60 THE EQUILIBRIUM OF ELASTIC SOLIDS. 



t> CVrt 



and since - - and r - - are small, we may assume 



X X 



The force, when resolved in the tangential direction, is approximately 



1 d d ( SdSx rVStfYl 



-ja j- M= Er \r - -- \ , 

 rdQdr \ x x 2\a5/J 



By eliminating - : between (54) and (55) we have 



...................... (56) . 



2 x 24 



When P = 0, -Jf 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 

 Young'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. 



