408 
PROFESSOR CAYLEY ON THE PROBLEM OF 
(B, D ) c 0 - y 0 - y' 0 by reciprocity, 
(a, 13) d -& + 2(c 0 — y 0 — y' 0 )-j-(X— 3)(b— (3 — ^3') = 0, 
(a, e ) e-« -s' +2(d -i -^)+(X-3)(c-y-y') = 0, 
(B, F) e 0 — s 0 — s' 0 by reciprocity, 
(« , F ) t — — <p' + 2(e 0 — g 0 — So ) + (X — 3)(d — ci — £/) = 0, 
(«> 9 ) g -X — x' + 2 ( f “P — <p')+(X— 3)(e— g -s' ) = 0, 
(B, FI) go— Xo — Xo b y reciprocity, 
and so on. 
26. The mode of obtaining these equations appears ante , Nos. 5 and 6, but for greater 
clearness I will explain it in regard to a pair of the equations, say those for (a, e), ( a , D). 
Regarding a as given, we draw from a the tangents «Bc, touching at B and besides inter- 
secting at c (viz. the number of tangents is =X — 2, and the number of the points c is 
= (X — 2)(a— 3)); from each of the positions of ewe draw to the curve the (X— 3) 
tangents cDe touching at D and intersecting at e ; the whole number of these tangents 
is =(X — 2)(a* — 3)(X— 3) ; and this is also the number of the points D, but the number 
of the points e is = (X — 2) (x — 3) (X — 3) (x — 3). Now this system of the 
(X — 2)(x— 3)(X— 3) tangents is the curve 0 of the general theory (ante, Nos. 3, 4), 
viz. the curve 0 (which does not pass through a) intersects the given curve in the 
three classes of points c, D, e, the number of intersections at a point e being =1, 
at a point D being =2, and at a point e being=X — 3. And we have thus the equation 
e — s — s'-l-2(d— A— ci') + (X — 3)(c — y — y') = 0, 
where e, cl, c are the numbers of united points and (s, s'), (ci, &'), (y, y') the correspondences 
in the three cases respectively. 
27. Observe that we cannot, starting from a, obtain in this manner the equation for 
the number of the united points (a, D) ; for we introduce per force the points e, and 
thus obtain the foregoing equation for (a, e). But starting from D, the tangent at this 
point besides intersects the curve in (a’— 2) points, each of which is a position of c ; and 
from each of these drawing a tangent cB« to the curve, we have the curve 0 consist- 
ing of these (a 1 — 2)(X — 3) tangents, not passing through D, but intersecting the given 
curve in the three classes of points c, B, a, viz. the number of intersections at each point 
c is —H — 3, at each point B it is =2, and each point a it is =1; and we have thus 
the equation 
(d— 5_S r )+2(c 0 — y,— y„)+(X— 3)(b — 0— /3')=0, 
where the numbers (d, S, &'), (c 0 , y 0 , y' 0 ), (b, (3, /3') refer to the correspondences (D, a), 
(D, B), and (D, c) (or what is the same thing (a, B)) respectively. 
28. Correspondence (a, B). 
We have 
(3=X-2, (3'=x—2, 
and thence 
b=a+X— 4+2A 
= — 3a— 3X-}-2|, 
