Surface reciprocal to a given one. 473 



In like manner twice the number of double points in a section of the reciprocal 

 surface is 



n(n-l)^jw(n-l)^-lj-n(n-l)-12n(n-l) (n-2) 



= n{n-l) (n-2) (n' -«' + n - 12). 



Hence, then, by the same reasoning, the reciprocal surface has a double line 



whose degree is 



lb' = n{n - 1) {n - 2) (n' - n"" + ?i - 12). 



The importance of these results justifies us in giving another and more 

 direct investigation of them. To every double or stationary point on the reci- 

 procal surface corresponds a double or stationary tangent plane on the original 

 surface. Let us then investigate directly the conditions fulfilled by the point 

 of contact of a double tangent plane to a given surface. We investigate these 

 by the same method by which we investigated the condition that a line should 

 touch a surface. If the co-ordinates of three points be a\yxZxWi., ^'ii/iZiW,, xyzic, 

 then those of any point on the plane through the three points will be 

 Xx ■'r fiXi -V va'2, Xy + fxyi + vy^, 7^ -V ixz^ + vz-^, Xiv + fxwi -1- viCi ; and if we 

 substitute these values for xyzw in the equation of the surface, we shall have 

 the relation which must be satisfied for every point where this plane meets the 

 surface. Let the result of this substitution be [t/^] = ; it may be written — 



\"t/+ \"-VA,f7-t- V-VAaC/'-f ^ (/xAi -f vAif U + &C., 



where 



/ d d d d\ d d d d 



^^ = [''^Tx^y'Ty^''Tz-^'"^d^} ''^=''^Tx^y'Ty^''dz^'''-'Tw 



Now since the tangent plane to a surface always meets it in a section having a 

 double point, the condition that the plane joining the three given points should 

 touch the surface, is found by eliminating X, ^, v between 



dUJ]_ d[lJ]_ d[U]_ 

 dX ~ ' dfX ~ ' du ~^' 



or, in other words, the discriminant of the equation [U]. If we suppose two 

 of the points fixed, and consider the third to be variable, then the condition so 

 found will be the equation of the tangent planes to the surface, which can be 



