414] 
ON POLYZOMAL CURVES. 
475 
all one of the two opposite values of Va + fii, calling it the positive value, and 
representing it by Ja + /3i, then, for any particular values of the coordinates, U being 
= a + f3i, the value of J U may be taken to be = J a + /3i; and the like as regards 
JV, &c. JU, JV, &c. have thus each of them a determinate signification for any 
values whatever, real or imaginary, of the coordinates. The coordinates of a given 
point on the curve V U + J V + &c. = 0, will in general satisfy only one of the equations 
J U + J V ± &c. = 0; that is, the point will belong to one (but in general only one) of 
the 2 1 ' -1 branches of the curve; the entire series of points the coordinates 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 Joe + ¡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 + ¡3i passes continuously to a! + ¡3'i, so Va + /3i should pass con 
tinuously to Va' + f3'i; but (as is known) it is not possible to satisfy universally this 
condition of continuity; viz., if for facility of explanation we consider (a, /3) as the 
coordinates 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 Va 4- ¡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 signi 
fication to be attached to Va + (3i 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 VZ7+VF=0, the branch 
VI7+\/T=0 is impossible. In fact, for any point whatever, real or imaginary, of the 
curve, we have U = V, and therefore V U = V V; the point thus belongs to the other 
branch V U — V V = 0, not to the branch V U + V V = 0 ; the only points belonging to 
the last-mentioned branch are the isolated points for which simultaneously V U = 0, 
\/V=0\ viz., the points of intersection of the two curves U=0, V—0. 
10. It is not clear to me whether the case is the same in regard to the branch 
\/Z7-f-VF+ViT = 0 of a trizomal curve. In fact, for each point of the curve 
VF+ VF + VF = 0 we have (U- V- IF) 2 = 4FIF, and therefore, U — V— W = ± 2VTVIf; 
there may very well be points for which the sign is +; that is, points for which 
U = V + IF+2\/FVJF, and for these points we have + U = VF+ VlF; for real values 
of the coordinates the sign on the left hand must be + (for otherwise the two sides 
60—2