METHOD OF DEDUCING A SURFACE FROM A PLANE FIGURE. 413 



The class of <£, that is to say, the number of the tangent planes drawn 

 through two arbitrary points in space, is equal to that of the intersections of 

 the Jacobians of two linear twofold systems of sextics K. The Jacobian of 

 such a system is of the order 3(6 - I) = 15, and passes 3.2 - 1 = 5 times through 

 each fundamental point ; but the curve is clearly included in the Jacobian, 

 therefore, this latter will break up into a fixed curve, 0, and a variable one, 

 being of the order 15 — 12 = 3, and possessing the multiplicity 5 — 4 = 1 at the 

 fundamental points. So the residual Jacobian is a cubic curve 123456. Two 

 such curves meet in 9 — 6 = 3 more points ; hence the class of is 3. 



The surface <I>, being of the twelfth order and third class, and having 

 twenty-seven nodal right lines and a cuspidal curve of the twenty-fourth degree, 

 is the reciprocal of the general cubic surface. It was very easy to foresee this 

 conclusion, in accordance with the (1, 1) correspondence between any surface 

 and its reciprocal. But I wished to give an instance of the method of deducing 

 a (unicursal) surface from a plane figure assumed as its representative. 



37 FEB 



