1915] MOTION OF A VISCOUS INCOMPRESSIBLE FLUID 345
the only difference being the change from D to - D' and the reversal of sign in 7T/6, equivalent to the introduction of a constant (complex) factor.
When 77 exceeds 2'5, the second term of the series within { } in Xx is less than 10~2, so that for rough purposes the { 1 may be omitted altogether. We then have
.................. (21)
*! = -Drj '"* e^ • ^ sin (V2 . ^--»r/24), .................. (22)
• S2= D;77"ie^/2-'?esm(V2.77^-'7r/24-7r/6), ......... (23)
4= .DV*^2' ^008^2.^-77/24-^/6) .......... (24)
Here D and D' are both positive — the logarithms have already been given — and we see that sl3 t$ are somewhat approximately in the same phase, and ti, S2 in approximately' opposite phases. When 77 exceeds a small integer, the functions fluctuate with great rapidity and with correspondingly increasing maxima and minima. When in one period \/2 . if increases by 27T, the exponential factor is multiplied by e2*", viz. 535f4. From the approximate expressions applicable when 77 exceeds a small integer it appears that s^ ^ are in quadrature, as also sz, tz-
.For some purposes it may be more convenient to take Sl5 S2, or (expressed more correctly) the functions which identify themselves with Sa, 22 when T/ is great, rather than Slt 82, as fundamental solutions. When 77 is small, these functions must be calculated from the ascending series. Thus by (15) (0=1, £=0)
^ir-irC^ft-Sir-irdf)^ ..................... (25)
and (0=0, D = l)
S2=7r-ira)e-^/e^ + 37r-ir(f)^/8^ ..... : ....... (26)
Some general properties of the solutions, of (5) are worthy of notice. If S= s + it, we have
Let R = i (s2 + f) ; then
dR • ds , , dt
-j— = S ~j- +t-j-dr) dr} dv)
/s\* ft\* *s . H
and "7-~r =» ( T- ) +T~ +*T~i+^T^'
(Lrf \dr)J \dt)/ drf drf
of which the two last terms cancel, so that dzR/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, «a (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 S± 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.