411] 
A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 
341 
w 3 (w + 2z) = 0, that is in the line x = 0, w = 0 three times, and in the line x=0, w+2^=0 
(that the section consists of right lines is of course a speciality, and it is clear that 
considering 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 = z* + &c., and the line 
w + 2z = 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, z = 0. And 
similarly the tangent plane at the other of the two points is y = 0, meeting the surface 
in the curve y = 0, w 3 (w + 2z) = 0, that is in the line y = 0, w = 0 three times, and in 
the line y — 0, w 4- 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 
/x 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 g the tangent of the cuspidal 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. 
Article Nos. 40 and 41. Application to a Class of Surfaces. 
40. Consider the surface PP 2 4- GR-Q 3 = 0, where f p, g, r, q being the degrees 
of the several functions, and n the order of the surface, we have of course 
v = f+2p=g + 2 r + 3 q. 
There is here a nodal curve, the complete intersection of the two surfaces P = 0, R = 0 ; 
hence b =pr, k = \pr (p - 1) (r- 1), = %b(b-p-r + 1); t = 0; whence (q) = p r (p + r- 2). 
There is also a cuspidal curve the complete intersection of the two surfaces P = 0, Q = 0; 
hence c = pq, h = %pq (p - 1) (q - 1), = \c (c-p - q + 1); whence (r) = pq (p + q - 2) : I 
have written for distinction (q), (r), to denote the q, r of the fundamental equations. 
The two curves intersect in the pqr points P = 0, Q = 0, R = 0, which are not 
stationary points on either curve; that is, /3 = 0, 7 = 0, i=pqr. 
There are on the nodal curve the j = (f+g)pr pinch-points P= 0, P = 0, P = 0, 
and 6r = 0, P = 0, P = 0. There are on the cuspidal curve 6 =fpq off-points F= 0, 
P = 0, Q = 0 ; and there the gpq singular points G = 0, P = 0, Q = 0. I find that these 
last, and also the 0 points each three times, must be considered as close-points, 
that is, that we have y = (g + ; I/)p?-