illustrated by the TJieory of Impulsive Torsion. 223 



to have been sufficiently considered in the light of practice. They 

 show that there can be a kinetic accumulation of stress at sections 

 where a wave of vibrations is reflected, and that in many cases this 

 kinetic stress can considerably exceed double the value of the statical 

 stress at that section due to the same load permanently applied. In 

 fact, every case of repeated loading or live-loading forms a kinetic 

 problem which must be independently solved, and which, if it presents 

 difficulties, still presents difficulties such as the mathematician is in 

 duty bound to overcome. 



While the papers to which I have referred give a fairly complete 

 solution of a number of problems in longitudinal and transverse loading 

 of bars and girders, the problem of torsional loading seems to have 

 been untouched up to the present. Yet the question of torsional 

 vibrations is one that arises very frequently, as in the cases of shafting 

 and axles. The present memoir endeavours to give from the mathemati- 

 cal standpoint a fairly comprehensive solution of the problem of the 

 kinetic accumulation of stress in bars, shafts, axles, &c., owing to 

 repeated, impulsive, or changing systems of torsional loading. The 

 attempt following the traditions of de Saint- Venant has been 

 made, however, not to leave the results in the form of long series 

 unintelligible to the average practical man. A very large amount of 

 numerical reduction has been undertaken, and many of the results are 

 shown graphically. The whole of the calculations, as well as the original 

 large diagrams, are my own work ; but I owe to Mr. G. Baker and 

 Mr. J. Longbottom, who have at one time or another been my assistants 

 at University College, the reduced diagrams which accompany this 

 paper, and I have to cordially thank them for the care and labour they 

 have given to the reproduction of my drawings. 



2. With regard to the general conclusions of the memoir, I should 

 wish to draw special attention to the following points : 



(rt.) When a load is repeated, or applied, reversed, and repeated, 

 there will in general be kinetic accumulations of stress ; the 

 amount of this accumulation varies with the times of incidence 

 and of release of the load, but it may easily exceed the double 

 of the stress due to the statical application of the load. 

 (b.) Wohler's empirical laws must either be considered as allowing 

 for this kinetic accumulation, or not. If they do not allow 

 for it, but are based merely on the assumption of a statical 

 load gradually applied and gradually removed, then they do 

 not cover an immense range of repeated loading which occurs 

 in practice. If they do allow for it, then the conception of 

 the material being worn out by a maximum stress lower than 

 the elastic limit is a false one, for the explanation of the 

 destruction of the material lies in the kinetic accumulation of 

 a much greater stress. 

 VOL. LXVII. R 



