in which: 
Summarizing: when the a, A, and g fulfil these conditions 1, II, 
and III, the hysteretical term, as VorLTERRA has assumed it, comes 
simply to damping of the vibrating motion. 
When we call the roots —p,, —p, £ q,?, the solution becomes: 
w= A,e mt + Ae —Patcosg,¢-+ A,e—Ptsng,t. . . (A) 
As regards initial conditions, 
c=, «= 0 
e.g. up to t=O (from t= — oo) by adaptation of the found solution 
(A) to these values. From the integral equation follows: 
t 
1 tr 
ae ——— 
ge time, fe de de 
— 6 
aa 
nme 0 asi eee 
PL 
and this value of ae must be used when we make A,, A,, and A, 
q 
by assuming the supposition that | ({—r) has only values differing from zero for 
values of t—r that are very small. 
When we put ¢ — Tr =S, our equation (1) reduces to: 
d*xz 
Te a= fvOreHe. @ 
0 
According to the equation mentioned : 
E = (Ey gE er yEee 
wat) d= + [Od |E Od 
0 0 0 
Let us now put: 
ao foo] 
[rouse m fewer 
0 0 
then we get after substitution in the equation (A): 
ie. a damped vibration. 
For & >a we have the damped exponential motion. 
As a special case also x= occurs in it. 
