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. 423
Then we have the result; the p functions
=-ji f_ i (^'-1, 28-1 +TK2i-1, 28s) (aH n- i ) dSn-2
J 5=1 + a 28-1 OX _ 2
extended over the infinite complex (n-2)-fold, are functions of the complex variables
r1,..., rp, and are the partial differential coefficients of a function: =f (i ~ +4- 3dr-2 +... + T n'-dTp).
13. We proceed now to put this result into a new form, from which it will
appear that the real part of the function 1' is equivalent with a result given by
Poincaré, being a generalised potential function, and that the imaginary part is a
generalised solid angle function.
Putting,2r-1 = 2r + 8.21-1,
we have, clearly,
tc2î, - 3I aHM aH+ i? +.2 s
82r-1 -— [ KC2r-1, i aX+ K-' a -l, 2 x- +.. + K2r-i, %n a) dSXn-2
8)2r = j\ - a-lH2 f74A2 - - _.. + K.c2r-l, n-l 8 _ dSn 2;
of these the latter, in virtue of the equations
C2i'-1, 2S-1 = K21', 2S, C2r-1, 2S - - K2r, 2S-1
can be written
2ar = JkK2r 1 a + KC2. 2 +2. + K2r'n an sn-2
We proceed now to shew that in fact
- afr dSn-2,2) -ja-, fn-2)
these integrals, like the others, being extended over the whole infinite complex (n - 2)fold, and supposed convergent. Take the first of the two forms given for 2,., namely
^2 _ [` /m 7^, ^ i( ^H, aHH aH
2r = - f H dnK2r-1, 2r 9H dS 2 m + KC2 -1, 2k-1 St-2k )2>
J C2r-1 k<=1 a%^2^1 > aX2k
where k does not assume the value r; over a finite portion of the (n - 2)-fold, bounded
by a closed (n-3)-fold, the integral
r( Ï alHM aWm aHm
J K2k, 2 -1. f_ + "K2r-1 2k-i - + -k, 2k d n2r- d 2
is by the results of Part I. equal to
jfCk-1, 2k, 2-i _ Hm dS&3_,