482 
ON POLYZOMAL CURVES. 
[414 
or, what is the same thing, it is 
V© (\U + &c.) + 4 -7=- (L V/ + &c.) — 4 — ^ (L 2 Vi + &c.) + &c. 
V© 7 *©V© 
which expansion may contain the factor <4, or a higher power of <4». For instance, if 
we have VZ + &c. = 0, the expansion will then contain the factor <4; and if we also 
have ZVi+&c. = 0 (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 + bQ + cR, with the same values of P, Q, R, but with different values 
of the coefficients a, b, c, then it implies the three equations aV(+&c. = 0, b Vi + &c. = 0, 
cVi + &c. = 0; and so in other cases), if I say ZV7+&c. be also =0, then the 
expansion will contain the factor <4 2 , and so on; the most general supposition being, 
that the expansion contains as factor a certain power <4“ of d>. Imagine each of the 
polyzomes expanded in this manner, and let certain of the expansions contain the 
factors <4“, <4>0, &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. = 00, the Norm will contain the factor d?" 
30. It has been mentioned that the form VZ(© + 2<4) is considered as including the 
form Vi© +Zd>, that is, when 1 = 0, the form VZ<4. If in the equation of the v-zomal curve 
there is any such term—for instance, if the equation be VXd> -f Vw(© + Jfd>) + &c. = 0, 
the radical VPd> contains the factor <4*; but if L contains as factor an odd or an 
even power of <4, then VZd> will contain the factor <4 a where a is either an integer, 
or an integer + |. Consider the polyzome VPd> + Vra (© + if/d>) + &c., belonging to any 
particular branch of the curve; the radical VXd> contains, as just mentioned, the factor 
d> a , and if the remaining terms Vra(© + M<t>) + &c., are such that the expansion 
contains as factor the same or any higher power of d>, then the expansion of the 
polyzome VZd> 4- Vra (© + il/d>) + &c., belonging to the particular branch will contain the 
factor 4?“; and similarly we may have branches containing the factors d> a , d>^, &c., 
whence, as before, if to = a + /3 + &c., the Norm will contain the factor d>“ ; the only 
difference is, that now a, /3, &c., instead of being of necessity all integers, are each 
of them an integer, or an integer+4; of course, in the latter case the integer may 
be zero, or the index be =4. It is clear that co must be an integer, and it is, in 
fact, easy to see that the fractional indices occur in pairs; for observe that a being 
fractional, the expansion of Vm (© + ilPd>) + &c., will contain not d> a , but a higher power, 
d>“ +3 , where a + q is an integer; whence each of the polyzomes VZd> + (Vm(© +T/d>j + &c.) 
will contain the factor <4°. 
31. Observe that in every case the factor <4 a presents itself as a factor of the 
expansion of the polyzome corresponding to a particular branch of the curve ; the 
polyzome itself does not contain the factor d> a , and we cannot in anywise say that the 
corresponding branch contains as factor the curve d> a = 0; but we may, with great 
propriety of expression, say that the branch ideally contains the curve <4“ = 0; and this