Problem in Two Dimensions. 435° 
regard X as given, and suppose « (the width of the plate) to 
diminish without limit, we see that (13) also diminishes without 
limit. 
The application of these results to the problem of the open 
pipe in two dimensions depends upon the imaginary intro- 
duction of a movable piston, itself without mass, at the 
mouth of the pipe—a variation without influence upon the 
behaviour of the air at a great distance outside the mouth, 
with which we are mainly concerned. The conclusion is that 
if the wave-length or pipe-length be given, the mouth or 
open end may be treated more and more accurately as a loop 
as the width is diminished without limit. Both parts of the 
pressure-variation, corresponding the one to inertia and the 
other to dissipative escape of energy, ultimately vanish. 
So far we have considered the case where in (3), (4) the 
width of the strip is small in comparison with the wave- 
length. It remains to say something as to the other extreme 
ease; and it may be well to introduce the discussion by a 
brief statement of the derivation of the semi-convergent series 
n (6) by the method of Lipschitz ™. 
ee 
bw)? where w is a complex 
Consider the integral {77 
J (1 
variable of the form uw+z2v. ibe we represent, as usual, 
simultaneous pairs of values of w and v by the coordinates of 
a point, the integral will vanish when taken round any closed 
circuit not including the points w=+7. The circuit at 
present to be considered is that enclosed by the lines u=0, 
v=1, and a quadrant at infinity. It is easy to see that along 
this quadrant the integral ultimately vanishes, so that the 
result of the integration is the same whether we integrate from 
w=i to w=iw, or from w=? to w=~ +2. Accordingly 
. py oe 1 cme au ae bere de 
1) Seis?) / (2ir) 8 
| Ee 
1? 37 eyo.) r 
=(5) a {a= i i: 4 ii 2. (8ir)? at eee (Sin) ae Re 
on expansion and integration by a known formula. This 
agrees with (6), if ka he written for r. 
In arriving at the ‘value of \\ Vde dy, we have to integrate 
(5) twice with respect to x between the limits 0 and 2. 
* Crelle, Bd. lvi. (1859). See also Proc. Lond. Math. Soc. xix. p. 504 
(1888) ; Scientific Papers, iii. p. 44. 
