380 
PEOFESSOE CAYLEY ON THE PEOBLEM OF 
points; that is, the number of the points c is =(A— 2)(a— 3), and so in other cases; 
the calculation is always immediate, and the only difference is that, instead of a factor 
a or A, we have such factor in its original form or diminished by 1, 2, or 3, as the case 
may be. Similarly starting from g, considered as a given point on the curve g(=a), we 
find yj the number of the corresponding points a ; thus in the case where the curves are 
all distinct curves, we have %'=F#D (==%;); and so in other cases we find the 
value of y\ The points (a, g) have thus a (%, y) correspondence, where the values of 
X, x are found as above. 
2. There will be occasion to consider the case where in the triangle aFcDeF (or say 
the triangle dBcDeFa) the point a is not subjected to any condition whatever, but is a 
free point. There is in this case a “ locus of a” which is at once constructed as follows : 
viz. starting with an arbitrary tangent aBc of the curve B, touching it at B and inter- 
secting the curve c in a point c ; through c we draw to the curve D the tangent cDe, 
touching it at D and intersecting the curve e in a point e ; and finally from e to the curve 
F the tangent eFci , touching it at F and intersecting the original arbitrary tangent aBc 
in a point a, which is a point on the locus in question. We can, it is clear, at once de- 
termine how many points of the locus lie on an arbitrary tangent of the curve B (or 
of the curve F). 
3. The general form of the equation of correspondence is 
_p(a— a — a!)-\-q(b — [3 — j3')-j- .... =kA* ; 
viz. if on a curve for which twice the deficiency is =A we have a point P corresponding 
to certain other points P', Q', . . . in such wise that P, P' have an (a, oi) correspondence, 
P, Q! a (f 3 , (3 1 ) correspondence, &c. ; and if (a) be the number of the united points (P, P'), 
(b) the number of the united points (P, Q'), &c. ; and if moreover for a given position of 
P on the curve the points P', Q' . . . are obtained as the intersections of the curve with a 
curve 0 (depending on the point P) which meets the curve Jc times at P, jp times at each 
of the points P', g times at each of the points Q 1 , &c. ; then the relation between the 
several quantities is as stated above : see my “ Second Memoir on the Curves which 
satisfy given conditions,” Philosophical Transactions, vol. 159 (1868), pp. 145-172. 1 
omit for the present purpose the term “ Supp.,” treating it as included in the other 
terms. 
4. In the present case we consider, as already mentioned, the unclosed trilateral 
aBcDeFg, where the angles a, g are on one and the same curve a{ — g) (the curve in the 
general theorem); and the curve 0 is the system of lines eFg which by their intersec- 
* To avoid confusion with the notation of the present memoir, I abstain in the text from the use of D 
as denoting the deficiency, and there is a convenience in the use of a single symbol for twice the deficiency ; 
hut writing for the moment D to denote the deficiency, I remark, in passing, that perhaps the true theoretical 
form of the equation is 
&(0 — D — D)+p(a — a — a!) + q(\>— (3 — /3')+ .. . =0; 
viz. the point P is here considered as having with itself a (D, D) correspondence, the number of the united points 
therein being=0. 
