PROFESSOR CAYLEY ON POLYZOMAL CURVES. 7 



imaginary, of the curve, we have U = V, and therefore JU— »JV; the 



point thus belongs to the other branch J U — J V = 0, not to the branch 

 JU + JV — 0; the only points belonging to the last-mentioned branch are the 

 isolated points for which simultaneously JU = 0, JV - 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 

 JU + J~V + J W = of a trizomal curve. In fact, for each point of the 

 curve JTJ + JT+ JW = we have {JJ—V— W) 2 = 4 VW, and therefore, 

 U— V— W — -±: 2 J Y J W ; there may very well be points for which the sign 

 is + ; that is, points for which U — V + W + 2 JV s/W, and for these points 

 we have ^= J U = JV+ J W -, for real values of the co-ordinates the sign on 

 the left hand must be -I- (for otherwise the two sides 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 co-ordinates, and if the sign be in 

 fact — , then the point will belong to the branch si U + JV + s/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 understood 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 y-zomal curve some of 

 the branches are or may be impossible, and that there is at least one impossible 

 branch, viz. the branch JU + y/V+&c. = 0. 



12. For the purpose of referring to any branch of a polyzomal curve it will be 

 convenient to consider >/Z7as signifying determinately + +/U, or else— JU; 

 and the like as regards J'y, &c, but without any identity or relation between 

 the signs prefixed to the JfJ-, */!/, &c, respectively ; the equation 

 J1T+ JV r + &c. = 0, so understood, will denote determinately some one (that 

 is, any one at pleasure) of the equations *JU±JV±kc. = 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 *JU+ +/T] &c. = 0, or else of the 

 branch s/U+ VT^+ &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 consi- 

 dered in connection with each other, it is necessary to employ the sign — 

 with one or more of the radicals Ju, J~V, &c. ; thus in the trizomal curve 

 JTJ + s/V + JW — 0, we may have to consider the branches 

 sITf + JT + JW= 0, JTJ + sjv — JW = ; 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. 



