Mr Dixon, An example of complex double integration. 389 
Now the lengths of the a-path and y - path are respectively 2 irR 
and :}> kR, so that the integral does not numerically exceed 
27 tR . kR . c/R 3 or 27 rck/R, 
and therefore tends to zero as R increases. 
An expression for the value of the integral can also be found as 
follows. In the first integration, with respect to y, the subject is 
uniform and has poles r) 1 , t ] 2 ... enclosed by the path. Hence the 
result of the first integration is 
n 
2ltt 2 
r = 1 
This expression is symmetrical in g 2 ... rj n , and is therefore a 
rational function of x which may become infinite through the 
vanishing either of U or of dV/dy, when y = rj 1 , rj 2 
When d V/dy = 0 we have either a branch-point of the functions 
7] x , r/ 2 ... or a node on the curve F=0; the latter cannot make U 
vanish also, by hypothesis, and we may if necessary change the 
axes so that U does not vanish at any of the branch-points. We 
shall in fact choose for axis of y a line that is not parallel to the 
tangent to V — 0 at any point where it meets TJ = 0 or to any line 
that contains two of the intersections of U = 0, V = 0 . 
Suppose first then that dV/dy = 0, when x = x l} y=rj li and 
therefore U f 0. Two or more terms in the summation are infinite, 
but their sum is finite as will now be shewn. Suppose to fix the 
ideas that p 2 comes into coincidence with rj 1 when x — x 1 . Then 
before integrating at all we may deform the field and make the 
«-path exclude x 1 . The form of the ?/-path must of course vary 
with x so as always to include rp, rj 2 ... and exclude y l , y 2 ..., but 
this is not made more difficult by a coincidence between r] 1 , 77 „ 
which are both included. The conclusion is then that we may 
suppose all such points excluded from the a?-path, and that we 
are only concerned with values of x such that U, V vanish 
together for some value of y. It is easy to calculate the residue 
in such a case, for only one term in the series 
becomes infinite, by supposition, and its residue is 
U 
dv 
dy 
w /dUdV 
/ dx dy' 
d U 
where — is formed on the understanding that y is a function of x 
given by V — 0. The residue is therefore 
W 
id(U, V) 
0 (x, y) 
VOL. XIV. PT. IV. 
26 
