PROFESSOR CAYLEY ON POLYZOMAL CURVES. 15 



<£ a ; and if any other values of the irrational function contain respectively <£#, &c, 

 then the Norm will contain the factor $« + + & c - 



33. A branch ideally containing $<* = may for shortness be called integral 

 or fractional, according as the index a is an integer or a fraction ; by what precedes 

 the fractional branches present themselves in pairs. If for a moment we consider 

 integral branches only, then if the y-zomal contain <£ = 0, this can happen in one 

 way only, there must be some one branch ideally containing $ = ; but if the 

 »-zomal contain $ 2 = 0, then this may happen in two ways, — either there is a 

 single branch ideally containing $ 2 = 0, or else there are two branches, each of 

 them ideally containing $ = 0. And generally, if the y-zomal contain <£" = 0, 

 then forming any partition « = a + (3 + &c. (the parts being integral), this 

 may arise from there being branches ideally containing cj>« = 0, $>& = 0, &c. 

 respectively. The like remarks apply to the case where we attend also to 

 fractional branches, — thus, if the y-zomal contain <£ = 0, this may arise (not 

 only, as above mentioned, from a branch ideally containing $ = 0, but also) from 

 a pair of branches, each ideally containing & — . And so in general, if the 

 y- zomal contain <j>" = 0, the partition « = a + (B + &c. is to be made with the 

 parts integral or fractional (= -£- or integer + | as above), but with the fractional 

 terms in pairs ; and then the factor $" = may arise from branches ideally con- 

 taining $ a = 0, $£ = 0, &c. respectively. 



34. Any zomal, antizomal, or parazomal of a y-zomal curve, */^(0 + Lq) + &c. 

 = 0, is a polyzomal curve (including in the term a monozomal curve) of the 

 same form as the i>-zomal ; and may in like manner contain <£ = 0, or more gene- 

 rally, $" = 0, viz., if w = a + (3 + &c. be any partition of w as above, this will 

 be the case if the zomal, antizomal, or parazomal has branches ideally contain- 

 ing $« = 0, <£ s = 0, &c. respectively. It is to be observed that if a zomal, anti- 

 zomal, or parazomal contain <£ = 0, or any higher power $" = 0, this does not 

 in anywise imply that the zomal contains even $ = 0. But if (attending only to 

 the most simple case)a zomal and its antizomal, or a pair of complementary 

 parazomals, each contain $ = inseparably (that is, through a single branch 

 ideally containing <£ = 0), then the y-zomal will have two branches, each ideally 

 containing $ = 0, and it will thus contain <£ 2 = 0. In fact, if in the zomal and 

 antizomal, or in the complementary parazomals, the branches which ideally con- 

 tain $ = are 



Jl(Q + X*) + &c. = 0, Jn(0 + N<t>) + &c. = 



respectively (for a zomal, the -I- &c. should be omitted, and the first equation be 

 written */J(9 + L<&) = 0), then in the ^-zomal there will be the two branches 



(JI{Q + Z*) + &c.) ± ( Jn(® + AT*) + &c.) = , 

 each ideally containing <l> = 0. 



