Barus and Strouhal— Viscosity of Steel. 29 
identical couples.* The absolute value of these couples during 
the experiment remains constant to 2 or 8 percent. If the 
sectional areas of the wires be identical our apparatus leads to 
results which come very close to Newton’s definition of vis- 
cosity. Newton suggested that the internal friction of liquids 
is cet. par. directly “proportional to the difference of velocity 
between nearly contiguous surfaces. In the bifilar apparatus 
the torsional viscosities of two substances are equal if for iden- 
tical strains and equal sectional areas torsional change of form 
occurs at like rates per unit of length This is the condition 
of rest or zero-motion of the suspended body. 
We do not at present wish to do more than advert to an im- 
portant deduction here: it is obvious that if the sections of the 
wires be so closen that the motion of the bifilar body is zero, 
the viscosity of the wires must be inversely related to those 
sections. This principle apparently enables us to arrange solids 
in a scale of viscosity. We may formulate it approximately 
thus: 
Consider an elementary ring of either wire, whose height is 
dz and whose right sectional area is 2ardr. Let df be the 
amount of tangential force uniformly distributed over this area. 
At the time ¢ Jet the velocity of the upper surface relatively -to 
the lower be cr, where c is a time-function and cet. par. a char- 
acteristic of the viscosity of the wire. Then if fe be the co- 
efficient of cet at the time ¢ we have 
, er2nrdr 
Tea Mia EM, yn dh 
If we multiply by 7 and then integrate between zero and @ 
(thickness = 2e), the numerical result is the part of the im- 
pressed torsional couple which corresponds to the length dz. 
A similar integral holds for the other wire, to distinguish 
which it is merely necessary to accentuate f, w, ¢, 7, p, The 
sum of the two integrals is zero. If, moreover, we put c=c’ 
in view of the state of rest of the bifilar body, we find that the 
viscosities (w4, 4’,) of the two substances (wires) are to each 
other inversely as the squares of the respective sectional areas 
by which the motion of the bifilar suspension is annulled. 
Unfortunately the problem is much more complex than it 
here appears. The dependence of torsional deformation on 
time in case of a single wire is obviously related to the char- 
acter of the molecule. When two different substances in wire 
form are twisted bifilarly against each other, the effect will 
* Many years ago we compared the longitudinal resilience of hard and soft 
steel by fastening one end of thin wires in a vise and bending them with a 
weight applied at the other. We found but insignificant differences. Hence the 
stored torsions of two steel wires, hard and soft, produced by equal couples are 
cet. par. of equal angular value. 
