Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

19

we must insure now however that 1 shall be really a functional
of C, and not depend on the manner in which C is approached
by the summing of successive variations from Co.
Let C,, be any two parameter family of curves containing Co
and C, in such a way, say, that Coo = Co, and C1 = C. In
order that b shall not be dependent on the path that (X/A) traces
in going from (0, 0) to (1, 1) it is necessary and sufficient that
0 0( 0 0b
0O AX 0X 69u
This condition must hold for every two parameter family which
can be formed from the class of curves we are considering.
If we write 52 = c5(5), we shall have for it the value
902
and the test for integrability, corresponding to the possibility
of interchanging the order of differentiation in regard to X and ju,
lies in the possibility of interchanging the two variations An
and b1n of C without changing 52b.
The variation of SD, as given by (28), due to the new variation
i1n(Mi), depends upon the variation of F, which we may write
as some linear functional E[1inlM] of 5in, upon the variation
of 8n itself (i. e., o9/,u(On/OX)dXdg,), which we call 52n, and on
the variation of ds. The last is obviously given by the formula
- kaln8s,
where k is the curvature, counted positively if the center of
curvature lies on the positive direction of n. We have, then:
2 = f E[61n I M]bn(M)ds
(29)
+ J F[C, 4 I M](5/2n - kanSin)ds.
In order to investigate more closely the quantity 52n consider
the special equation,9[C] = fff(M)dxdy,
(C)
which defines I as a functional of curves. We have