PROFESSOR CAYLEY ON POLYZOMAL CURVES. 13 



r(r — s) common points, a 2" -2 -tuple point, counting as 2 2v ~ 5 — 2"~ 3 dps., to- 

 gether as (2 2 " -5 — 2"- 3 ) r(r — s) dps. ; whence, taking account of the nodes, the 

 total number of dps. is = 2 v - i r[(2"- 1 — 2)r — {v—l)s}- 



Depression of Order of the v-zomal Curve from the Ideal Factor of a Branch or Branches — 



Art. Nos. 28 to 37. 



28. In the case of the r(r — s) common points as thus far considered, the 

 order of the y-zomal curve has remained throughout = 2"~ 2 r, but the order admits 

 of depression, viz., the constants /, m, &c, and those of the functions L, M, &c, 

 may be such that the norm contains the factor q> m ; the y-zomal curve then con- 

 tains as part of itself (<£ w = 0) the curve $ = taken w times, and this being so, 

 if we discard the factor in question, and consider the residual curve as being the 

 i/-zomal, the order of the y-zomal will be = 2 v ~ 2 r — w (r — .<?) . 



29. To explain how such a factor $ w presents itself, consider the polyzome 

 Jl(g + £<£) + &c, or, what is the same thing, s/l Jq + £$ + &c, belonging 

 to any particular branch of the curve, we may, it is clear, take Jq + £<$, &c. 

 each in a fixed signification as equivalent to >Jq + L^ &c, respectively, and the 

 particular branch will then be determined by means of the significations attached 

 to JT, s/m, &c. Expanding the several radicals, the polyzome is 



Jl{ J& + \L~- \L* JS= + &c } + &c; 



or, what is the same thing, it is 



Je (ji + &c.) + I *=(LJi + &c.) - 8 -^(^V^+ &c) + &c 



which expansion may contain the factor $ , or a higher power of $ . For in- 

 stance, if we have *JT + &c. = 0, the expansion will then contain the factor $ ; 

 and if we also have L JJ + &c. = (observe this implies as many equations as 

 there are asyzygetic terms in the whole series of functions L, M, &c. ; thus, if 

 L, M, &c, are each of them of the form aP + IQ + cfi, with the same values 

 of P, Q, B, but with different values of the co-efficients a, b, c, then it implies the 

 three equations a s /J+ &c. = , bJJ+ &c. = , cJJ + &c. = 0; and so in 

 other cases), if I say LJJ+ &c. be also = , then the expansion will contain 

 the factor $ 2 , and so on ; the most general supposition being, that the expansion 

 contains as factor a certain power <£* of $ . Imagine each of the polyzomes 

 expanded in this manner, and let certain of the expansions contain the factors 

 <|> a , &, &c, respectively. The produce of the expansions is identically equal to 

 the product of the unexpanded polyzomes — that is, it is equal to the Norm ; 

 hence, if a + /3 + &c. = ©, the Norm will contain the factor <£*\ 



VOL. XXV. PART I. D 



