52     ON THE EFFECT OF THE INTERNAL  FRICTION  OF FLUIDS
M\, M_e) and 1 - M'_e by *r/4, and by L, as well as in reducing the right-hand member of (105) to the form k + J~^l ti. The terms of the series (113) were calculated to five places of decimals. That these series are sufficiently convergent to be employed when m-1'5, might be presumed from the numerical values of the terms, and is confirmed by finding that they give k = 1*952, and Jc' ==1*153. For m = r5 and a few of the succeeding values, the second and third of the series (113) were summed directly as far as m"^5 inclusively, and the remainders were calculated from the formulae (114). Two columns are annexed, which give the values of m% and ttl^2&', and exhibit the law of the variation of the two parts of the force F, when the radius of the cylinder varies, the nature of the fluid and time of oscillation remaining unchanged. Four significant figures are retained in all the results.
m | k | k^f | m^2/c | m^2# | m | Jc | W | m^2& | m^2/c'
0 | 00 | oo | 0 | 0 | 2-1 | 1-677 | •7822 | 7*395 | 3-450
•i | 19-70 | 48-63 | •1970 | •4863 | 2-2 | 1-646 | •7421 | 7-966 | 3-592
•2 | 9-166 | 16-73 | •3666 | •6691 | 2-3 | 1-618 | •7059 | 8-557 | 3-734
•3 | 6-166 | 9-258 | •5549 | •8332 | 2-4 | 1-592 | •6730 | 9-168 | 3-877
•4 | 4-771 | 6-185 | •7633 | •9896 | 2-5 | 1-568 | •6430 | 9-799 | 4-019
•5 | 3-968 | 4-567 | •9920 | 1-142 | 2-6 | 1-546 | •6154 | 10-45 | 4160
•6 | 3-445 | 3-589 | 1-240 | 1-292 | 2-7 | 1-526 | •5902 | 11-12 | 4-303
•7 | 3-082 | 2-936 | 1-510 | 1-439 | 2-8 | 1-507 | •5669 | 11-81 | 4-444
•8 | 2-812 | 2-477 | 1-800 | 1-585 | 2-9 | 1-489 | •5453 | 12-52 | 4-586
•9 | 2-604 | 2-137 | 2-110 | 1-731 | 3-0 | 1-473 | •5253 | 13-25 | 4-728
1-0 | 2-439 | 1-876 | 2-439 | 1-876 | 3-1 | 1-457 | •5068 | 14-01 | 4-870
1-1 | 2-306 | 1-678 | 2-790 | 2-021 | 3-2 | 1-443 | •4895 | 14-78 | 5-012
1-2 | 2-194 | 1-503 | 3-160 | 2-164 | 3-3 | 1-430 | •4732 | 15-57 | 5-154
1-3 | 2-102 | 1-365 | 3-552 | 2-307 | 3-4 | 1-417 | •4581 | 16-38 | 5-296
1-4 | 2-021 | 1-250 | 3-961 | 2-450 | 3-5 | 1-405 | •4439 | 17-21 | 5-437
1-5 | 1-951 | 1-153 | 4-389 | 2-595 | 3-6 | 1-394 | •4305 | 18-06 | 5-580
1-6 | 1-891 | 1-069 | 4-841 | 2-739 | 3-7 | 1-383 | •4179 | 18-93 | 5-721
1-7 | 1-838 | •9965 | 5-312 | 2-880 | 3-8 | 1-373 | •4060 | 19-82 | 5-863
1-8 | 1-791 | •9332 | 5-804 | 3-024 | 3-9 | 1-363 | •3948 | 20-73 | 6-005
1-9 | 1-749 | •8767 | 6-314 | 3-165 | 4-0 | 1-354 | •3841 | 21-67 | 6-145
2'0 | 1-711 | •8268 | 6-845 | 3-307 | 00 | 1 | 0 | 00 | oo
The numerical calculation by means of the formulas (103), (104), (105) becomes very laborious when many values of the functions are required. The difficulty of the calculation increases with the value of m for two reasons, first, the calculation of the functions M"_0, &c. becomes longer, and secondly, the moduli of the numerator and denominator of the fraction in the right-hand