476 
ON POLYZOMAL CURVES. 
[414 
would have opposite signs), but there is no apparent reason, or at least no obviously 
apparent reason, why this should be so for imaginary values of the coordinates, and 
if the sign be in fact —, then the point will belong to the branch \/U-\-^V+^/W = 0. 
11. But the branch in question is clearly impossible for any series of real points; 
so that, leaving it an open question whether the epithet “impossible” is to be under 
stood to mean impossible for any series of real points (that is, as a mere synonym of 
imaginary), or whether it is to mean impossible for any series of points, real or 
imaginary, whatever, I say that in a v-zomal curve some of the branches are or may 
be impossible, and that there is at least one impossible branch, viz., the branch 
V U + V V + &c. = 0. 
12. For the purpose of referring to any branch of a polyzomal curve it will be 
convenient to consider V U as signifying determinately + V U, or else - V U; and the 
like as regards V V, &c., but without any identity or relation between the signs pre 
fixed to the V U, V V, &c., respectively; the equation V U + f V+ &c. = 0, so understood, 
will denote determinately some one (that is, any one at pleasure) of the equations 
VT? ± V V ± &c. = 0, and it will thus be the equation of some one (that is, any one at 
pleasure) of the branches of the polyzomal curve — all risk of ambiguity which might 
otherwise exist will be removed if we speak either of the curve Vi/" + VV, &c. = 0, or 
else of the branch V JJ q- V V + &c. = 0. Observe that by the foregoing convention, when 
only one branch is considered, we avoid the necessity of any employment of the sign +, 
or of the sign —; but when two or more branches are considered in connection with 
each other, it is necessary to employ the sign — with one or more of the radicals 
V U, V V, &c.; thus in the trizomal curve V U + V V + V W = 0, we may have to consider 
the branches VC r +VF+VTF=0, VCf+VF— VTF=0 ; viz., either of these equations 
apart from the other denotes any one branch at pleasure of the curve, but when 
the branch represented by the one equation is fixed, then the branch represented by 
the other equation is also fixed. 
Article Nos. 13 to 17. The Points common to Two Branches of a Polyzomal Curve. 
13. I consider the points which are situate simultaneously on two branches of 
the y-zomal curve V U + V V + &c. = 0. The equations of the two branches may be taken 
to be 
V U + &c. + (V W -h &c.) = 0, 
V? + k.-(Vf+k) = 0, 
viz., fixing the significations of V U, V V, V W, &c. in such wise that in the equation 
of one branch these shall each of them have the sign +, we may take V U, &c. to 
be those radicals which, in the equation of the other branch, have the sign +, and