Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

420 MR BAKER, ON THE THEORY OF FUNCTIONS
taken over the limited portion of the (n - 2)-fold; we have
afr a8$ f ( aR1 aRl aRi aMRn
at, -"ag =-J 1 F(Kri + +* rn a., — A/cs a - Ks n a J dn -2,
at8 atr ax8 ax axax
and therefore, adding to the right hand the vanishing quantity
laRI aR2 aRn
rax ax, a, ( Raxanx(\ 7
we have
ar as - f(ir aRi +rsaR aRi)S
ats atr i-iJX 1 axi +Ksj
where i does not take the values r, s. Thus by the formula proved in Part I. we
have
aar _ a-8a 1=E t sriRi. d Sn-3)
where the integral is taken over the (n - 3)-fold bounding the limited (n - 2)-fold over
which the integrals j,..., n are taken.
11. It follows that if the (n - 3)-fold integral vanishes, the expression
dt1 +... +,ndtn
is a perfect differential; on grounds further considered below (~~ 12, 22) we suppose that
this (n-3)-fold integral does vanish; we suppose also that L2r -iL2r-l and that the
(n - 2)-fold is a complex (n - 2)-fold; then from the equations
/C2r-1, 2-1 = 2r, S l, 2 Cr-l, 2 - -C2r, 28-1
it follows that
Ç'2r = f {(K2r, + igK2r,,) L1 +4... + (K2.,2,_r-2) r-i +... + (Ka:r, n-. + iK2r,n) Ln-} dS_-2
-= i f {(Krc_, * iC2r-_, 2) L +... + (iCKr-i, 2r) L2r-.i +... + (tCr-l, n-i + k2r1-l, n) Ln-i} dS.-2
so that
2r-= i2r-1,
and therefore
a~a2r-l = a = a i 8s-= i a2r-_,
at2s atr2r-. at-2r-i at2s-1
which gives
(at, +ia)2-l=o 0.
Thus, under the hypotheses introduced, all the functions C,,..., n are functions of
the complex variables Ir,..., rp, and there is a function
< = f((dt2 + ^A2 +... + dtn) = f(idTi + ^d72 +... + n-ildTp)