TEANSACTIOFS. 



I. — On Polyzomal Curves, otherwise the Curves 

 \/u + Vv + &c. = 0. 

 By Professor Cayley. Communicated by Professor Tait. 



(Eead 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 co-ordinates (a, y, z), then */Jj + VV+ &c. is a poly- 

 zome, and the curve Vjj + */~V + &c. = a polyzomal curve. Each of the 

 curves VT7 = 0, W = 0, &c. (or say the curves U = 0, V = 0, &c.) is, on account 

 of its relation of circumscription to the curve s/JJ + \/~P + &c. = 0, considered 

 as a girdle thereto (^«m«), and we have thence the term " zome " and the derived 

 expressions "polyzome," " zomal," &c. If the number of the zomes VJJ^Vv' 

 &c. be = v, then we have a y-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 consideration ; it is in general 

 assumed that v is = 3 at least. It is sometimes convenient to write the general 

 equation in the form Vffi + &c. = 0, where /, &c. are constants. The Memoir 

 contains researches in regard to the general y-zomal curve ; the branches thereof, 

 the order of the curve, its singularities, class, &c. ; also in regard to the y-zomal 

 curve Vl(Q + L <£>) + &c. = 0, where the zomal curves -f L $ = 0, all pass 

 through the points of intersection of the same two curves O = 0, $ — of the 

 orders r and r—s respectively ; included herein we have the theory of the depres- 

 sion 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 VlU + VmV + VnW + VpT = ; then if 

 U, V, W, T are in involution, that is, connected by an identical equation 

 a(7 + bK + cW -f- d:Z 7 = 0, and if I, m, n, p, satisfy the condition 



/ 771 7Z 7) 



— +^-4-— +4r = 0, the tetrazomal curve breaks up into two trizomal curves, 

 abed 



each expressible by means of any three of the four functions U, V, W, T ; for 



VOL. XXV. PART I. A 



