68 
INVESTIGATIONS IN CONNEXION WITH CASEY’S EQUATION. 
[395 
1 
point of J. The number of the binodal curves il is consequently equal to that of 
the nodes of A, and the number of the cuspidal curves il is equal to that of the 
cusps of A; we have consequently to find the Pltickerian numbers of the curve A ; 
and this Prof. Cremona accomplishes by bringing it into connexion with the foregoing 
curve 2, and making the determination depend upon that of the number of the conics 
which satisfy certain conditions of contact in regard to the curve 2. 
Consider, as corresponding to any given point (/, g, h) whatever, the conic 
f q h> 
- + “+- = 0 which passes through three fixed points, the angles of the triangle 
x y z 
x = 0, y = 0, z = 0. For points (/, g, h) which lie in an arbitrary line Af+Bg+Ch = 0, 
the corresponding conics pass through the fourth fixed point x : y : z = A : B : G. 
Assume for the moment that to the points (f, g, h) which lie on the foregoing curve 
A, correspond conics which touch the foregoing curve 2. Then 1°. to the points of 
intersection of the curve A with an arbitrary line, correspond the conics which pass 
through four arbitrary points and touch the curve 2; or the order of the curve A 
is equal to the number of the conics which can be drawn through four arbitrary 
points to touch the curve 2; viz. if in be the order, n the class of 2, the number 
of these conics is =2m + n, or substituting for m, n the values 3(ft — l) a and 3ft (ft — 1) 
respectively, the number of these conics, that is the order of A, is = 3 (ft — 1) (3ft — 2). 
2°. To the nodes of A correspond the conics which pass through three arbitrary points 
and have two contacts with 2, viz. if m be the order, n the class, and k the number 
of cusps of 2, then the number of these conics is = ^ (2m + n) 2 — 2in - on — §/c, or 
substituting for m, n their values as above, and for tc its value = 12 (ft — 1) (n — 2), 
the number of these conics, that is, the number of the nodes of A, is found to be 
= f ( n ~ 1) (27 ft 3 — 63?i 2 + 22ft + 16). 
3°. To the cusps of A correspond the conics which pass through three arbitrary 
points, and have with 2 a contact of the second order; the number of these (m, ft, k 
as above) is = 3ft -f k, or substituting for n and k their values as above, the number 
of these conics, that is the number of the cusps of A, is = 3 (ft — 1) (7ft — 8). We 
have thence all the Pliickerian numbers of the curve A, viz. these are 
Order = 3 (ft — 1) ( 3ft — 2), 
Class = 6 (ft — l) 2 , 
Nodes = | (ft — 1) (27ft 3 — 63ft 2 + 22ft + 16), 
Cusps = 3(ft—1)( 7ft —8), 
Double tangents = § (ft — 1) (12ft 3 — 36ft 2 + 19ft + 16), 
Inflexions =12 (ft — 1) ( ft — 2), 
and as a verification it is to be observed, that the deficiency of the curve A is equal 
to that of the curve J, viz. it has the value £ (3ft - 4) (3ft — 5). The foregoing 
numbers include the result that the number of the binodal curves 
J(fU)+V(gV)+<j(hW) = o, 
is 
= | (ft — 1) (27ft 3 — 63ft 2 + 22ft + 16).