|88] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION 



than k. Hence, if the formulae of that section applied to such fine wires, the effect of the wire 

 on the arc of vibration would be much greater than its effect on the time of vibration, and 

 therefore would be quite sensible. But it has been shewn in Section IV., that the effect of the 

 wire in diminishing the arc of vibration is probably greater than would be given by the 

 formula, and therefore the uncertainty depending on the wire is likely to amount to a very 

 sensible fraction of the whole amount. Again, since Bessel's experiments were all made in air, 

 no data are afforded whereby to eliminate the portion of the observed result which was due to 

 friction at the point of support, imperfect elasticity of the wire, or gradual dissipation of vis 

 viva by communication of motion to the supporting frame. Moreover in the case of the long 

 pendulum the observations were made with rather too large arcs, for the law of the decrease of 

 the arc of vibration deviated sensibly from that of a geometric progression. In Baily's 

 experiments, only the initial and final arcs are registered, and not even those in the case of the 

 " additional experiments." Hence these experiments do not enable us to make out whether it 

 would be sufficiently exact to suppose the decrease to take place in geometric progression. 

 Moreover, the final arc was generally so small, that a small error committed in the measure- 

 ment of it would cause a very sensible error in the rate of decrease concluded from the 

 experiment. For these reasons it would be unreasonable to expect a near accordance between 

 the formulae and the results of the experiments of Bessel and Baily. Still, the formulas might 

 be expected to give a result in defect, and yet not so much in defect as not to form a large 

 portion of the result given by observation. On this account it will not be altogether useless to 

 compare theory and observation with reference to the decrement of the arc of vibration. 



73. Let us first consider the case of a sphere suspended by a fine wire. Let the notation 

 be the same as was used in investigating the expression for the effect of the air on the time of 

 vibration, except that the factors k', k\ come in place of k, &,. Considering only that part of 

 the resistance which affects the arc of vibration, we have for the portions due respectively to 

 the sphere and to the element of the wire whose length is ds, and distance from the axis of 

 suspension s, 



kMnil+a) — , «, ds.ns—-, 



v * dt I dt 



and if we take the moment of the resistance, and divide by twice the moment of inertia, the 



dQ 

 coefficient of — in the result, taken negatively, and multiplied by t, will be the index of e in 



the expression for the arc. Hence if a be the initial arc of vibration, and a t the arc at the 

 end of the time t 



k'M'jl + ay+^k^M.'P vt 



log < a °- log « a ' — jf(i+«)«4*»*» •*? ' ' ' ( m 



M' (I + ay being as before taken for the moment of inertia of the sphere, which will be 

 abundantly accurate enough. If then we put I for the Napierian logarithm of the ratio of the 



