Colloquium publications.

18

THE CAMBRIDGE COLLOQUIUM.

If our expression is of the form
(27) E[uI s]= ff(ssi)u(sl)ds + Ao(s))u) + A,(s) d(
Jo7 77 18] Z Ats) dsi=1
where s and si replace M and M1, the adjoint expression has the
form
E2[vIs] = fcf(sls)v(sl)dsl + Ao(s)v(s)
(27') c di
di
+ Z (- 1) i- (Ai(s)v(s)).
i=l 1 S
If Ei[uls] and Fl[u s] have for adjoints E2[v s] and F2[vls]
respectively, then Ei[Fl[u]] (that is, El[F[u Is'] Is]) has for
adjoint F2[E2[v]] (that is, F2[E2[v Is'] Is]). In fact, by (26),
f (s)ElFi[u l'] I s]d = f F[ ls]E2[ I s]ds
= F2[E2[vl s Is]u(s)ds.
Jc
Similarly, more than two expressions can be compounded, and
G2[F2{E2[v]}] shown to be the adjoint of E1[Fi{Gl[u]}]. In particular, if E[u M] is self adjoint, and F2 is the adjoint of F1,
then the expression F1[E{F2[u]}] is also self adjoint. This fact
we shall make use of, later.
15. The Conditions of Integrability. We wish to find the functional [CI A, B...] which will satisfy an equation of the type
(28) 4[C AB.. *] = F[C,, |A, B... M]6n(M)ds,
where F depends upon C and perhaps also d, as functional arguments (i. e., for instance upon all the values of 4 when A, B, * *
range independently over the curve C*), and on M as a point
argument. According to (28) we should expect that given
([Co I AB * ] the functional 1 would be determined for all
other curves C. This existence theorem will be considered later;
* In (25) F depends upon the particular values of P when A and B take the
position M on C.