420 Mr. R. V. Southwell on a Graphical Method 



applicable to rods o£ any given shape, is beyond the range 

 of exact analysis : at all events, mathematical investigations 

 have in general been restricted to rods of fairly simple form, 

 the case most frequently discussed being that of a rod with 

 one free and one clamped end, in which the flexural rigidity 

 at any section varies as some power of the distance from the 

 free end. 



If it be conceded that the interest of this problem lies 

 principally in the quantitative results obtained, then a 

 reasonably accurate method of solution which is unre- 

 stricted in its application would appear to possess advantages 

 over any isolated analytical solution, however rigorous. No 

 great mathematical interest can attach to exact results when 

 these are based upon a theory (of thin rods) which ad- 

 mittedly is only an approximation to the truth, and a slight 

 decrease in accuracy will be more than compensated by 

 ability to take account of any specified end conditions, and 

 to deal with rods of which the cross-sections vary in any 

 specified way, or with continuous rods supported at several 

 points. This is more especially true in relation to the engi- 

 neering applications of the theory. The "whirling speeds" 

 of a rotating shaft, as has frequently been pointed out *, 

 are identical with its natural frequencies of vibration, and in 

 many cases in which their values are of practical importance 

 the variation of flexural rigidity along the length of the shaft 

 is incapable of expression in mathematical form, whilst the 

 bearings impose constraint upon the direction of the central 

 line at two or more points : no great accuracy is required in 

 the result, but it is essential that the method employed shall 

 not break down in any particular instance by reason of 

 purely mathematical difficulties. 



The graphical processes now to be described constitute 

 a simple extension of methods which I have recently pro- 

 pounded t for finding the critical load of a strut of varying 

 cross-section. They seem preferable, as regards ease and 

 quickness of application, to any general method of solution 

 which is based upon the use of infinite series J; and the 

 accuracy with which they reproduce the results of exact 

 calculation in examples which can be treated analytically 



* Cf., e.g., C. Chree, Phil. Mag. vol. vii. (1904) p. 504; the ex- 

 pressions obtained in this paper for the frequencies of lateral vibration 

 in a rotating shaft have, however, been shown by Pidduck (loc. cit.) to 

 be erroneous. I hope shortly to publish a paper dealing with this point 

 at greater length. 



t 'Aircraft Engineering,' April 1920, pp. 113-114. 



\ Cf., e.g., the discussions of W. L. Cowley and H. Levy, and of 

 J. Moms, quoted above. 



