322 
PHYSICS: A. A. MICHELSON 
the corresponding return displacement at the time, /, counted from the 
instant of release, will be 
To account for the viscous term assume* 
F=eS''S m- 
whence 
S, = \- {Fd?\ , p = — 
Lpe J J n -\- \ 
li F = constant, and Fo the constant value of F during the preceding 
stress during the time, to , 
Ss= - [(Ft + FotoY- (FotoY] 
counting from the actual zero. 
As shown by the formula, if the previous strain be considerable the 
new strain is relatively small. This strengthening by previous strain 
is one of the striking features of the behavior of every substance which 
exhibits viscous yield. 
If, in this expression, F represents the actual stress, it assumes that 
the viscous force is proportional to the velocity, which is true for fluids ; 
but for ^solid friction,' the force is independent of the velocity. 
It may be assumed in the present case of internal viscosity of solids 
that the actual law may be between these two extremes, e.g. 
P = a {Sf in which K < 1 
This would give instead of P, or, in better agreement with experiment, 
F' = PCe''' 
The elastico- viscous term is readily obtained by making the viscosity 
coefficient a function of the time. 
Thus, if the restoring force be represented hy a s and the viscous re- 
sistancet by ei!"S, the integration gives S2 = Sq{1 — e r ) where « =~ 
andt r = —m-\-l. 
* Experiment gives p = 3^, (0.3 to 0.6) which makes n = 1, The usual assumption, n = 0 
gives p = 1. 
t The assumptions in both viscous and elastico-viscous hypotheses made the viscosity 
coefficient (that is the coefficient of S) zero at the beginning of the motion and infinite at 
I = CO which is, of course, inadmissible. Instead of S"- and t"^ we might substitute 
(iS + S"") / (6 + 5^) and (7 4- r) / (c + 1"^) in which fi/b and y/c are very small; but the 
resulting equations are far lei;s simple and are not appreciably more accurate in expressing 
the results than those here given. 
I The usual assumption, m = 0, gives r = \. 
