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ON POLYZOMAL CURVES.
473
Part I. (Nos. 1 to 55).—On Polyzomal Curves in General.
Article Nos. 1 to 4. Definition and Preliminary Remarks.
1. As already mentioned, U, V, &c. denote rational and integral functions (*][x, y, z) r ,
all of the same degree r in the coordinates (x, y, z), and the equation
fill + VF+&C. = 0
then belongs to a polyzomal curve, viz., if the number of the zomes fill, fiV, &c. is
= v, then we have a i/-zomal curve. The radicals, or any of them, may contain rational
factors, or be of the form P fiQ; but in speaking of the curve as a v-zomal, it is
assumed that any two terms, such as P fiQ + P' fiQ, involving the same radical fiQ,
are united into a single term, so that the number of distinct radicals is always = v;
in particular (r being even), it is assumed that there is only one rational term P.
But the ordinary case, and that which is almost exclusively attended to, is that in
which the radicals fi U, fi V, &c. are distinct irreducible radicals without rational factors.
2. The curves U — V = 0, &c. are said to be the zomal curves, or simply the
zomals of the polyzomal curve fiU+ fiV& c . = 0; more strictly, the term zomal would
be applied to the functions U, V, &c. It is to be noticed, that although the form
fi U + fi V + &c. = 0 is equally general with the form fill] + fimV-\- &c. = 0 (in fact, in
the former case, the functions U, V, &c. are considered as implicitly containing the
constant factors l, m, &c., which are expressed in the latter case), yet it is frequently
convenient to express these factors, and thus write the equation in the form filtl + fimV+ &c.
For instance, in speaking of any given curves U = 0, V = 0, &c., we are apt, disregarding
the constant factors which they may involve, to consider U, V, &c. as given functions:
but in this case the general equation of the polyzomal with the zomals U= 0, V = 0,
&c., is of course filU + fimV + &c. =0.
3. Anticipating in regard to the cases v=l, v=2, the remark which will be
presently made in regard to the y-zomal, that fi £7 4- fi V + &c. = 0 is the curve represented
by the rationalised form of this equation, the monozomal curve fi U = 0 is merely the
curve £7=0, viz., this is any curve whatever £7=0 of the order r; and similarly, the
bizomal curve fi £7 + V V = 0 is merely the curve £7 — V = 0, viz. this is any curve
whatever 0 = 0, of the order r; the zomal curves £7=0, V = 0, taken separately, are
not curves standing in any special relation to the curve in question 0 = 0, but £7=0
may be any curve whatever of the order r, and then V = 0 is a curve of the same
order r, in involution with the two curves 0 = 0, £7=0; we may, in fact, write the
equation 0=0 under the bizomal form V £7 + Vil + £7 = 0. In the case r even, we
may, however, notice the bizomal curve P + fi £7 = 0 (P a rational function of the degree
|-r); the rational equation is here 0= £7 — P 2 = 0, that is £7=0 + P 2 , viz., P is any
curve whatever of the order ^r, and £7 = 0 is a curve of the order r, touching the
given curve 0 = 0 at each of its -§-r 2 intersections with the curve P = 0. I further
c. vi. 60