PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



19 



not points of intersection of the two tetrazomal curves. The number of inter- 

 sections of the two curves is thus 8 x 2r 2 , = 16r 2 , as it should be. 



The Theorem of the Decomposition of a Tetrazomal Curve — Art. Nos. 42 to 45. 

 42. I consider the tetrazomal curve 



JlU + Jm~V + JnW + JpT = , 



where the zomal curves are in involution, — that is, where we have an identical 



relation. 



&U + bV + cW + dT=0 ; 



and I proceed to show that if /, m, n, p, satisfy the relation 



abed 



the curve breaks up into two trizomals. In fact, writing the equation under the 

 form 



( JlU + Jm~V + Jn~Wf -pT = , 



and substituting for T its value, in terms of U, V, W, this is 



(Id + pa) U + (md + pb) V + (ml + pc) W 

 + 2 J 7 ^d JTW + 2 JnJd JWU + 2 JJ^d JTfV = ; 



or, considering the left-hand side as a quadric function of {JU, *JV~, s/W), the 

 condition for its breaking up into factors is 



that is 



Id + pa, d Jim, d Jin 



d J ml, md + pb, d Jmn 



d Jnl, d Jnm, nd + pc 



= , 



;j 2 (/bcd + meda 4- ndab + joabc) = 

 or finally, the condition is 



+ Vt + n +l> = 



43. Multiplying by Id + pa, and observing that in virtue of the relation we 

 have 



(Id t- pa) (md + pb) = Imd 2 



abd 



-pn 



acd 



(Id 4- pa) (nd + pc) = lnd' 2 - ^ -pm 



the equation becomes 



((Id + pa) JJJ + d J fa JT+d JU JwY = ~ pfbJnJY-? Jm JW) 



