56 Proceedings of the Royal Irish Academy. 



iXow, "vre know that to each double point correspond two values of the 

 parameter, that is to say, one corresponding to each branch of the 

 cui've on which the double point lies. The number of double points 

 is consequently equal to half the number 



{m - 1) {m - 2) or ^{m - 1) {?n - 2). 



Having found the equation of the curve, we are then to find the 

 greatest common measui'e of Z, M, and i\^ which will give us the 

 function A, and consequently the double points on the curve. This 

 is most simply done by dividing Z by J{J.2, ^s)- 



"We now turn to the discussion of the cuiwe given by the 

 equations 



y = A2 + JB^ yR. 

 & = A3 + £3 ^/ R. 



Proceeding in the same manner as that adopted in the case of the 

 unicursal curve, we find 



Z is proportional to the determinant, 



dX ^"^ dX ^ ■ d\ 'dX ' ^ ^^ ^ ' ^^ dX 



'^W^'-^^BJ-i "^.yu'^.B/4 



^y R i^R 



d}!. dfj. d/x d/ji 



2 yR 2 yR 



dit. dfj. dfj. dfx ^ dfx. ^ dfji 



(6) Z = K{2RIJ{A,, R.) -r JiR„ A,)] + R,J{A„ R) + £,J{R, A,), 



+ yRl2J{A„ A,) + 2RJ{B, £,) + £oJ{R„ R) + B,J{R, A,)-]}. 



jS^owthe degree of R is 2«, that of J{A^ £2) is 



m - 1 + m - w - 1, or Im - n - 2, 



and consequently the function multiplied by A is of the degree 

 2in + n - 2 in A and [x. 



