518 On the Skew Surface of the Third Order. 



and the second form puts in evidence the simple line 



(X-*=0, Y-£=0). 



But to obtain a more convenient form, write for a moment 

 X— wzZ = P, Y— nZ = Q. The equation is 



(a 3 + /3 3 )PQ-*/?(P 3 + Q 3 ) 



+ Z{(P*u-Q?/3){ncc-m/3)-VQ l {mc*? + nP*)}=0, 



or, as this may be written, 



= (« 3 + /3 3 )PQ + ( a 2 P- / S 2 Q)Z(P/i-Qm) 



+ a/3{-P 3 -Q 3 -Z(mP 2 + rcQ 2 )}=0; 



or, observing that X=P + mZ, Y=Q + wZ, and thence 

 PY-QX=Z(P/i-Qm), 

 XP 2 + QY 2 = P 3 + Q 3 + Z (mP 2 + wQ 2 ), 



the equation becomes 



(a 3 + /3 3 )PQ+ (a 2 P-^Q)(PY-QX) -«/3(P 2 X+ Q 2 Y) =0, 

 or, what is the same thing, 



( a P 2 -/3Q 2 )(aY-/3X) + PQ(« 3 + /3 3 - a 2 X--/3 2 Y)=0 j 



whence, making a slight change in the form, and restoring for 

 P, Q their values, the equation is 



{ a (X-mZ) 2 -/3(Y-rcZ) 2 } |a(Y-/3) -/3(X-«) } 

 -(X-mZ)(Y-7iZ){ a 2 (X-a)+/3 2 (Y-/5)} =0, 

 a form which puts in evidence as well the simple line 



(X-*=0, Y-/3=0) 

 as the nodal line (X— twZ=0, Y — wZ=0). If Z=0, we have 



(*X 2 -/3Y 2 )(«Y-/3X)-XY{a 2 (X-«) +/3?{Y-/3)\ =0, 

 which is in fact the cubic curve 



(«?+/3 3 )XY-*l3(X 3 +Y 3 )=0. 

 Reverting to a former system of equations 



nx — my — B# -f- Ay = 0, 



B(*-«)-A(y-/S)=0, 

 or, as these may be written, 



Bx — Ay = nx — my, 



B« — A/3 = nx — my, 

 we find 



B{(3x— ay) = (/3—y)(nx—my), 



A(j3x — cty) = (« — x) {nx — my) ; 



