EussELL — Geometry of Surfaces derived from Cuhics. 475 

 the equations of the cones being 



{x + y + %) {oci/z + cazx + abxxj) - abc ^-^ — — ^::^ — xy% = 



de 



{x^-y + z) {lcy% + catx + alxy) - ale — — r^— ^ys = 



(aj6 



(11) 



These cones hare three generators in common 



0, 1,-1; -1, 0, 1; 1,-1,0, 



and touch along the three corresponding to them 



1, 0, 0; 0, 1, 0; 0, 0, 1. 



15. Tlie sections of the litanyent surface hy the li-pla7ies. — Let P be 

 any poiat on the curve (8), P' its correspondent, Z7the third point in 

 which the line joining them meets the cuxye, and U' the point in 

 which it meets v = 0, w = of the two contacts of PP' with C, Tia 

 on the Cayleyan (Art. 14), and T' is the harmonic conjugate of If', 

 and it is not difficult to prove that the locus of T' for points Z7' 

 situated on the line v = 0, w = is a cubic curve ^ which passes 

 through the nodes 



0,1,-1,0,0; -1,0,1,0,0; 1,-1,0,0,0. 



The tangent plane to (7 at T' is the plane PP'V{Ait. 7), which 

 in the case under consideration reduces to 



vjd-wje= 0; 



we see, therefore, that this bi-j)lane touches C along the whole length 

 of the curve (11). 



"We have now accounted for a sextic section and the square of a 

 cubic, and the remaining curve of the 12th degree is the locus of 

 points in the plane 



vj^ -^vJe = 0, 



1 This cubic curve may also be obtained by expressing that the polar of x, y, z, v, 



-^—3: V touches the IcYZ + caZX + ahXY=0, the tangent cone to the Hessian 



at the node. The result is 



'^ he {ax"^ — dv'^) + \/ ca [by''' — dv-) + \f ab [cz^ — dv-) = 



/ ^/.i^ J. (12) 

 ■which, -when rationalized, has v as a factor and leaves a cubic. 



