PROFESSOR CAYLEY ON RECIPROCAL SURFACES. 
213 
Application to a Class of Surf aces. Article Nos. 40 & 41. 
40. Consider the surface FP 2 + GR 2 Q 3 = Q, where/, p, g, r, q being the degrees of the 
several functions, and n the order of the surface, we have of course n =f-\- 2p =g + 2r + 3q. 
There is here a nodal curve, the complete intersection of the two surfaces P=0, R=0 ; 
hence b=pr, 1c=^pr(p—V){r—l), =\b(b-p—r-\- 1); whence (q)*=pr{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—l)(q—\) = \c(c-p—q-\-l) ; whence (r)*=pq(p-\-q— 2). 
The two curves intersect in the pqr points P=0, Q=0, R=0, which are not stationary- 
points on either curve; that is, j3 = 0, <y= 0, i=pqr. 
There are on the nodal curve the j=(f-\-ff)W pinch-points F=0, P = 0, R=0, and 
G=0, P = 0, R=0. There are on the cuspidal cuxveb=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 6 points each three times, must be considered as close-points, that is, that we have 
X={(jAof)pq. 
41. We ought then to have 
b{n- 2) =*, 
c{n— 2) =2a-\-6; 
2{q) + 3i+j =2g, 
3(r) + c +2«+%=5<r-f 43 ; 
the first two of which give §, <r, and then, substituting their values, the other two 
equations should become identities. In fact, attending to the values pr=b, pq=c, the 
equations become 
2b(p+r-2)+Sbg+b(f+g)=2b(n-2), 
3c(p-\-q-2)-\-c+2cr-jrb(g+3f)=%{c(n-2)—cf} +4(f. 
The first of these is 
2n=2p + 2r+Sq+f+g, =(2p+f)+(2r+3q+g), 
and the second is 
ln=3p+3q+2r-\-g+%f, =%{2p+f)+{2r-\-3q+g), 
so that the equations are satisfied. 
The Flecnodal Curve. Article No. 42. 
42. A point on a surface may be flecnodal, viz. the tangent plane may meet the 
surface in a curve having at the point a flecnode, that is, a node with an inflexion on one 
of the branches. Salmon has shown that, for a surface of the order n without singu- 
larities, the locus of the flecnodal points, or flecnodal curve, is the complete intersection 
of the surface by a surface of the order 11 n — 24, which may be called the flecnodal 
surface, the order of the curve being thus =n{Y\.n— 24). I have succeeded in showing, 
in a somewhat peculiar way by consideration of a surface of revolution, that if the surface 
of the order n has a nodal curve of the order b, and a cuspidal curve of the order c, then 
* I have written for distinction (q), (»•), to denote the q, r of the fundamental equations. 
