6 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



tively of the polyzome. The separation of the branches depends on the precise 

 fixation of the significations of JTJ, >JV, &c., and in regard hereto some further 

 explanation is necessary. 



6. When U is real and positive, ' J if may be taken to be, in the ordinary 

 sense, the positive value of JJJ, and so when U is real and negative, JJJ may 

 be taken to be = i into the positive value of J^JJ •, and the like as regards 

 JV, & c - The functions U, V, &c, are assumed to be real functions of the 

 co-ordinates ; hence, for any real values of the co-ordinates, U, V, &c. are real 

 positive or negative quantities, and the significations of JJj, Jv, &c. are com- 

 pletely determined. 



7. But the co-ordinates may be imaginary. In this case the functions 

 U, V, &c. will for any given values of the co-ordinates acquire each of them a 

 determinate, in general imaginary, value. If for all real values whatever of a, 0, 

 we select once for all one of the two opposite values of J a + /3z, calling it the 

 positive value, and representing it by J a + fti, then, for any particular values of 

 the co-ordinates, U being = a + /3i, the value of JJJ may be taken to be 

 = J a + (3i; and the like as regards J V, &c. JJj t Jv, &c. have thus each 

 of them a determinate signification for any values whatever, real or ima- 

 ginary, of the co-ordinates. The co-ordinates of a given point on the curve 

 JIT + JY + &c. = 0, Avill in general satisfy only one of the equations 

 JJJ ± jy~± &c. = 0; that is, the point will belong to one (but in general 

 only one) of the 2" -1 branches of the curve ; the entire series of points the 

 co-ordinates of which satisfy any one of the 2 V_1 equations, will constitute the 

 branch corresponding to that equation. 



8. The signification to be attached to the expression J a + (3i should agree with 

 that previously attached to the like symbol in the case of a positive or negative 

 real quantity; and it should, as far as possible, be subject to the condition of 

 continuity, viz., as a + fii passes continuously to a + fi'i, so Ja + fii should pass 

 continuously to Ja! + fti\ but (as is known) it is not possible to satisfy univer- 

 sally this condition of continuity ; viz., if for facility of explanation we consider 

 (a, /3) as the co-ordinates of a point in a plane, and imagine this point to describe 

 a closed curve surrounding the origin or point (0, 0), then it is not possible so to 

 define J a + (3i that this quantity, varying continuously as the point moves along 

 the curve, shall, when the point has made a complete circuit, resume its original 

 value. The signification to be attached to J a + pi is thus in some measure 

 arbitrary, and it would appear that the division of the curve into branches is 

 affected by a corresponding arbitrariness, but this arbitrariness relates only to the 

 imaginary branches of the curve : the notion of a real branch is perfectly definite. 



9. It would seem that a branch may be impossible for any series whatever of 

 points real or imaginary. Thus, in the bizomal curve JU 4- JV^O, the 

 branch JU + ^/F^Ois impossible. In fact, for any point whatever, real or 



