PROFESSOR CAYLEY ON POLYZOMAL CURVES. 5 



order r ; and similarly, the bizomal curve JTl + *JV= is merely the curve 

 U — V = 0, viz. this is any curve whatever Q = 0, of the order r ; the zomal 

 curves U = 0, F" = 0, taken separately, are not curves standing in any special 

 relation to the curve in question = 0, but U = may be any curve whatever 

 of the order r, and then V — is a curve of the same order r, in involution with 

 the two curves = 0, U — ; we may, in fact, write the equation = under 

 the bizomal form JIT + +/Q + U = 0. In the case r even, we may, however, 

 notice the bizomal curve P + JJJ '= (P a rational function of the degree \r) ; 

 the rational equation is here Q = U — P 2 = 0, that is U = + P 2 , viz., P is any 

 curve whatever of the order \r, and U = is a curve of the order r, touching the 

 given curve = at each of its lr 2 intersections with the curve P = 0. I fur- 

 ther remark that the order of the y-zomal curve */F + & c - = is =2 y_2 r; this is 

 right in the case of the bizomal curve Jjj + J V = 0, the order being = r, but 

 it fails for the monozomal curve JlJ = 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 JJj + Jf + &c. =0 

 is meant the curve represented by the rationalised equation — 



Norm (\/U + */V + &c.) = 



viz. the Norm is obtained by attributing to all but one of the zomes JlJ, s/V, &c., 

 each of the two signs +, — , and multiplying together the several resulting 

 values of the polyzome ; in the case of a v-zomsl curve, the number of factors is 

 thus =2"- 1 r (whence, as each factor is of the degree \r, the order of the curve 

 is 2"- 1 . Jr, = 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 

 JTJ + +/y + &c = 0, is a mere symbol for the rationalised equation, Norm 

 (s/U+ JT+ &c.) = 0. 



The Branches of a Polyzomal Curve — Art. Nos. 5 to 12. 



5. But we may in a different point of view attend to the signs of the radicals ; 

 if for all values of the co-ordinates we take the symbol J~, and consider JJJ, Jv, 

 &c. as signifying determinately, say the positive values of Jzf, J~V, &c; then each 

 of the several equations ± s/U ± */T + &c. = 0, or, fixing at pleasure one of the 

 signs, suppose that prefixed to s/~U, th en eacn of the several equations 

 sJU ± J V ± &c. = 0, will belong to a branch of the polyzomal curve : a 

 y-zomal curve has thus 2"- 1 branches corresponding to the 2"- 1 values respec- 



VOL. XXV. PART I. B 



