1041 
d\ es 
(=) En) 
; : Cae 5 a : 
the lines, along which 7 is constant, are straight lines which are 
tangents to the boundary curve; a few of these lines are shown in 
Ö 
the diagram (they are dotted : aS 4, 1 and 2). *) 
4. We will now discuss the general case, in which R'—R is not 
infinitely small. For an aperiodic motion it is necessary that é in 
equations (8) and (8') be real, i. e. &' =O, and as in our physical 
Ree u 
problem 4’ is necessarily negative, it follows that 5 = (/ — 
N 
— — —m is purely imaginary; therefore 5 = 6" 7, where 5 
d is 
= pe a In that case the expression for u (equation 17") becomes: 
7 
RR? 
— aa LU! (R'—r) cos b" (R'—r) — (6? R'r +1) sin D(R'—r)], (56) 
i 
1) Properly speaking the quantities 3 and 7’ have no physical meaning in this 
io eh: 
case, but the ratio T represents the logarithmic decrement of the amplitude per 
unit of time. 
*) These lines are really only needed on one side of the point of contact, viz. 
above it, for the following reason. To every point of the aperiodic region two 
NI 
values of T belong, viz. the two roots of equation (26); calling these kj, and k, 
which are both negative the equation of the aperiodie motion will be 
garebian ete 7s) Gee te (DO) 
At least this would be the case, if from the beginning the motion of the sphere 
satisfied the equation 
da ze EM 0 (24 
— a el ea) oer re ee ot 
de | dt ) 
as in the case of a periodic motion (comp. § 4 of comm. 148d, note) ; our theory, 
however only applies after an infinite time, so that for practical purposes we need 
only consider one of the terms of (56), namely the one with the smaller value of 
N 
ps that term must become predominant after a sufficiently long time 
We may thus confine ourselves to equations of the same form as (8) and (8’) 
and the previous calculations which were based on these equations are applicable 
in the present case. 
