Roberts — On Bicursal Curves. 



55 



But these equations show us that L, M, and iVare proportional to 

 the determinants 



rfy 



dz 





dz 



dx 



dz 



dy 



dX 



dX 





dX 



dX 



dX 



dX 



dy 



dz 



> 



dz 



dx ' 



dx 



dy 



dfx 



dfx. 





dfx 



dfji 



dfx 



dfx 



Now we may write 



dy dz_dy ^^j^^^^j. 

 dX dfji dfj. dX 



where J stands for the Jacobian of the quantics Ao and A3, and 

 where it is to be remembered that 



J(A„A,) = -J(A3, A,). 

 Consequently we may write 

 (5) L =A J{A„ A,), 



M=AJ{A3,A,), 

 Ii = A J{A„ Ao), 

 where A is some quantity yet to be determined. 



Now the equation of the curve being of the m*'* degree, 



dU 

 dx 



is of 



the {m - 1)"* in x, y, and s, and, as these are each of the degree m in 

 the parameter, L, regarded as a function of the parameter, is of the 

 m{m - 1 )''' degree. 



The system of equations (5), however, shows us that L, M, and iV 

 are proportional to functions of the parameter of the degree 



2m - 2, for J [A,, A^), J(A„ A,), J{A„ A^) 



are of the degree 2m - 2. 



Hence, it follows that L, 21, N, when expressed in tenns of the 

 parameter, contain a common factor A whose degree in A and jx must 

 be equal to the difference between ot(»^-1) and 2(w-l), or 

 m^ - Sin + 2. 



L, M. and i\^ can thus simultaneously vanish for the 



{m - 1) {m - 2) roots of A = 0. 



K.I. A. PROC, VOL. Tin., SEC. A.] G 



