On determining Frequencies of Lateral Vib 



ration. 419 



and the changes in the field to be two connected aspects of a 

 single solution regarded as a whole, as to suppose that one 

 is antecedent to the other. The adoption of this point of 

 view would not affect the character of the field of being 

 expressible in terms of: retarded potentials, but it would 

 permit other boundary conditions than those of a simple 

 moving boundary to be applicable for the representation of 

 real motions. The example which has been discussed in this 

 paper seems to show, in one case at least, the legitimacy of a 

 solution with less prescribed boundary conditions. 



October 10th, 1920. 



XXXVI. On a Graphical Method for determining the Fre- 

 quencies of Lateral Vibration, or Whirling Speeds, for a Rod 

 of N on- Uniform Cross-Section. By R. V. Southwell, 

 M.A., Fellow and Lecturer of Trinity College, Cambridge* \ 



[Plate VII.] 



rjPHE determination of the normal modes and frequencies 

 X of lateral vibration for a rod of varying cross-section 

 is a problem which has attracted the attention of many 

 elasticians, and several papers on the subject have appeared 

 in this Magazine f . It would seem that a complete solution, 



* Communicated by the Author. 



t References to the investigations of Bernouilli, Euler, Kirchhofr", 

 Sturm, Liouville, and others will be found in Lord Ra}deigh's ' Theory 

 of Sound,' or in the ' Dynamical Theory of Sound ' of H. Lamb. The 

 following is believed to be a fairly complete list of recent papers bearing 

 on the problem : — 



W. J. M. Rankine, ' The Engineer,' vol. xxvii. (1869), p. 219. 



S. Dunkerley, Phil. Trans. Roy. Soc. (A), vol. clxxxv. (1894), p. 279. 



C. Chree, Phil. Mag., vol. vii. (1904), p. 504, and vol. ix. (1905), 



p. 132. 

 C. A. B. Garrett, Phil. Mag., vol. viii. (1904), p. 581. 

 J. Morrow, Phil. Mag., vol. x. (1905), p. 113 ; vol. xi. (1906), p. 354; 



and vol. xii. (1907), p. 233. 

 A. Morley, 'Engineering,' July 30th and August 13th, 1909. 

 P. F. Ward, Phil. Mag., vol. xxv. (1913), p. 85. 

 A. Fage, ' Engineering,' July 20th. 1917. 



J. W. Nicholson, Proc. Roy. Soc. (A), vol. xciii. (1917), p. 506. 

 H. H. Jeffcott, Proc. Roy. Soc. (A), vol. xcv. (1918), p. 106. 

 W. L. Cowley and H. Levy, Advisory Committee for Aeronautics, 



R. & M. 485 (1918). 

 J. Morris, Advisory Committee for Aeronautics, R. & M. 5o] (1918). 

 G. Greenhill, Advisory Committee for Aeronautics, 1\. & M. 560 



(1918). 

 F. B. Pidduck, Lond. Math. Soc. Proc, vol. xviii. (1920), p. 393. 



