344 
A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 
[411 
46. The problem is as follows: given a curve of the order m with 8 nodes and 
k cusps; it is required to find the number of the conics, centre on a given line, 
and an axis coincident in direction with this line, which have with the given curve 
a 4-pointic intersection, or contact of the third order. This may be solved by means 
of formulae contained in my “ Memoir on the Curves which satisfy given Conditions,” 
Phil. Trans, vol. clviii. (1868), pp. 75—144; see p. 88; [406]. 
Taking x = 0 for the given line, the conic (a, b, c, f g, li\x, y, 1) 2 = 0 will have 
its centre on the given line and an axis coincident therewith, if only h = 0, g = 0; and 
denoting these two conditions by 2X, it is easy to see that we have 
(2X= 1, (2X:/) = 2, (2X • //) = 2, (2X///) = 1. 
But in general if the conic satisfy any other three conditions 3Z, then the number of 
the conics (2X, 3Z) is 
= a' ( - + £8) 
+ ( — t a + + — I$) 
+ i ( 2 a - i/ 3 )> 
where a, ¡3, 7, 8 denote (2X.\), (2X :/), (2X •//), (2X ///), viz. in the present case the 
values are 1, 2, 2, 1 respectively, and where a, /3y denote (3 Z:), (3 Z •/), (3 Z//) 
respectively. 
47. Substituting for a, /3, 7, 8 their values, the number of the conics in question 
is =^(3', that is = ^(SZ ■ /). Suppose that 3Z, or say 3, denotes the condition of a 
contact of the third order with a given curve (m, 8, k), or say with a given curve 
(m, n, a) (m the order, n the class = m 2 — m — 28 — 3/c, a = 3n + k), then we have 
(3 : ) = — 4m — Sn + 3a, 
(3 • /) = — 8m — 8n + 6a, 
(3 // ) = — 3m — 4>n + 3a; 
and from the second of these the number of the conics in question is = — 4m — 4n + 3a, 
that is, it is = — 4m + bn + 3/c, or finally it is = om 2 — 9m — 108 — 12/c. 
Hence, assuming that the 2m lines each counts 6 times(*), 
Order of surface = 2m 
Order of flecnode surface =11 (2m —24) or 22m — 24 
Order of intersection = 44m 2 — 48m 
Nodal curve, \{w? — m)+8 circles, 11 times 
Cuspidal curve k circles, 27 times 
Circles of contact m 2 — m — 28 — 3/c, 
Lines of contact 2m , 6 times 
Flecnodal curve, 5m 2 — 9m — 108 — 12/c circles each twice 
22m 2 - 22m + 448 
+ 54 k 
2m 2 — 2m — 48 — 6/c 
4-12 m 
20m 2 - 36 m - 408 - 48* 
44?/i 2 — 48m. 
1 See foot-note p. 343: the like remark applies to the present terms in m, which cannot be got rid of 
by an alteration of the numbers 22 and 27 to which the investigation relates.