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.

OF SEVERAL COMPLEX VARIABLES. 421
for which
ar - = 2r-1~
12. We now give a more special value to the functions L1,..., Ln. Let
1 n-2
(lt) = T- [(- tl)2 +. + (n - t)2] 2
and
Hm=,(x lt)- P(xO()+ jt )(xO)-... + - t () (i 0o),
m being a finite positive integer or zero, and
t(t D- = t + 3... + t',
When r2, =t +... + t2, is sufficiently small in regard to R2, = 1+... + x2, we have
(- 1m+l / a \mf+i (_ 1)m+2 / a \M+2
-(n+)!t ) ( +I )! t; g() 0(o)+... to o,
rm+i y
R-l Kin+i () + En K1+2 (+ ) +. to 00,
where u denotes (xt, +... + xt,)/Rr and is numerically less than unity, and K, s(U) denotes,
save for a factor independent of,/ but depending on n and s,
U 2n-3 ds r n-3 -( -) —()h )~ 2- i
As we can find a real angle 0=cos-1/, we have
1. 1 / r \-"rn-2 n-2
(xlt>) 2 R=-2 - Qei) 2 (1-R ( e i) 2;
by expanding the binomials and considering the explicit expression for the coefficient
of rk+/1Rn+k-l it is immediately obvious that this coefficient is not greater than if 0
were zero. Thus when r < R the absolute value of Hm is of the form
rm+l (- _ 2) (n- 1)... (n + n- 2) 1
Rn+m-i lm+ 1 1 - e'
where, supposing n, m, r fixed, e is only unity when R =r, but for R>r has zero for
limiting value as R increases indefinitely. It follows that a value R0 can be chosen
such that for fixed n, m, r and all values of R > R, we may have
I TmT 1 R
with B a finite constant independent of R.