PROFESSOR CAYLEY ON POLYZOMAL CURVES. 45 



101. Now writing 



ff'l x — — g(c'm + b'm x ) + h(b'n + c'n x ) , 



ff'Pi = ~ c ((l' m ~ h'm x ) + Hh'n — g\) , 

 and 



ff'p = c(c'm + b'm x ) + b(b'?i + c'n x ) , 



ff'l — g{g'm — h'mj + h(h'n — g'n x ) , 

 we find 



f 2 f 2 (h v i ~ ^V) — — (pg + ch)[(c'm + b'm x )(h'n — g'n x ) + {g'm — h'm x )(b'n + c'nj] 

 = (bg + ch) (b'g' + c'h' )(m x n x — mn) 



— aa'ff'(m x n x — mn) 



that is, 



viz., this equation is satisfied identically by the values of l x , m v n v p x determined 

 as above. 



102. Hence if m x n x = mn, we have also l x p l = Ip, and we can determine m x , n x , 

 so that m x n x shall = «, viz., in the first or second of the four equations (these 

 two being equivalent to each other, as already mentioned), writing m x = On, and 



therefore n x — ^ m , we have 



— ff'l + (/g'm + hh'n — gh'nd — g'hm - = , 



cc'm + bb'n — ff'p + cb'nd + bc'm - = , 

 which are, in fact, the same quadric equation in 0, viz., we have 



—ff'l + ggfin + hh'n _ gh' _ _ gh' 

 cc'm + bb'n — ff'p ch' be 



The final result is that there are two sets of values of l x7 m v n v p x , each satisfying 

 the identity 



— IA + mB + nC — pD + l x \ — m x B x — n l C l + p l D 1 = , 



and for each of which we have 



/, , m, , n, , p, n , , 



-i + _J + _i + -O = 0, Lp. = Ip , m,n x == mn . 



a x \\ c dj 



103. Consider, in particular, the case where p — ; the relation 



here becomes 



abed 



, aq' a'h 



I = — -jL-m — — n . 



VOL. XXV. PART I. M 



