1888.] 



On JEolotropic Elastic Solids. 



217 



from that of a right circular cylinder. The corresponding form of 

 the solution obtained for the spheroid, when the ratio of the polar to 

 the equatorial diameter becomes infinite, may thus be expected to 

 apply very approximately to the portions of a rotating cylinder not 

 too near the ends, and thus for a long thin cylinder to be for all 

 practical purposes satisfactory. This is verified directly, and it is 

 shown that this solution is in all respects as approximately true as 

 that universally accepted for Saint- Venant's problem. According to 

 the solution the cylinder shortens, and every cross-section increases 

 in radius but remains plane. The shortening and the increase in the 

 radius are, of course, proportional to the square of the angular 

 velocity. 



Part IV treats of the longitudinal vibrations of a bar of uniform 

 circular section and of material the same as in Part II. Assuming 

 strains of the form — 



radial = r ^(r) cosQ?2— a) ooskt, 

 longitudinal = 0(r) sin(pz — a) cos M, 



<fi(r) is found in terms of ^(r) by means of the equations established 

 in Part II. Prom these equations is deduced a differential equation 

 of the fourth order for YK r )» an( i f° r this a solution is obtained con- 

 taining only positive integral even powers of r. A relation exists 

 determining all the constants of the solution in terms of the co- 

 efficients a and a 2 of r° and r 2 . In applying this solution to the 

 problem mentioned, terms containing powers of r above the fourth 

 are neglected, and it is shown to what extent the results obtained are 

 approximate. 



On the curved surface the two conditions that the normal and 

 tangential stresses must vanish determine a 2 in terms of a , and lead 

 to the following relation between h and p — 



k = p{~J^-iP w '}- 



Here p denotes the density and a the radius of the beam, while M 

 is Young's modulus, and a the ratio of lateral contraction to longi- 

 tudinal expansion for terminal traction. This agrees with a result 

 obtained by Lord Rayleigh* on a special hypothesis. 



Proceeding to the terminal conditions, it is shown how p is deter- 

 mined from the conditions as to the longitudinal motion at the ends 

 being either quite free or entirely non-existent. Since a depends 

 only on the amplitude of the vibrations, we are left with no arbitrary 

 constant undetermined. If the bar be so " fixed " at its ends, that 

 the radial motion is unobstructed, this leads to no difficulty, but if an 



* « Theory of Sound,' vol. 1, § 157. 



