805] 
note on abel’s theorem. 
41 
here the 4 points of intersection are on a line y = ax + 6; we have therefore 
y x = ax x + b,, y 4 = aXi + b ; the equations between the dm’s give 
(y x — ax x — b) dm x + ... + (y 4 — ax 4 - b) dm A = 0, 
that is, is a single relation 0 = 0; or the 3 equations thus reduce themselves to 
2 independent equations. 
Again, if the fixed curve be a quintic, m = 5, there are here between the displace 
ments the 6 equations 
lx 2 dm = 0, Hxy dm = 0, 2y 2 day — 0, 
'¡Lx dm = 0, 2y dm =0, 2 dm = 0; 
the two cases in which the number of independent equations is less than 6 are (i) 
when the variable curve is a line, and (ii) when the variable curve is a conic. For 
the line n= 1, and the number should be =3. We have the above 6 equations; but 
the equation of the line is ax + by + c = 0, that is, we have ax x + by 1 + c = 0, &c. ; we 
deduce the 3 identical equations 
%x (ax + by + c) = 0, % (ax + by + c) — 0, 2 (ax + by +c)= 0, 
and the number of independent equations is thus 6 — 3, =3 as it should be. 
So when the variable curve is a conic, n = 2; the number of independent equations 
should be = 5. The points of intersection lie on a conic (a, b, c, f, g, h]\x, y, l) 2 = 0; 
we have therefore the several equations (a, b, c, f g, li§x x , y x , 1) 2 = 0, &c.: we have 
therefore the single identical equation 
2 (a, b, c, f g, li$x, y, If dm = 0, 
and the number of independent equations is 6 — 1, =5 as it should be. 
Obviously the like considerations apply to the case where the fixed curve is a 
curve of any given order whatever.