516 Mr. A. Cay ley on the Skeiv Surface 



we have 



\{na-ml3)Z+l3X-aY\(x-X)-Z(x~*){nX-mY)=:0, 



\{n*-mP)Z+l3X-aY\(y-Y)-Z{y-f3){nX--mY)=0, 



which give 



{2tf=X{(wa-m£)Z + /3X- a Y} -«Z(nX-mY) 



= - mZ/3X + X(/8X - «Y) + mZ«Y 



= (X-mZ)(/3X-«Y), 

 and 



%=Y{(rc*-m/3)Z + /3X- a Y} -0Z(nX-roY) 



=/iZ«Y+ Y(/3X-aY) -nZ£X 

 = (Y-nZ)(/3X-aY), 

 where 

 12 = (w*-m/3)Z -f (/3X-aY) - Z(rcX-mY) 



=/3(X-mZ) -a(Y-nZ) -Z {rc(X-mZ) -m(Y-wZ) } 



= 08-fiZ)(X-mZ)-(a-roZ)(Y-nZ). 



So that 



(X - mZ) oax - «Y) 



a?= 



y= 



(/3-7lZ)(X-mZ)-( a -tt^Z)(Y-rcZ) , 

 (Y-wZ)(/3X-aY) 



(/3-7iZ)(X-mZ)-(a-mZ)(Y-7iZ)' 



which equations give the coordinates (a?, y) of the point in which 

 the generating line through the point (X, Y, Z) of the surface 

 meets the cubic 



(a 3 + /3 3 )#?/- {x 3 + ?/ 3 )a/3 = 0. 



Substituting these values of (x } y) in the equation of the cubic, 

 we obtain the equation 



(a 3 + /3 3 ) (X - mZ) ( Y - nZ) { 08 - »Z) (* - mZ) - (a - roZ) (Y - nZ) }■ 



-a^(/3X-aY){(X-7wZ) 3 + (Y-wZ) 3 }=0; 



or, as it may be written, 



(a 3 4-/8 3 )(X-mZ)(Y-nZ){/3(X-mZ)-a(Y-7iZ)J 



+ ( a 3 + /3 3 )(X-mZ)(Y-7iZ)Z(mY-rcZ) 



- a) e(/3X-aY){(X~mZ) 3 +(Y-7iZ) 3 } =0. 



This equation contains, however, the extraneous factor • 



/3(X-mZ)-a(Y-7iZ). 



which, equated to zero, gives the equation of the plane through 



