414] 
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