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@';