120 Prof. Cayley, Note on Abel's Theorem. [Oct, 31, 



the equation becomes 



\ax ay ay ax J T 



or say — J{f, <f>) dco + 8cf> = 0, 



that is dco = T : ; 



^ (j> 9) 



and then multiplying each side by the arbitrary function (%, y, l) m ~ 3 , 

 we have 



t(x,y,ir- 3 dco=t ( y { ^~ 3 s6, 



where 8<j> being of the order n in the variables, the numerator is a 

 rational and integral function of (x, y) of the order m + 7i — 3: 

 hence by a theorem contained in Jacobi's paper Theoremata nova 

 algebraica circa sy sterna duarum cequationum inter duas variabiles, 

 Crelle t. xiv. (1835) pp. 281 — 288, the sum on the right-hand side 

 is = 0: hence the required result X (x, y, l) m ~ 3 dco = 0. 



Observing that (x, y, \) m ~ z is an arbitrary function, the equa- 

 tion just obtained breaks up into the equations 



Sdco = 0, Zxdco = 0, tydco = 0, . . . tx m ~ 3 dco = 0,... ty m ~ 5 dco = 0, 



viz. the number of equations is 



1 + 2 + . . . + (m - 2), = \ (m - 1) (m - 2), 



which is = p, the deficiency of the curve. 



Suppose the fixed curve f(x, y, 1) = is a cubic, m = 3, and 

 we have the single relation 2 dco — 0, where the summation refers 

 to the Sn points of intersection of the cubic and of the variable 

 curve of the order n, <f>(x, y, 1) = 0. 



In particular if this curve be a line, n = 1, and the equation is 

 dco 1 + dco 2 -\-dco 3 = 0; here the two points (x 1} y t ), (x 2 , y 2 ) taken at 

 pleasure on the cubic, determine the line, and they consequently 

 determine uniquely the third point of intersection (x 3 ,t/); there 

 should thus be a single equation giving the displacement dco in 

 terms of the displacements dco v dco 2 ; viz. this is the equation just 

 found dco t + dco 2 + dco 3 = 0. 



So if the variable curve be a conic, ?i=2; and we have be- 

 tween the displacements of the six points the relation 



dco 1 4- dco 2 ... + dco 6 = : 



here five of the points determine the conic, and they therefore 

 determine uniquely the sixth point; and there should be between 

 the displacements a single relation as just found. 



