PROFESSOR CAYLEY ON POLYZOMAL CURVES. 9 



rationalised equation of the curve is therefore FG = 0. The derived equation is 

 GdF + FdG = 0; at the point in question JTJ = 0, JY -v JW + JT= 0; 

 G and dG are each of them finite (that is, they neither vanish nor become 

 infinite), but we have 



F=0,dF=dU- (Jy + JW + JT) (dV+ Jy + dW + Jw + dT + jT), = dU, 



and the derived equation is thus GdU — 0, or simply dU — 0. It thus appears 

 that the point in question is an ordinary point on the tetrazomal curve ; and, 

 farther, that the tetrazomal curve is at this point touched by the zomal curve 

 U = 0. And similarly, each of the points of intersection of the two curves 

 JJJ — 0, JT + JW + JT — 0, is an ordinary point on the tetrazomal curve ; 

 and the tetrazomal curve is at each of these points touched by the zomal curve 

 U=0. 



16. Consider, secondly, a point for which JlJ + JT= 0, JW + Jf = 0; 

 to form the Norm, taking in this case the two factors 



JT/ + JT+ JW + JT, 



JV + JT- JW- JT, 

 let their product 



= (JV + JT) 2 -(JW + JT? 



be called F, and the product of the remaining six factors be called G ; the 

 rationalised equation is FG = 0, and the derived equation is FdG + GdF = 0. 

 At the point in question G and dG are each of them finite (that is, they neither 

 vanish nor become infinite), but we have 



F=0,dF=(JU+ JT) {dU+ JTT+dV+ JT)-(JW+ jT)(dW+ JW+ dT+ JT),=0, 



that is, the derived equation becomes identically = 0; the point in question is 

 thus a singular point, and it is easy to see that it is in fact a node, or ordinary 

 double point, on the tetrazomal curve. And similarly, each of the points of 

 intersection of the two curves JJj + JT = 0, J~W + JT 1 = is a node on 

 the tetrazomal curve. 



17. The proofs in the foregoing two examples respectively are quite general, 

 and we may, in regard to a v-zomal curve, enunciate the results as follows, viz., 

 in a y-zomal curve, the points situate simultaneously on two branches are either 

 the intersections of a zomal curve and its antizomal curve, or else they are the 

 intersections of a pair of complementary parazomal curves. In the former case, 

 the points in question are ordinary points on the y-zomal, but they are points of 

 contact of the v-zomal with the zomal ; it may be added, that the intersections of 

 the zomal and antizomal, each reckoned twice, are all the intersections of the 

 i/-zomal and zomal. In the latter case, the points in question are nodes of the 

 v-zomal ; it may be added, that the y-zomal has not, in general, any nodes other 

 than the points which are thus the intersections of a pair of complementary para- 

 zomals, and that it has not in general any cusps. 



VOL. XXV. PART I. C 



