670 
As analogy to the solution (A) p. 668 we now get: 
K 
a—Aq : 
eA” Ae Poos gt + Age °° sin gat + (A) 
which must also again be adapted to the initial conditions. 
Finally also the remarks on the energy and on the limiting cases 
of the footnote on p. 665 and seq. may be repeated here. In 
the energy now another term appears, Aw, which represents the 
work done by the force A. And as regards the limiting movements, 
the same limiting movements also occur here again. 
And now I will again start from the integro-differential equation 
dx 
pramen. . = 
with general w and inquire into the condition on which this equation 
of vibration revised for hysteresis by VorTERRA, represents a damped 
movement. . 
First I state that there can always be given a point of time t 
so that the history of before this moment + may be neglected. On 
physical grounds the function w must be such that: 
lim w (t—r) = 0, 
t—t= © 
for the influence of what took place very long ago, must become 
small. Now this wy under the integral sign in the righthand member 
of (1) must, however, still be multiplied by a(x), the a at this moment 
tT in the past, and when it is very large the product e(t) (t—t) 
may not be neglected after all. Now | observe that the number of 
times that in the previous history an x(t) can occur lying above a 
definite « which may be chosen arbitrarily great, must be decidedly 
finite, for the simple reason that we have to do with a physical 
problem. And then I go so far back in time that I have passed 
this finite number. 
Now equation (1) reduces to: 
t 
dx 
aa + an = | @ (t)W(t—r) dr. 
0 
In Fonet. de |. p. 97—99 Vourrerra gives as solution of this 
equation : 
«aS, (t/a) + OS, (t/@) 
in which S, and S, are two transcendental functions of @ as follows: 
