44 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



This equation will in fact be identically true if only 



— ffi + y f J m + Mm ■ — <jh'i"x — u'li «, = o , 



cc'm + IV n — ff'p . + cb'm 1 + be n x = , 



gc'vi — hb'n + ffh + ( J^ m \ — he n x = 0, 



cg'm — bh'n . + clim x + bg n A + ff'p x = . 



From the first and second equations eliminating m 1 or n x , the other of these 

 quantities disappears of itself, and we thus obtain two equations which must be 

 equivalent to a single one, viz., we have 



Wffl + c'g'afm + bha'f'n + g'hff'p = , 

 bW l + cga'f'm + b'h'afn + gh'ffp = ; 



which equations may also be written 



c 'f i c ' ( f "f 1 V 

 ak bit «f ab * 



cf , co af fa _ 



i h ,l + j/r,™ + -f-.,n + [;,,/> =0; 



and it thus appears that the equations are equivalent to each other, and to the 



assumed relation 



I m n p n 



— + -r + — + ' , = 0. 

 a i> c a 



100. Similarly, from the third and fourth equations eliminating m or n, the 

 other of these quantities disappears of itself, and we find 



'.'//A - cga'f' m i + a f r '<Ji'i ~ fyffPi = 0. 

 bk'ffl t - afb'h'm x + bha'f'n x - b'hff'p x = , 



equations which may be written 



-r.l r-i-m + -4-.,n - ^-,p = 0, 



ac eg af ga 



f'h' , b'h' a'f b'q' n 



^l- -r-m + -^n ~p = 0, 



all bg af ab r 



where we see that the two equations are equivalent to each other and to the 



equation 



/, m. n. p. n 



a i b i c i d i 



It thus appears that the quantities l^ m^ n u p u must satisfy this last equation. 

 It is to be observed that the first and second equations being, as we have seen, 

 equivalent to a single equation, either of the quantities m u n^ may be assumed 

 at pleasure, but the other is then determined ; the third and fourth equations 

 then give 4, pi ; and the quantities / 1} m^ n^ p u so obtained, satisfy identically the 



equation -1 + ^ + — 1 +5 1 = 0- 

 ^ & x bj c x d 1 



