n{n-\); 2t'=:n(?z-l) (ra-2) (w'-?i^ + ra- 12); c' = 4n(w- 1) (n-2); 



p' = n{n - 2) (Ji' - n^ + n- 12) ; a = in{n - 2) ; 



k' = 3n(n - 2) ; 2c'=n{n - 2) (w^ - 9) ; 



!' = 0, p'=2n{n-2) (nn-24:), y' = in(n-2) (n- 3) (n^ + 3n-lG) ; 



2h' = n{n - 2) (16n^ - 64?i' + 80n^ - 108n + 156). 

 VOL. xxin. 3 R 



