
Ae , 
EQUILIBRIUM OF ELASTIC SOLIDS. 111 
By eliminating oe between (54.) and (55.) we have 
v2 OO OMG) 
Rey OOS tte a ye Adee 4 (56.) 
When P=0, M depends on the sixth power of the radius and the cube of the 
angle of torsion, when the cylinder is composed of separate filaments. 
Since the force of torsion for a homogeneous cylinder depends on the fourth 
power of the radius and the first power of the angle of torsion, the torsion of a 
wire having a fibrous texture will depend on both these laws. 
The parts of the force of torsion which depend on these two laws may be 
found by experiment, and thus the difference of the elasticities in the direction of 
the axis and in the perpendicular directions may be determined. 
A calculation of the force of torsion, on this supposition, may be found in 
Youne’s Mathematical Principles of Natural Philosophy ; and it is introduced 
here to account for the variations from the law of Case II., which may be observed 
in a twisted rod. 
Case VIII. 
It is well known that grindstones and fly-wheels are often broken by the 
centrifugal force produced by their rapid rotation. I have therefore calculated 
the strains and pressure acting on an elastic cylinder revolving round its axis, and 
acted on by the centrifugal force alone. 
The equation of the equilibrium of a particle (see Equation (21.)), becomes 
where qg and p are the tangential and radial pressures, / is the weight in pounds 
of a cubic inch of the substance, g is twice the height in inches that a body falls 
in a second, ¢ is the time of revolution of the cylinder in seconds. 
By substituting the value of g and = in Equations (19.), (20.), and neglect- 
ing 0, 
pee 4 dp _40?k | dp) l(adp_ 402k | dp 
a Comat RBS Wie: r+rSh) +7, (3-3 aie SS) 

5 : le Wek E\ , 
which gives p= tpl 2 + a + ¢, 
lL) Wek 2E). s 
ae as a ray (- 4 rat )r Sa 5 (57.) 
I=" et IGE 
If the radii of the surfaces of the hollow cylinder be a, and a,, and the pres- 
sures acting on them 4, and /,, then the values of ¢, and c, are 
VOL. XX. PART I. 26 
