PROFESSOR CAYLEY ON POLYZOMAL CURVES. 17 



On the Trizomal Curve and the Tetrazomal Curve — Art. Nos. 38 and 39. 



38. The trizomal curve 



Ju+ Jv+ JW= o 



has for its rationalised form of equation 



U 2 + V 2 + W 2 -2VW - 2WU-2UV=0; 



or as this may also be written, 



(1,1,1, -1, -1, -1)(U,V, TF) 2 = 0; 



and we may from this rational equation verify the general results applicable to 

 the case in hand, viz., that the trizomal is a curve of the order 2r, and that 



U = 0, at each of its r 2 intersections with V — W= 

 V = 0, „ „ W'- U = 



W = 0, „ „ U - V = 



respectively touch the trizomal. There are not, in general, any nodes or cusps, 

 and the order being = 2r, the class is = 2r(2r — 1). 



39. The tetrazomal curve 



J~C+ JT+ JW+ JT = o 

 has for its rationalised form of equation 



(0-a + F 2 +W 2 + T 2 -2UV- 2UW-2UT- 2VW- 2VT - 2WTf - &WVWT = , 



and we may hereby verify the fundamental properties, viz., that the tetrazomal 

 is a curve of the order 4r, touched by each of the zomals U — 0, V= 0, W — 0, 

 T = in 2r 2 points, viz. by U = at its intersections with JJJ + *JW + JT 

 = 0, that is, V 2 + W 2 + T 2 - 2 VW - 2 VT - 2WT = ; and the like as 

 regards the other zomals), and having Sr 2 nodes, viz., these are the intersections 

 of UV + *JT= 0, JW + JT = 0), {Ju + JW =0, Jv+ JT= 0), 

 (s/U + »JT= 0, > V /T F + s/W = 0), or, what is the same thing, the intersections 

 of(U-V=0, W—T=0), (U-W=0, V-T= 0), (U-T=0, V-W=0). 

 There are not in general any cusps, and the class is thus = 4r(4r — 1) — 6r 2 , 

 = 10 r 2 -4r. 



On the Intersection of two V-Zomals having the same Zomal Curves — Art. Nos. 40 and 41. 



40. Without going into any detail, I may notice the question of the intersec- 

 tion of two y-zomals which have the same zomal curves — say the two trizomals 

 JW+ JT + */W= 0, JTU+ VmF + JnW = 0, or two similarly related 



VOL. XXV. PART I. E 



