PROFESSOR CAYLEY ON RECIPROCAL SURFACES. 
209 
or, what is the same thing, xQ-\-y 2 l{=Q ; the Hessian has therefore along the line x— 0, 
y = 0 the same tangent plane x=Q as the surface; or it touches the surface along this 
line ; that is, the line counts twice in the intersection of the two surfaces. 
28. If instead of the right line we have a plane curve, say if the equation be 
x<p+ PkJ/=0, then the value of the Hessian is tf^-l-PT^O (viz. the second term divides 
by P only, not by P 2 ), so that, as before mentioned in regard to a conic of contact, the 
surface and the Hessian merely cut but do not touch along the curve x—^), P=0. To 
show this in the most simple manner take the equation to be x<p-\-^P 2 =0 ; let A', B', C', I)' 
be the first derived functions of <p, and (A, B, C, D), ( a , A, c, d, /* y, A, l, m, n ) the first 
and second derived functions of P ; then if in the equation of the Hessian we write for 
greater simplicity x =0, the equation is 
2A' + P«+A 2 , B'+PA +AB, C'+P^+AC, 
B'+PA+AB, P b + B 2 , P/+BC, 
C' + Py+AC, P/+BC, P<? + C 2 , 
D' + PZ+AD, Pm+BD, Pw+CD, 
The equation contains for example the term 
D'+PZ +AD 
Pm+BD 
P n + CD 
PfZ+ D 2 
= 0 . 
- (D f + PZ + AD) 2 { P \bc -/ 2 ) + P(AC 2 + cB 2 - 2/BC) } , 
dividing as it should do by P, but not dividing by P 2 ; and considering the portion 
hereof — D ,2 P(AC 2 +cB 2 — 2/BC), there are no other terms in D ,2 P which can destroy 
this, and to make the whole equation divide by P 2 ; which proves the required negative. 
29. For the off-point or singularity 0—1; this is a point on the cuspidal curve at 
which the second derived functions all of them vanish. In further explanation hereof 
consider a surface U=0, and the second polar of an arbitrary point (a, /3, y, b) ; viz. 
this is (aB a .+j3d 2/ +yB*+^d M ,) 2 U=0, or say for shortness A 2 U=0, where the coefficients 
of the powers and products of (a, /3, y, c>) are of course the second derived functions of U ; 
this equation, when reduced by means of the equations of the cuspidal curve, may acquire 
a factor A, thus assuming the form A(«P+/3Q-j-yE -)-SS) 2 =0, and if so the intersections 
of the cuspidal curve with the second polar ( = 2o- + fl, if, as for simplicity is supposed, 
there is no nodal curve) will be made up of the intersections of the cuspidal curve with 
the surface A = 0, and of those with the surface aP + |3Q+yIt + c$S = 0 each twice; the 
latter of these, depending on the coordinates (a, /3, y, ci) of the arbitrary points, are the 
points a each twice ; the former of them, or intersections of the cuspidal curve with the 
surface A=0, are the points 0, or off-points of the cuspidal curve. If there is a nodal 
curve, the only difference is that the off-points are such of the above points as do not lie 
on the nodal curve. 
30. As the most simple instance of the manner in which this singularity may present 
itself, consider a surface FP 2 + GQ 3 =0, where the degrees of the functions are/*, p, g, g, 
and therefore n =/+ 2p=g J r oq, if n be the order of the surface. This has a cuspidal 
curve P=0, Q=0 of the order pq ; the equation A 2 (FP 2 +GQ 3 ) = 0 of the second polar, 
mdccclxix. 2 Gr 
