142
ON THE MECHANICAL DESCRIPTION OF CERTAIN SEXTIC CURVES. [504
— i. +1
then, if u = ± (i + e), cos a = —, sin a = “ , and similarly if v = ±(i + £), then cos /3 =
6 €
sin ¡3 = • Hence the points at infinity of the sextic curve are as follows :
1°. u = + (i + e), v not = ± (i + £),
— ai ± c — a'i + c' « , . „
x — , y = , first pair ot points ;
2°. v = + (i + £), u not = + {% + e),
— bi + d — h'i + d' . . „
x = , y = — , second pair ot points ;
3°. u = ± (i + e), v = ± (i + Ç),
— eci ± c — hi + cl el'î + c' — h'i + d
ae + 6^=0, as above, third pair of points;
which six points are in general distinct from each other, and from the circular points
at infinity.
The foregoing values of x, y may be said to be “ circular quoad a,” if a = c,
a' = — c ; and similarly to be “ circular quoad ¡3,” if b = d', b' = — d.
And we see at once that if the values are circular quoad a, then the first pair
of points coincide with the circular points at infinity ; and that, in like manner,
if the values are circular quoad /3, then the second pair of points coincide with the
circular points at infinity; but if the values are circular quoad a and ¡3 respectively,
then each of the three pairs of points coincides with the circular points at infinity :
so that these are then triple points on the curve ; or the curve is tricircular, having
besides the two triple points, 3 dps.
The relation between u, v gives
{v (C + Du 2 ) + 4abu} 2 = 16a 2 b 2 . u 2 — {A + Bu 2 ) (G + Did),
and it thus appears that if any one of the functions A, B, G, D is =0, the function
under the radical sign is a mere linear function of u 2 , say it is L + Mu 2 ; introducing
a new parameter 6 such that u ~ \J \jyjj ^ + > we have ^L+Mu 2 = ^L^—^, and
consequently u, v are each of them a rational function of 6. Hence, when any one of
the relations in question is satisfied, or say, whena + c2=6 + c, b + d = c + a, or c + d = a +b,
the curve becomes unicursal: there is no diminution of the order, and the curve is
consequently a unicursal sextic, or sextic with 10 dps.
