NOTE ON ABELS THEOREM.
39
[805
(1883),
of integrals,
be stated as
is a relation
¡s of a point
(X, y) on the
out nodes or
i intersections
m_s denote an
ie theorem is
>ints of inter
longing to the
btain
805]
where 8(f> is that part which depends on the variation of the coefficients, of the whole
variation of <f>; viz. if (f> = ax 11 + bx n ~' i y + then 8(f = x n da + x n ~ Y y db + ...; 8<f> is thus,
in regard to the coordinates (x, y), a rational and integral function of the order n.
Writing in this equation
dx,
d, J = % dm •
the equation becomes
/d(f> df d(f> df\
\dx dy dy dx)
dco + 8(f> = 0,
or say
that is,
— J (/, (f>) dw -j- 8(f> = 0,
dco =
8(f)
J(f, <t>)’
and then multiplying each side by the arbitrary function (x, y, l) m ~ 3 , we have
2 (*, y, 1 )“-* dco = t 8*.
where 8(f) being of the order n in the variables, the numerator is a rational and
integral function of (x, y) of the order m + n — 3: hence by a theorem contained in
Jacobi’s paper “Theoremata nova algebraica circa systema duarum sequationum inter
duas variabiles,” Crelle, t. xiv. (1835), pp. 281—288, [Ges. Werke, t. III., pp. 285—294],
the sum on the right-hand side is = 0: hence the required result £ (x, y, ]) m ~ 3 dtu = 0.
Observing that (x, y, l) m 3 is an arbitrary function, the equation just obtained
breaks up into the equations
Xdco = 0, Xxd(o = 0, %y do) = 0,..., Xx m ~ 3 do) = 0,..., %y m ~ 3 do) = 0,
viz. the number of equations is
1 + 2 + ... + (m — 2), = ^ (m — 1) (to — 2),
which is =p, the deficiency of the curve.
Suppose the fixed curve f (x, y, 1) = 0 is a cubic, m = 3, and we have the single
relation 2 do = 0, where the summation refers to the 3n 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 do) 1 + do). 2 + d(o 3 = 0 ;
here the two points (ir x , y)), (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 , y s );
there should thus be a single equation giving the displacement d(o 3 in terms of the
displacements dw 1 , dco 2 ; viz. this is the equation just found
do) 1 + do) 2 + d(o s = 0.