The collected mathematical papers of Arthur Cayley.

206

ON THE CURVES WHICH SATISFY GIVEN CONDITIONS.

[406

the two expressions of the same quantity being of course equivalent in virtue of the
relations between (jp, v, p, a) and (/p', v', p', a') respectively.
The characteristics of (X, Z), (X, 2Z), (X, 3Z) are at once deducible from the
before-mentioned expression ap + O3v of (X, 4Z).
39. Zeuthen's investigations are based upon the before-mentioned theorem, that
in a system of conies (4X), characteristics (p, v), there are 2pJ-v point-pairs and
2v - line-pairs. If in the given system the number of point-pairs is =X and the
number of line-pairs is =w, then, conversely, the characteristics of the system are
I= X vs), z,=4( i+2 ).
And by means of this formula he investigates the characteristics of the several systems
of conies which satisfy four conditions (4X) of contact with a given curve or curves,
viz. these are the conies
(1) (1) (1)(1), (1, 1)(1)(1), (1, 1) (1, 1), (1, 1, 1)(1), (1, 1, 1, 1),
(2) (1)(1), (2)(1, 1), (2, 1) (), (2, 1, 1),
(2) 2), (2, 2)
(3), (3, 1),
(4)
where (1) denotes contact of the first order, (2) of the second order, (3) of the third
order, (4) of the fourth order, with a given curve; (1)(1) denotes contacts of the first
order with each of two given curves, (1, 1) two such contacts with the same given
curve, and so on. A given curve is in every case taken to be of the order vn and
class n, with s nodes, Kc cusps, T double tangents, and t inflexions (mi, n,, 8,, ci, T7, ^1;
m2, n2, &c., as the case may be). The symbols (1), &c. might be referred to the
corresponding curves by a suffix; thus (l), would denote that the contact is with a
given curve of the order m (class n, &c.); but this is in general unnecessary.
40. In a system of conies satisfying four conditions of contact, as above, it is
comparatively easy to see what are the point-pairs and line-pairs in these several
systems respectively; but in order to find the values of X and ws, each of these pointpairs and line-pairs has to be counted not once, but a proper number of times; and
it is in the determination of these multiplicities that the difficulty of the problem
consists. I do not enter into this question, but give merely the results.
41. For the statement of these I introduce what I call the notation of Zeuthen's
Capitals. We have to consider several classes of point-pairs and the reciprocal classes
of line-pairs. A point-pair may be described (ante, No. 31) as a terminated line, and
a line-pair as a terminated point; and we have first the following point-pairs, viz.:
A, line terminated each way in the intersection of two curves or of a curve with
itself (node).
B, tangent to a curve, terminated in a curve, and in the intersection of two(
curves or of a curve with itself.