PROFESSOR CAYLEY ON POLYZOMAL CURVES. 21 



45. The equation by which the decomposition was effected is, it is clear, one 

 of twelve equivalent equations ; four of these are 



(o >J™+~k'^ *-& )( » )' = 



cda m\ / 



dab «\ ^ / 





d %/ 



abc 



and the others may be deduced from these by a cyclical permutation of ( U, V, W), 

 (a, b, c), (I, m, n), leaving T 7 , d, p unaltered. 



Application to the Trizomal ; the Theorem of the Variable Zomal — Art. Nos. 46 to 51. 



46. I take the last equation written under the form 



which, putting therein p = 0, is 



(a J mil - b Jiff = ^- nT , 



which is in fact the trizomal curve, 



aJmlT- b JTV + J— nT= , 

 c 



viz., the trizomal curve s/fU + JmV + JnW = 0, — if a, b, c be any quantities 

 connected by the equation 



a be 



(the ratios a, b, c thus involving a single arbitrary parameter) ; and if we take T 

 a function such that a6 r + bF + cW ir +dr=0; that is, T = 0, any one of the 



VOL. XXV. PART I. F 



