4 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



Academy, 10th February 1867, and published in their "Proceedings," pp. 44, 45, 

 contains the theorems mentioned iu the letter of 30th April, and some other 

 theorems. It is not necessary that I should particularly explain in what manner 

 the present Memoir has been, in the course of writing it, added to or altered in 

 consequence of the information which I have thus had of Mr Casey's researches ; 

 it is enough to say that I have freely availed myself of such information, and that 

 there is no question as to Mr Casey's priority in anything which there may be in 

 common in his memoir on Bicircular Quartics and in the present Memoir. 



Part I. (Nos. 1 to 55). — On Polyzomal Curves in General. 

 Definition and Preliminary Remarks — Art. Nos. 1 to 4. 



1. As already mentioned, U, V, &c. denote rational and integral functions 

 (*) (x, y, z) r , all of the same degree r in the co-ordinates {oc, p, z), and the equation 



VCl + VV + &c. = 

 then belongs to a polyzomal curve, viz., if the number of the zomes VW, *J^V, &c., 

 is = v, then we have a i-zomal curve. The radicals, or any of them, may con- 

 tain rational factors, or be of the form PjQ; but in speaking of the curve as a 

 y-zomal, it is assumed that any two terms, such as PjQ + P WQ, involving the 

 same radical *JQ, 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 ex- 

 clusively attended to, is that in which the radicals Vu, Vlf, &c. are distinct irre- 

 ducible 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 */U + */ V + &c. = ; more strictly, the term 

 zomal would be applied to the functions £/, V, &c. It is to be noticed, that al- 

 though the form */U + JV + &c. = is equally general with the form 

 JlU+ Jm V+ &c. = (in fact, in the former case, the functions U, V, &c, are con- 

 sidered as implicitly containing the constant factors /, 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 JfU + Jm V + &c. For instance, in speak- 

 ing 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 JfU + JmV + &c. = 0. 



3. Anticipating in regard to the cases v = 1, v = 2, the remark which will be 

 presently made in regard to the y-zomal, that JJj + JV + &c. = is the curve 

 represented by the rationalised form of this equation, the monozomal curve 

 J J} = is merely the curve U = 0, viz., this is any curve whatever U = of the 



