On the effect of the internal friction of fluids on the motion of pendulums / by G.G. Stokes.

  • Stokes, George Gabriel, 1819-1903.
Date:
1851
Catalogue details

Licence: Public Domain Mark

Credit: On the effect of the internal friction of fluids on the motion of pendulums / by G.G. Stokes. Source: Wellcome Collection.

Provider: This material has been provided by The Royal College of Surgeons of England. The original may be consulted at The Royal College of Surgeons of England.

    44/104 (page 40)
    required in the calculation of the functions &c., and of the products LMq, &c,, in order to ensure a given degree of accuracy in the result. The calculation by the descending series (113) is on the contrary very easy. It will be found that the first differences of Vd^k' and of in* (^ - 1) are nearly constant, except near the very beginning of the table. Hence in the earlier part of the table the value of k or k' for a value of tn not found in the table will be best got by finding — m® or tn^Ar' by interpolation, and thence passing to the value of k or k'. Very near the beginning of the table, interpolation would not succeed, but in such a case recourse may be had to the formulas (103), (104), (105), the calculation of which is comparatively easy whem ttl is small. It did not seem worth while to extend the table beyond tn = 4, because where ttl is greater than 4, the series (113) are so rapidly convergent that k and k' may be calculated to a sufficient degree of accu- racy with extreme facility. 38. Let us now examine the progress of the functions k and k'. When 111 is very small, we may neglect the powers of ttl in the numerator and denominator of the fraction in the right-hand member of equation (105), retaining only the logarithms and the constant terms. We thus get whence A; + \/ - lA;'=l 1tt-®\/ - 1 4 7T m- {k -1) VH^k' = (115) L being given by (102) and (104), or (104) and (106). When 111 vanishes, L, which involves the logarithm of ttl“', becomes infinite, but ultimately increases more slowly than if it varied as 111 affected with any negative index however small. Hence it appears from (115), that A; — 1 and k' are expressed by 1tt“^ multiplied by two functions of III which, though they ultimately vanish with ttl, decrease very slowly, so that a considerable change in ttl makes but a small change in these functions. Now when the radius a of the cylinder varies, everything else remaining the same, ttl varies as a, and in general the parts of the force F on which depend the alteration in the time of vibration, and the diminution in the arc of oscillation, vary as a^k, a^k’, respec- tively. Hence in the case of a cylinder of small radius, such as the wire used to support a sphere in a pendulum experiment, a considerable change in the radius of the cylinder produces a comparatively small change in the part of the alteration in the time and arc of vibration which is due to the resistance experienced by the wire. The simple formulae (115) are accurate enough for the fine wires usually employed in such experiments if the theory itself be appli- cable ; but reasons will presently be given for regarding the application of the theory to such fine wires as extremely questionable. From III = ‘3 or ’4 to the end of the table, the first differences of each of the func-