PROCEEDINGS 
OF THE 
ROYAL SOCIETY OF EDINBURGH. 
vol. vi. 1867-68. No. 75. 
Monday , l&th December 1867. 
Professor LYON PLAYFAIR, C.B., Vice-President, 
in the Chair. 
The following Communications were read : — 
1. On Polyzomal Curves, otherwise the Curves 
JU+JV+ &c. = 0. 
By Professor Cayley. Communicated by Professor Tait. 
If U, V, &c., are rational and integral functions (*) (x, y, z ) r , 
all of the same degree r, in regard to the co-ordinates (x, y, z), then 
U + \/F+ &c. is a polyzome, and the curve \/ U + \/V + &c. 
= 0 a polyzomal curve. Each of the curves J U =0, JV = 0, &c. 
(or say the curves TJ = 0, V —■ 0, &c.), is on account of its rela- 
tion of circumscription to the curve JU + JV + &c. = 0, con- 
sidered as a girdle thereto (£w/xa), and we have thence the term 
“ zome ” and the derived expressions “ polyzome,” “ zomal,” &c. 
If the number of the zomes J U, V, &c. be = v, then we have 
a j/-zome, and corresponding thereto a v-zomal curve ; the curves 
U = 0, V = 0, &c., are the zomal curves or zomals thereof. The 
cases v = 1, v = 2, are not, for their own sake, worthy of consider- 
ation ; it is in general assumed that v is = 3 at least. It is some- 
times convenient to write the general equation in the form 
VTu + &c. = 0, where l , &c., are constants. The memoir con- 
tains researches in regard to the general v- zomal £urve ; the 
2 F 
VOL. VI. 
