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. 443
aQ1 aQ2 aQ3 aQ4
therefore + + + =,
ax, 2ax, a3 ax4
which is a necessary condition for the consistence of the four equations just written. It is
satisfied for instance by
Q2=iQ1= af +i af Q4=Q3= f + a
f being any function of x,, x,, x3, x4; corresponding to these values the four equations just
written are satisfied by
P1 = P34=0, P13 =, P14=f, P2 = if, P24=-f.
But it does not thence follow that
(C13 + i~14 + tK23 - 24) f. dSn-2
= f{( +iK,) (f - )) ( K- + iK4) (f + i )}dS
for the first integral vanishes for a complex (z - 2)-fold, and the second integral does not
necessarily vanish, as we see by taking for instance
i (1x -tl)2 + (X2- t2)2
f 2 (. - 21) ( 2) - tl)2 + (X2 - t2)2 + (X3 - t3)2 + (X4 - t4)2
when we get
+ = (a) (lt) af + a a
~ —+^.*/-=.-a ^. =t, +?- v d} (x\tg
ax, a/ X 4 ax,3 dx4 vax a/x,
whereby the second integral becomes
fd&s + {(Kl+i~) ( -i ) ( X+) (X t) +( +i 4) ( -i (1 t)},
which is not always zero.
In explanation it may be noticed that on the (n-2)-fold there are points where
e: = 72; and for these f is infinite.
25 Julz 1899.
56-2