181
[509
510] 181
tied mitre
placed (as
loving the
s A passes
e fixed in
510.
ON BICURSAL CURVES.
(say these
ir manner,
spur-wheel
nicating a
[From the Proceedings of the London Mathematical Society, vol. IV. (1871—1873),
pp. 347—352. Read May 8, 1873.]
m b
n
A CURVE of deficiency 1 may be termed bicursal: there is some distinction
according as the order is even or odd, and to fix the ideas I take it to be even.
A bicursal curve of the order n contains
as in the
was made
iratus con-
iratus, and
noving the
\ n (n + 3) — [%(n — 1) (n — 2) — 1}, = 3n constants;
hence, if the order is = 2n, the number of constants is = 6n; such a curve is normally
represented by a system of equations
(x, y, *) = (1, 0) n + ( 1, 0) n ~ 2 V®,
where © is a quartic function, which may be taken to be of the form (1 — O' 2 ) (1 — kr& 2 ),
or otherwise to depend on a single constant; viz. (x, y, z) are proportional to n.-thic
, supported
'see fig. 1)
lentagraph-
works upon
functions involving such a radical: since in the values of (x, y, z) one constant divides
out, the number of constants is 3 {(n + 1) 4- (n — 1)} — 1 + 1, = 6n, as it should be.
But the curve of the order 2 n may be abnormally or improperly represented by
a system of equations
(x, y, z) = (1, 0) n+k + (1, 0) n+k ~ 2 V®,
nent where
teel of one
\ the plane
tie same or
dieel is in
nt of gear
viz. these equations, instead of representing a curve of the order 2n + 2k, will represent
a curve of the order 2n, provided only there exist 2k common values of 6 for which
each of the three functions vanish. The passage to a normal representation is effected
by finding 0' a function of 0, (viz. an irrational function of 6) such that the
foregoing equations become
(x, y, z) = ( 1, 0') n + ( 1, 0') n ~V@';