474 
ON POLYZOMAL CURVES. 
[414 
remark that the order of the v-zomal curve V V+ &c. = 0 is = 2 1 ' -2 r; this is right in 
the case of the bizomal curve V U + V V = 0, the order being = r, but it fails for the 
monozomal curve V TJ = 0, the order being in this case r, instead of \r, as given by 
the formula. The two unimportant and somewhat exceptional cases v = 1, v = 2, are 
thus disposed of, and in all that follows (except in so far as this is in fact applicable 
to the cases just referred to), v may be taken to be =3 at least. 
4. It is to be throughout understood that by the curve V U -\- V V+ &c. = 0 is 
meant the curve represented by the rationalised equation 
Norm (Vi7+ fV + &c.) = 0, 
viz. the Norm is obtained by attributing to all but one of the zomes V U, V V, &c., 
each of the two signs +, —, and multiplying together the several resulting values of 
the polyzome; in the case of a y-zomal curve, the number of factors is thus = 2 1 "' 1 r 
(whence, as each factor is of the degree \r, the order of the curve is 2 V_1 • \r, 
= 2*'— 2 r, as mentioned above). I expressly mention that, as regards the polyzomal curve, 
we are not in any wise concerned with the signs of the radicals, which signs are and 
remain essentially indeterminate ; the equation V U + V V 4- &c. = 0, is a mere symbol for 
the rationalised equation, Norm (f U + VF+ &c.)= 0. 
Article Nos. 5 to 12. The Branches of a Polyzomal Curve. 
5. But we may in a different point of view attend to the signs of the radicals; 
if for all values of the coordinates we take the symbol J „, and consider J U, JV, 
&c. as signifying determinately, say the positive values of f U, V V, &c.; then each of 
the several equations ±JU±JV + &c. = 0, or, fixing at pleasure one of the signs, 
suppose that prefixed to JU, then each of the several equations JU ± JV + &c. = 0, 
will belong to a branch of the polyzomal curve: a y-zomal curve has thus 2 1 ' - " 1 
branches corresponding to the 2" -1 values respectively of the polyzome. The separation 
of the branches depends on the precise fixation of the significations of JU, JV, &c., 
and in regard hereto some further explanation is necessary. 
6. When U is real and positive, JU may be taken to be, in the ordinary sense, 
the positive value of V U, and so when U is real and negative, J U may be taken 
to be = i into the positive value of V — U; and the like as regards J V, &c. The 
functions U, V, &c. are assumed to be real functions of the coordinates; hence, for 
any real values of the coordinates, U, V, &c. are real positive or negative quantities, 
and the significations of jTJ, JV, &c. are completely determined. 
7. But the coordinates may be imaginary. In this case the functions U, V, &c. 
will for any given values of the coordinates acquire each of them a determinate, in 
general imaginary, value. If for all real values whatever of a, /3, we select once for