1915]
MOTION OF A VISCOUS INCOMPRESSIBLE FLUID
343
If we take as ^ and S2 the two series in (8), the real and imaginary parts of each are readily separated. Thus if
S1 = s1+itl, S2 = s3 + it2, .....................(10)
Ave have on introduction of irj
- 9y 9y2
6>1~ " 27076 + 2. 3 . 5. 6 .""879" 11.12""'' ......
__ 9^ 9V.............
1~~2".~3"1"2T3T5T6T8T9~>"' ...........................
*2 = O~ 3.4:6.7:9. io + "":...............'............(13)
_9Y_ _______9V8_______I (14.
2-77 3".47677 + 3.4.6.7.9.10.12.13 '*"' ......( }
in which it will be seen that sls s2 are even in 97, while tls ts are odd.
When 77 < 2, these ascending series are suitable. When 77 > 2; it is better to use the descending series, but for this purpose it is necessary to know the connexion between the constants A, B and G, D. For x = ir) these are (Stokes) '
A =7r-^r(|){(7+De-^(ij, B = Sir-lT(%) {-C + Dei7r/0|. ...(15) Thus for the first series S, (A = 1, B •= 0 in (8)) , ;
log JD = 1-5820516, 0 = DeJ''r/0;......l............(16)
and for £2 (J. = 0,5=1) ;
logD' = 1-4012366, - C"= DV^/0,................(17)
so that if the two functions in (9) be called Sj and Z2,
These ;values may be confirmed by a comparison of results calculated first from the ascending series and secondly from the descending series when i) = 2. Much of the necessary arithmetic has been given already by Stokes*. Thus from the ascending series
5: (2) = - 13-33010, t, (2) = 11-62838;
s2(2) = - 2-25237, tz (2) = - 11-44664. In calculating, from the descending series the more important part is Si, since
For t] = 2 Stokes finds
Sj = - 14-98520 + 43-81046*,
of 'which the' log. modulus is i'6656036, and the phase +108° 52' 58"-99. When the multiplier 0 or C" is introduced, there will be an addition of + 30° to this phase. Towards the value of Sl I find
-13-32487 + 11-63096 i;
* Loc. cit. Appendix. It was to take advantage of this that the " 9," was introduced in (5).
1ard it well, said it was no s.