The collected mathematical papers of Arthur Cayley.

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406 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 207
C, common tangent of two curves, or double tangent of a curve, terminated each
way in a curve.
D, inflexion tangent of a curve terminated each way in a curve:
and the corresponding line-pairs, viz.:
A', point terminated each way in the common tangent of two curves or the double
tangent of a curve.
B', point of a curve terminated by the tangent of a curve, and by the common
tangent of two curves or double tangent of a curve.
C', intersection of two curves, or of a curve with itself (node), terminated each
way by the tangent to a curve.
D', cusp of a curve terminated each way by the tangent to a curve:
all which is further explained by what follows; thus in the case (1) (1)(1) (1),
= (l)m (l)n, (1)m (1)4,, the value of A is given as Snm2n. m3m (= 3mm2m3m4). Here
A is the number of the point-pairs terminated one way in the intersection of any
two mn, m2 of the four curves, and the other way in the intersection of the remaining
two m,, m4 of the four curves. But in the case (1, 1)(1)(1), =(1, 1),m(1)n,(1l), the
value of A is given as = Smim,2+ mm1.lmm2. Here A denotes the number of the
point-pairs, which are either (Smnm,) terminated one way at a node of m, and the
other way at an intersection of nm, m2, or else (mm1.mmm) terminated one way at an
intersection of m, mn, and the other way at an intersection of m, bn2: and so in other
cases.
42. This being so, we have
(1) () (1) (1), = (1), (1)Ml (1)M 3 (1),.
A= mnimM2.mnm4 (=3 mlnmm3rn24), 1 A' =lnZ.n1 n (= 3 nln2n3n4 ),
B = 1mln'2.. 1n 4 (= 3Ymm2m3n4 ), 2 B' = z1n2. 12. * 24 (= 3Z1n2n3nm4),
0 = mCl. mn. 7312z4 (= Zmlmn3A4 ). 4 C' = nl. n2. m3m4 (= nlz2m3m4).
(1, 1) (1)(), = (1, 1) (1)Z (1).2
A = 8mnm,2 + m m. nmm2, 1 A' = 712?32 + nnz.12 nn2,
B = 8n1m2 + 8n2m1 2 B' = rm2n2 + rmn2nl
+ mmi (n - 2) m2 + mm2 (n - 2) ml + nn (m - 2) n + nn2 (m - 2) n,
+ mm^n2 (m - 1) + mm2 (m - 1) + nnm2 (n - 1) + nn2m1 (n - 1)
+ nm,2n (m - 2), + nn2m (n - 2),
C = m1m2, 4 C' = Slnn
+ nn, (m -2) zm + nn (m - 2) ml + mm, (n - 2) n2 + mm2 (n - 2),
+ n,1m. 3- n (m-1), + mm. n (n - 1),

)D = ")m- m

3

D' = /cn, n2.