18 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



tetrazomals. For the trizomals, writing the equations under the form 



JU + JT + JW= 0, Jl JJ/ + J m JT+ Jn JW=0, 

 then, when these equations are considered as existing simultaneously, we may, 

 without loss of generality, attribute to the radicals JJJ, V 'V J W the same 

 values in the two equations respectively; but doing so, we must in the second 

 equation successively attribute to all but one of the radicals Jf, Jm, Jn, each 

 of its two opposite values. For the intersections of the two curves we have thus 



J U'- J V '■ J W — Jm — Jn '■ Jn — JJ : Jl — Jm , 

 viz., this is one of a system of four equations, obtained from it by changes of sign, 

 say in the radicals ^/^Tand Jn. Each of the four equations gives a set of r 2 

 points; we have thus the complete number, = 4r\ of the points of intersection of 

 the two curves. 



41. But take, in like manner, two tetrazomal curves; writing their equations 



in the form 



JW+ JT+ JW + JT = , 



J I JJT+ Jm JT^+ Jn JW+ JpjT = 0, 



then JjJ, JV, J W, JT may be considered as having the same values in the 

 two equations respectively, but we must in the second equation attribute succes- 

 sively, say to Jm, J ii, Jj>, each of their two opposite values. For the inter- 

 sections of the two curves we have 



( Jm - Jl) JV+ (Jn- Jl) JJV+( Jp- Jl ) JT = 

 ( Jl - Jm) JIT • + (Jn- Jm) JW + ( Jp - Jm ) J T = 



viz., this is one of a system of eight similar pairs of equations, obtained therefrom 

 by changes of sign of the radicals Jm, Jn, Jp. The equations represent each 

 of them a trizomal curve, of the order 2r ; the two curves intersect therefore in 

 4r 2 points, and if each of these was a point of intersection of the two tetrazomals, 

 we should have in all 8 x 4r 2 = 32>* 2 intersections. But the tetrazomals are 

 each of them a curve of the order 4r, and they intersect therefore in only 16r 2 

 points. The explanation is, that not all the 4r 2 points, but only 2r 2 of them are 

 intersections of the tetrazomals. In fact, to find all the intersections of the two 

 trizomals, it is necessary in their two equations to attribute opposite signs to one 

 of the radicals JTT, JT; we obtain 2r 2 intersections from the equations as they 

 stand, the remaining 2r 2 intersections from the two equations after we have in the 

 second equation reversed the sign, say of J f. Now, from the two equations as 

 they stand we can pass back to the two tetrazomal equations, and the first men- 

 tioned 2r 2 points are thus points of intersection of the two tetrazomal curves — 

 from the two equations after such reversal of the sign of JT, we cannot pass 

 back to the two tetrazomal equations, and the last-mentioned 2r 2 points are thus 



