Colloquium publications.

48

THE CAMBRIDGE COLLOQUIUM.

(42) J R[C IM].r = 0,
(2 S[C I MM,1].r = 0, Sy[C I MM1]i-r = O, Sz[C I MMi].r 0,
where r represents the unit vector in the direction of the curve at M, and r,
the same quantity at M1;
(43) SX[C I MM1] = Sx[C I M1M],
(44) Sy[C I MM1] = Szy[C I IiM], etc.,
where Syz represents the z-component of Sy, etc.
A complex vector functional R = R1 + iR2 can represent the vector
derivative of a complex functional F = F1 + iF2, only if R1 is the derivative
of F1 and R2 is the derivative of F2. In the special case just considered, the
relations (42), (43), (44) will be satisfied for both real vectors R1 and R2, if
they are satisfied for the complex vector R, and vice versa.
Let us speak of a scalar functional O[C | M] as an integrand for a scalar
functional F[C], if the vector O[C I M]R[C I M] satisfies the conditions of integrability. In the special case mentioned in the preceding paragraph, the
equations (42). are automatically satisfied, so the conditions of integrability
refer merely to (43) and (44).
If O[C I M] is an integrand for F[C], there is a functional q[C] whose vector
derivative is O[C M]R[C I M]. In fact:
(45) I[C] = ff 0[C I M]R[C I M] * da + h,
where the integration is extended over a cap of C, or over a cap joining a fixed
curve Co to the variable curve C, and h is an arbitrary constant.
The equation (45) may be written in the form
(45') 4[C] - 4Co] = ff o[C I M] d.d, = ' OdF,
where dF/do- denotes R[C [ M]. If we define:
d4,
d _ da
dF dF '
dowhere I[C] and F[C] are any two isogenous functionals, we shall have dT/dF
a scalar functional of C, M. Hence we have the formula
(46) fedF=fe dF.
If 0 is an integrand for F, and I and F are isogenous, then 0(dF/dTr) will be
an integrand for,f.
The relations (45'), (46) are invariant of a transformation of spaces. For a
closed surface, which does not contain singularities:
(47) f OdF = 0.
t Volterra, Rendiconti della R. Accademia dei Lincei, vol. III (1887), 2e
semestre, p. 229.