470 
[414 
414. 
ON POLYZOMAL CURVES, OTHERWISE THE CURVES 
J U+J V+ &c. = 0. 
[From the Transactions of the Royal Society of Edinburgh, vol. xxv. (1808), 
pp. 1—110. Read 16th December 1867.] 
If U, V, &c., are rational and integral functions (*$#, y, z) r , all of the same 
degree r, in regard to the coordinates (x, y, z), then V U + V V + &c. is a polyzome, 
and the curve fU + Vl r + &c. = 0 a polyzomal curve. Each of the curves fU= 0, 
V V = 0, &c. (or say the curves U = 0, V—0, &c.) is, on account of its relation of 
circumscription to the curve V17 + V T r +&c. =0, considered as a girdle thereto (^doga), 
and we have thence the term “ zome” and the derived expressions “ polyzome,” 
“ zomal,” &c. If the number of the zomes V U, f V, &c. be = v, then we have a 
¡/-zome, and corresponding thereto a ¡/-zomal curve; the curves U = 0, V= 0, &c., are 
the zomal curves or zomals thereof. The cases i/=l, v = 2, are not, for their own 
sake, worthy of consideration; it is in general assumed that v is = 3 at least. It is 
sometimes convenient to write the general equation in the form flU+kc. = 0, where l, 
&c. are constants. The Memoir contains researches in regard to the general ¡/-zomal 
curve; the branches thereof, the order of the curve, its singularities, class, &c.; also 
in regard to the ¡/-zomal curve fl (© + L<t>) + &c. = 0, where the zomal curves ® + L<P = 0, 
all pass through the points of intersection of the same two curves © = 0, = 0 of 
the orders r and r — s respectively; included herein we have the theory of the 
depression of order as arising from the ideal factor or factors of a branch or branches. 
A general theorem is given of “ the decomposition of a tetrazomal curve,” viz. if the 
equation of the curve be flU+ fmV+ fnW+ fpT= 0; then if U, V, W, T are in 
involution, that is, connected by an identical equation aZ7 +bF + cW + dT = 0, and if 
l, m, n, p, satisfy the condition ^ + ^ + “ + tetrazomal curve breaks up into