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XXVII. 



GEOMETEY OE SIJREACES DEEIYED EEOM CrBICS. 

 By EOBEET EUSSELL, M.A., E.T.C.D. 



[Eead June 26, 1899.] 



1 . It is well known that the locus of a point P, whose polar quadric- 

 with regard to a cubic surface is a curve haying its vertex at P', is a 

 surface of the fourth degree — the Hessian, and that the polar quadric 

 of P' is a cone having its vertex at P. Such points are called corre- 

 sponding points on the Hessian, and several elegant properties of this 

 surface are to he found in Salmon's " Geometry of Three Dimensions." 

 If the equation of the cubic surface be written in Sylvester's 

 canonical form 



ax^ + ly'^ + c%^ + dv'^ + ew'^ = 0, 



where x-^y-v%-\-v-\-io = 0, 



the equation of the Hessian is 



11111^ 



ax by cz dv ew 



and if the coordinates of P are x, y, %, v, w, those of P' are 



1 1 1 J_ ^ 



ax^ ly^ c% dv^ ew 



It is easy to see that the line joining PP' belongs to a congruency^ 

 that is, moves in space subject to two conditions ; and it has been 

 shown by Sir "William Eowan Hamilton that such lines are in general 

 bitangents to a surface. Several of the properties of this surface which 

 we shall denote by the symbol G are discussed in the following pages. 



2. Points on the surface. — In order to determine points on this 

 surface, we have to consider where PP' is met by consecutive lines of 

 the congruency. 



These are the points of contact of PP' with C. 



