Vibrations in Solid and Hollow Cylinders. 337 



paper (B) ; viz., that the general solution of the elastic solid 

 equations of motion in which the terms contain cos pz or 

 s'mpz consists of two independent parts. In the first, which 

 alone applies to longitudinal vibrations, a. and j3 are odd and 

 7 an even function of x and y. 

 We may thus assume 



a = cos kt cos (pz — e) { A^ 4- A/y + A 3 # 3 4- A 3 'x*y 



+a b 'v+a, ,, v +...}, 



/3=cos^cos(^.~-e){B 1 A< + B/# + B 3 ^ 3 + ...}, I * W 



7 = cos kt sin (pz - e) {C + C 2 x 2 + QJxy + C 2 ff f + •••}> ) 



where e is a constant depending on the position of the origin 

 of coordinates and the terminal conditions. It is obvious 

 from various considerations that the same e occurs in the 

 values of a, /3, y. Certain relations must subsist between the 

 constants A, B, C in the above expressions, in virtue of the 

 body- stress equations, but we do not require to take any heed 

 of these for our present purpose. 



§ 6. As the validity of solutions in series has been a subject 

 of contention in other elastic solid problems, some doubt may 

 be entertained as to the results (6). I would be the last to 

 deny the reasonableness of this, because I do not myself 

 regard (6) as universally applicable. 



According to my investigations, quantities such as 

 (A 3 # 3 /A.i#) are of the order (greatest diameter/nodal length) 2 , 

 and the series become less rapidly convergent as (greatest 

 diameter/nodal length) increases. In other words, increase 

 either in (greatest diameter/rod length) or in the order of the 

 " harmonic " of the fundamental note reduces the rapidity of 

 convergence. The proper interpretation, however, to put on 

 this is not that (6) is a wrong formula for longitudinal 

 vibrations, but simply that under the conditions specified the 

 vibrations tend to depart too widely from the longitudinal 

 type. If we apply this solution the results deduced from it 

 themselves tend to show the degree of rapidity of the con- 

 vergence, and what we have to do is to keep our eye on the 

 results and accept them only so long as they are consistent 

 with rapid convergency. 



Perhaps the following resume 1 of my views on this point 

 may be useful : — 



1. In obtaining (6) originally I employed the complete 

 elastic solid equations for isotropic materials. In other cases 

 where difficulties have arisen over expansion in series, they 

 seem mainly due to the fact that the elastic solid equations have 

 been whittled down in the first instance for purposes of 



