94 
PROFESSOR CAYLEY ON THE CURVES 
44. Thirdly, we have the point pairs : — 
K, common tangent of two curves or double tangent of a curve, terminated at points 
of contact. 
L, line from cusp of a curve touching a curve, and terminated at cusp and point of 
contact. 
M, line joining cusp of a curve with cusp of a curve, and terminated by the two 
cusps. 
N, inflexion tangent terminated each way at inflexion, viz. this is a line-pair -point. 
O, cuspidal tangent terminated each way at cusp, viz. this is a line-pair-point : 
and the corresponding line-pairs : — 
K', intersection of two curves or of curve with itself (node), and terminated by the 
two tangents. 
L', intersection of inflexion tangent of a curve with a curve, and terminated by the 
inflexion tangent and the tangent at the intersection. 
M', intersection of inflexion tangent of a curve with inflexion tangent of a curve, and 
terminated by the two inflexion tangents. 
N', =0, line-pair-point as above. 
O', =N, line-pair-point as above: 
which being so, we have 
(2X2), =(2) m (2) m , 
K =nn x , 
L —zn x -\-K x n^ 
M = /S/S 1 . 
( 2 , 2 ), =( 2 , 2 ) m . 
9 K'=mm„ 
3 L' 
1 M l =u l ’ 
K=r, 
L = z(n — 3), 
M=**(*-l), 
N =i, 
0=z. 
9 k'= a, 
3 L'= /(m— 3), 
1 - 1 ), 
2 N'= *, 
1 0 '= /. 
45. Fourthly, we have the point-pairs: — 
P, tangent of a curve at its intersection with another curve or itself, terminated each 
way at the point of contact — line-pair-point. 
Q, common tangent of two curves or double tangent of a curve, terminated each way 
at one of the points of contact — line-pair-point. 
J, ut supra. 
R, cuspidal tangent terminated at cusp and at a curve : 
and the corresponding line-pairs : — 
