338 JDr. C. Ohree on Longitudinal 



simplicity. When one omits terms in a differential equation for 

 diplomatic reasons, the results may be perfectly satisfactory 

 under certain limitations. Owing, however, to the mutilation 

 of the differential equations, the resulting solutions are unlikely 

 to contain within themselves any satisfactory indication of the 

 limits to their usefulness. It is very much a case of running 

 a steam-engine without a safety-valve. 



2. When the bar is of circular section and isotropic, the 

 series occurring in (6) are Bessel's functions of a well-known 

 type, whose rapidity of convergence appears well ascertained 

 under the normal conditions of the problem. When the 

 section is circular, and the material not isotropic but sym- 

 metrical round the axis, the series, whose mathematical law of 

 development I have obtained, converge to all appearance quite 

 as rapidly as in the case of isotropy. The other cases of 

 isotropic material — sections elliptical or rectangular — which 

 I have considered present similar features ; the only difference 

 being that the rate of convergence diminishes with increase 

 in any one dimension of the cross section. 



3. If we suppose k = Q, or the vibrations to be of infinite 

 period, the solution must reduce to that for the equilibrium of 

 a rod under uniform longitudinal traction. ISTow, in the case 

 of equilibrium 7/0 reduces to a constant, while a and ft are 

 linear in x and y for all forms of cross section. The commencing 

 terms in series (6) are thus of the proper form under all 

 conditions, and the form of the differential equations shows 

 that if a, for instance, contains a term in 00 it must contain 

 terms in a? 8 , xy 2 , and other integral powers of % and y. 



4. The general type of the differential equations is the 

 same for all kinds of elastic material, isotropic or ?eolotropic, 

 and the surface conditions are identical in all cases ; thus the 

 type of solution must always be the same. The results may 

 become enormously lengthy for complicated kinds of seolo- 

 tropy, but by putting a variety of the elastic constants equal 

 to one another we must reduce the most complicated of these 

 expressions to coincidence with the corresponding results for 

 isotropy. Of course it does not follow that the convergence 

 will be equally rapid for all materials. A large value of a 

 Poisson's ratio in conjunction with an elongated dimension 

 of the cross section may reduce the convergency so much as 

 to throw the higher " harmonics " outside the pale of longi- 

 tudinal vibrations. 



5. The more the section departs from the circular form 

 the less rapid in general is the convergence, and the larger 

 the correction supplied by the second approximation. In 

 fact the size of the correction is probably the best criterion 



