1080 
characteristic length of the body (in the case of the sphere the 
radius). Let 7’ represent a time characterising the motion: in the 
case of an oscillating moment we naturally select as such the time 
of oscillation which, as we have seen, is also the periodic time of 
the liquid, at least for a body of revolution. *) 
These quantities we shall take, in each special case, as the units 
of length and time and shall put t= Tt, /= Ri, so that t and { 
will now represent the reduced time and the reduced length. As 
unit of mass we shall further take the mass of the (new) unit of 
volume of the liquid. The equations then retain their form’) 
op Santee Akagi ae ou a; il 
mob vee lak oF ie oere os (1) 
The only difference is, that 1 in the new system of units has a 
different numerical value from before, namely in view of its 
dimensions (L—! MT): 
gC? mk 
ult aR 
The viscosity has not one definite value in the new system any 
more than in the old units; the coefficient 1%’ in. equation (1) 
may thus assume various values and a first condition therefore for 
internal similarity of the liquid motion is, that this coefficient has 
the same value (e.g. in the C.G.S. system) in all the cases considered; 
in other words: by ascribing to 7’ all possible values from O to oo 
we characterize an infinite series of different liquid motions. 
as 
oe 
2. We have thus found that for two similar conditions of motion 
y'/ -must have the same value. If the motion of the body were an 
undamped harmonie one (or a uniform rotating motion) this condition 
would be the only one for internal similarity of two motions; but 
when the motion is a damped harmonic motion, the other condition 
is to be added, that the logarithmic decrement d of the swings (a 
dimensionless number, and therefore independent of the units chosen) 
must obtain the same value in similar cases; and as all possible 
values from 0 to oo ®) may be assigned to the decrement, the quantity 
d characterizes a second infinite series of different liquid motions. 
1) This is probably true in the general case too. With a uniform rotation the 
time of revolution would be taken for 7’. 
2) The action of gravity as having no influence on the motion is here left out 
of account. 
3) Even from —o to +0, if one chose to also consider motions with arti- 
ficially increasing amplitudes. 
