

of a Solid Body in a Viscous Fluid. 633 



denominator, we obtain the same solution (21). The com- 

 parison brings out the connexion between the method of 

 solution of the particular form of integral equation and the 

 use of normal functions in dynamical problems. The latter 

 methods would not be available if we had not a complete 

 knowledge of the differential equations of the problem : for 

 instance, if it were stated directly as an integro-differentia! 

 equation like (3) in some problem of 'heredity/ 



4. The nature of the roots of (19) may be studied most 

 easily by graphical methods, or by using the form (18) or 

 equivalent expansions. It appears that, leaving aside the 

 possibility of multiple roots, there is an infinite series of 

 real negative roots and, in addition, a pair of roots which 

 may be complex, or real and negative. In the latter case 

 the motion is aperiodic ; in the former, the two complex 

 roots give the damped harmonic vibration while the re- 

 maining roots complete the solution according to (21) for 

 the given initial conditions. In the theory of determinations 

 of viscosity by oscillating cylinders or spheres it is usual to 

 assume a damped harmonic vibration, neglecting all the other 

 terms. 



Verschaffelt remarks that for a motion that is not purely 

 damped harmonic, the proportionality of the resistance to 

 the velocity no longer exists, and that it would then probably 

 be impossible to establish a general differential equation for 

 the motion. We have seen, however, that it may be ex- 

 pressed by an integro-differential equation as in (3). It 

 seems that in experiments under usual conditions, the final 

 state of a damped harmonic motion is practically reached 

 after a comparatively short time (a few minutes). 



With numerical values of the usual order, it is easy to see 

 that the lowest real negative root of (19) is much larger 

 numerically than the real (negative) part of the complex 

 roots. The matter would require closer examination if the 

 motion were ent : rely aperiodic, as in some experiments. 

 In the case of a sphere making oscillations of finite ampli- 

 tude, Verschaffelt has studied small damping effects due to 

 approximations involving the quadratic terms in the hydro- 

 dynamical equations ; this introduces damping coefficients 

 of three or five times the first approximation, and it may bo 

 that in such cases the purely aperiodic terms in the solution 

 should also be taken into account. 



5. It may be of interest to record the complete solution. 

 neglecting quadratic terms, for a sphere oscillating in a 

 liquid enclosed within a fixed concentric shell. 



Let co be the angular velocity in the liquid. the angular 

 Phil Mag. Ser. 6. Vol. 42. No. 251. Xoc. 1921. 2 U 



