8 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



The Points common to Tivo Branches of a Polyzomal Curve — Art. Nos. 13 to 17. 



13. I consider the points which are situate simultaneously on two branches 

 of the y-zomal curve JJJ + Jv + &c. = 0. The equations of the two 

 branches may be taken to be 



JU + &c. + ( JW + &c.) = 0, 

 JU + &c. - ( JW + &c) = 0, 



viz., fixing the significations of JJf y Jv, JW, &c. in such wise that in the 

 equation of one branch these shall each of them have the sign + , we may take 

 Jfj, &c. to be those radicals which, in the equation of the other branch, have the 

 sign + , and JW, &c. to be those radicals which have the sign — . The fore- 

 going equations break up into the more simple equations 



JU + &c. = 0, JW + &c. = 0, 



which are the equations of certain branches of the curves JJj + &c. = 0, and 

 JW + &c. = 0, respectively, and conversely each of the intersections of these 

 two curves is a point situate simultaneously on some two branches of the 

 original v-zomal curve J J/ + JV + &c. = 0. Hence, partitioning in any 

 manner the y-zome JJ/ + JV + &c. into an a-zome, JW+ &c. and a /3-zome 

 J^y + &c. (a + 6 = i/), and writing down the equations 



JU + &c = 0, JW + &c = 



of an a-zomal curve and a /3-zomal curve respectively, each of the intersections 

 of these two curves is a point situate simultaneously on two branches of the 

 y-zomal curve ; and the points situate simultaneously on two branches of the 

 c-zomal curve are the points of intersection of the several pairs of an a-zomal 

 curve and a /3-zomal curve, which can be formed by any bipartition of the f-zome. 



14. There are two cases to be considered : — First, when the parts are 

 \ t v — l (v—l is > 1, except in the case v=2, which may be excluded from 

 consideration), or say when the v-zome is partitioned into a zome and antizome. 

 Secondly, when the parts a, (3, are each > 1 (this implies v = 4 at least), or say 

 when the v-zome is partitioned into a pair of complementary parazomes. 



15. To fix the ideas, take the tetrazomal curve JU + JY + JW+ J~T 

 — 0, and consider first a point for which JU=0, JV + JW + JT = 0. 

 The Norm is the product of (2 3 = ) 8 factors ; selecting hereout the factors 



Ju+ Jr+ JW+ JT, 



Ju- Jv- JW- Jt, 



let the product of these 



= u-(Jv+ Jw+ Jt) 2 



be called F, and the product of the remaining six factors be called G ; the 



