633 Prof. T. H. Havelock on the Decay of Oscillation 
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-differential 
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 
stato 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 coefticients 
of three or five times the first approximation, and it may be 
that in such eases 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 @ be the angular velocity in the liquid, 0 the angular 
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