664 
§ 2. Let us now proceed to the treatment of linear elastic vibra- 
tion revised for hysteresis according to VorTERRA. 
Then the following equation holds for this movement : 
dx 
t 
ca ter = feed dr, 
(ee) 
Now VorrprraA shows that in general in case of elastic hysteresis 
W(t‚r) must be a function only of (t—t), though there are hereditary 
phenomena in which this need not be the case’). Disregarding the 
latter, we, therefore, write henceforth y(¢—r). The fact that w has 
this form, is, indeed, easy to understand; if it is assumed that for 
what ensues it is only of importance how long ago certain forces 
acted, and not at what absolute points of time this took place 
precisely, only the difference of time ¢—-r will appear, or in other 
words we imagine that the effect for what follows will be the same 
when we subject a certain previous history (with its consequence) 
to a translation in time. 
Let us now further assume in particular for w the form of the — 
function : 
t—T 
w(t—1t)=Ae 7%. 
in the above equation (1). The supposition is plausible, for the term 
t 
fe (rt) w (tr) dr 
—% 
accounts for the influence of the previous history on the condition 
at the moment {; or expressed more definitely : 
the element «x (rt) y (—rt) dr of that integral represents the contri- 
bution of the condition at the moment rt to the value of the accelera- 
ob ha 
da. shail 
tion at the moment ¢ (the term de in equation (1) ). It will be clear 
; 
that as the moment rt is longer ago with respect to ¢, this influence 
must be smaller; this is really in agreement with our supposition 
for w: 
bt 
y= A en EN 
for this becomes zero for t—rt — infinite, and increases with decreasing 
Lr. 
When we now work with this y, and solve the equation (1), we 
can at once derive 
Wass. Vouterra Lc. p. 114. 
