504] ON THE MECHANICAL DESCRIPTION OF CERTAIN SEXTIC CURVES.
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hence a curve for which the coordinates x, y are rational functions of u, v, is a curve
having a deficiency D = 1, or, what is the same thing, having a number of dps. less
by unity than the maximum number {= \ (n — 1)(n- 2), if n be the order of the curve}.
It will further appear that the relation 1 ) 2 (v, l) 2 = 0 is satisfied by the
values u = v = i and u = v= —i (if, as usual, i = V — I).
In the curve in question, the coordinates (x, y) are given linear functions of the sines
and cosines of a, ¡3; and if we make the curve meet an arbitrary line Ax + By + 0=0,
we obtain between the sines and cosines of a, /3 a linear relation which, substituting 1
therein the expressions in terms of u, v, takes the form
(*$iq 1) 2 .(1 + ?; 2 )+ (*;$+, l) 2 . (1 + u 2 ) = 0,
viz. this is a relation of the form 1 ) 2 (v, 1) 2 = 0, such that it is satisfied by
the four sets of values u — ±i, v = ± i, and therefore in particular by the values
u — v = i and u = v = — i.
Hence, considering the intersections of the curve by the arbitrary line, the values
of (u, v) are given by the two equations (*]£«, l) 2 (v, 1) 2 = 0, l) 2 (v, 1) 2 =0; these,
regarding for a moment u, v as ordinary rectangular coordinates, represent each of them
a quartic curve having two dps. at infinity on the axes u = 0, v = 0 respectively: each
of these points reckons therefore as 4 intersections, and the number of the remaining
intersections therefore is 4.4 — 2.4, =8. But, by what precedes, the two quartic curves
have also in common the points u = v = i and u=v = —i; and rejecting these, there
remain 8 — 2, =6 intersections.
The conclusion is, that the curve is a sextic curve of deficiency 1, that is, having
9 dps. The reasoning may be presented under a slightly different form as follows:
regarding u, v as coordinates, we have the curve (*$/+ 1 ) 2 (v, 1) 2 =0, a binodal quartic
curve, and having therefore the deficiency 1 ; the curve passes, as above-mentioned,
through the points u = v — i and u = v = — i. The required curve is obtained as a
transformation of the quartic curve by formulae of the form x : y : z(=l)~ P : Q : R,
where P, Q, and R {= (1 + w 2 ) (1 + v 2 )} are quartic functions of the coordinates u, v,
such that P — 0, Q — 0, R = 0 are each of them a quartic curve passing twice through
each of the nodes and once through each of the before-mentioned points, (u = v = i)
and (u = v=-i), of the binodal quartic curve. Hence the curve in question is a curve
of the order 4.4-2.4-2.1, =6, and having the same deficiency as the binodal
quartic, that is, the deficiency is = 1.
I observe that the sextic curve does not, in general, pass through the circular
points at infinity, but it intersects the line at infinity in three distinct pairs of points;
one of these, or all three of them, (but not two pairs only,) may coincide with the
circular points at infinity, the circular points at infinity being, in the latter case, triple
points, or the curve being tricircular: this will appear presently.
To obtain the foregoing equation (*$>, l) 2 (v, 1) 2 =0, the elimination of y gives
(a cos a + b cos /3 + d cos S) 2 + (ci sin a + b sin /3 + d sin h)- = c 2 ,
18—2
