1915] MOTION OF A VISCOUS .INCOMPRESSIBLE FLUID 34?
and from the imaginary part .
[ - f r \ r -A j * j.a - J ' ]
If we introduce the notation of double integrals, these equations become
_/Y I Isinh X (77 77') fa (77). 52 (77') - ti (77). £2 (T?')} ^ <&/ = 0> ..... .(35)
jjsinh X (77 - 77') fa (77). 4 (77') - s2 (77). t, (77')} dr)dr,' = Q} ......(36)
the limits for 77 and 77' being in both cases 77! and r/z. In these we see that the parts for which 77 and 77' are nearly equal contribute little to the result.
A case admitting of comparatively simple treatment occurs when X is so large that the exponential terms e^, e"^ dominate the integrals. As we may see by integration by parts, (31) then reduces to
$1 (^2) $2 (*7a) $1 (^i) $2 (^2) 0? ..............(37)
or with use of (29)
a o /~~\ ~ "......................\"°/
We have already seen that Si (77) cannot vanish; and it only remains to prove that neither can the integral do so. Owing to the character of S1, only moderate values of 77 contribute sensibly to its value. For further examination it conduces to clearness to write 77^ = a, % = I), where a and b are positive. Thus ,
[a (g a _ ^ 2\ ^
and it suffices to show that /" 2~ii\r cannot vanish. A short table makes this apparent [see p. 348].
The fifth column represents the sums up to various values of 77. The ap-
r00 (s a _ ^ 2^ $
proximate value of ^^f~^ is thus '2 'x 2'834 or '567. The true value J o (*i2 + tf)*
of this integral is (D'/D) sin 60° or '571, as we see from (30) and (19), (20). We conclude that (37) cannot be satisfied with any values of 773 and 77^
When the value of X is not sufficiently great to justify the substitution of (37) for (31) in the general case, we may still apply the argument in a rough manner to the special ca|e (773 4- % = 0) of (32), at any rate when 773/d^ is always positive. In the case of Slt when 77 = 0,^(0) = !, ^i(0)-0; 5/(0) = 0, so that 5(0) = ^, Rf (0) = 0. Again, when 77 = 0, Ğa (0) = 0, *, (0) = 0, so that E (0) = 0, R' (0) = 0. In neither case can R vanish for a finite (real) value of ?;, and the same is true of Sħ and 8Z. being blown from a loaded bag, charged beforehand with a foot blower. In this respect they are not fully comparable with those of Prof. Titchener, whose whistle was actuated by squeezing a rubber bulb. However, I have also tried a glass tube, 104 in. long, supported at the middle and rubbed with a resined leather. This should be of the right pitch, but the squeak heard did not suggest an s. I ought perhaps to add that the thing did not work particularly well.