212 
PROFESSOR CAYLEY OUST RECIPROCAL SURFACES. 
and the equation thus is 
(9 4 , Qhv, — p 2 -\-q 2 -{-2Qqiv-\-d 2 w 2 X2> 2 — q 2 — 2Qqw, 23 2 ^)=0, 
or, what is the same thing, 
(1, dtp, -f+f+Mqw+QWXf-tf-2 Qgw, 2qf=0; 
viz. this is 
il )2 ~~ Q 2 — ■2Qqw) 2 -\-iQqw{p 2 — q 2 — 2Qqw)-{-iq 2 (— jq 2 -{-q 2 -\-2Qqw+Q 2 w 2 ) = 0 ; . 
or reducing, it is 
(q> 2 — <f)( qf — 5 q 2 ) + 8 Qq 3 w — 0, 
the equation of the section in terms of the coordinates q>, q , w. The equation is satis- 
fied by the values q>=0, ^=0 which belong to the assumed point (1, O 2 , 0, 9) of the 
conic, and in the vicinity of this point we have y> 4 +89y 3 w=0, which is a triple branch of 
the form y z =x i , the tangent ^=0 being, it will be observed, the tangent of the conic. 
But at the close-points, or when 9 = 0 or 0=co , the transformation fails ; and these points 
must be considered separately. 
37. At the first of these, viz. the point 2=0, w= 0, x=0, the tangent plane of the 
surface or cuspidal tangent plane is x=0, and this meets the surface in the curve #=0, 
w 3 (w+2;z)=0, that is in the line x=0, w=0 three times, and in the line x=0, w-{-2z=0 
(that the section consists of right lines is of course a speciality, and it is clear that con- 
sidering in a more general surface the section as defined by an equation in ( w , z, y), the 
line w=0 represents the tangent to a triple branch w 3 = 2 4 +&c., and the line w + 22=0 
the tangent to a simple branch) ; these lines are each of them, it will be observed, 
distinct from the tangent to the cuspidal conic, which is x=0, 2 = 0. And similarly 
the tangent plane at the other of the two points is y= 0, meeting the surface in the 
curve y— 0, zt? 3 (w -f- 2^) = 0 , that is in the line y= 0, w=0 three times, and in the line 
y— 0, w-\-2z=0. 
38. The close-plane or reciprocal singularity ^=1 is (like the pinch-plane) a torsal 
plane, meeting the surface in a line twice and in a residual curve ; the distinction is that 
the line and curve have an intersection P lying on the spinode curve ; the close-plane is 
thus a spinode plane ; it meets the consecutive spinode plane in a line passing through 
P, and which is not the tangent of the residual curve. In the reciprocal figure, the 
reciprocal of the close-plane is on the cuspidal curve, and is a close-point ; the reciprocal 
of the point P is the cuspidal tangent plane ; that of the line u> the tangent of the cus- 
pidal curve ; that of the tangent of the residual curve the cotriple tangent ; that of the 
torsal line the cotangent. 
39. The torsal line of a close-plane is not a mere torsal line ; in fact by what precedes 
it appears that the surface and the Hessian intersect in this line, counting not twice 
but three times, and it is thus that the reduction in the order of the spinode curve 
caused by the close-plane is =3. 
