1881.] Prof. Cayley, Note on Abel's Theorem. 119 



during the past twelve months— as during the preceding year; and 

 I trust that my Presidency which closes to-day will not in future 

 be thought to have been detrimental to the prosperity of the 

 Society. 



Mr Balfour having taken the chair the following communica- 

 tion was made to the Society : 



Note on Abel's Theorem. By Professor Cayley. 



Considering Abel's theorem in so far as it relates to the first 

 kind of integrals, and as a differential instead of an integral 

 theorem, the theorem may be stated as follows : 



We have a fixed curve f(x, y, 1) = of the order m; this implies 

 a relation f'(x)dx+f'(y)dy = Q, between the differentials dx, dy 

 of the coordinates of a point on the curve ; and we may therefore 



write 



dx dy_ 



do>== f¥)~~f¥)' 



and, instead of dx or dy, use dco to denote the displacement of a 

 point (x, y) on the curve. 



Taking for greater simplicity the fixed curve to be a curve with- 

 out nodes°or cusps, and therefore of the deficiency \{m -l)(m- 2), 

 we consider its ran intersections by a variable curve <f>(x, y,l)=Q 

 of the order n. And then, if (x, y, l) m ~ 3 denote an arbitrary ra- 

 tional and integral function of (x, y) of the order ra - 3, the theorem 

 is that we have between the displacements dta lt dco 2 ...dco mn of the 

 mn points of intersection, the relation 



t (x, y, l) m - 3 da> = 0, 



where the left-hand side is the sum of the values of (x, y, l) m ~ 3 do), 

 belonging to the mn points of intersection respectively. 



For the proof, observe that, varying in any manner the curve <j>, 

 we obtain 



fdx + fdy + H=0, 

 dx dy 



where S</> is that part which depends on the variation of tiie co- 

 efficients, of the whole variation of <f> ; viz. if </> = ax tt + bx n «+..., 

 then 8d> = x n da + x n - y ydb + ..., 8<f> is thus, in regard to the co- 

 ordinates (x, y), a rational and integral function ol the order n. 

 Writing in this equation 



dx,dy = j y dco,--da>, 



9—2 



