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X. On the Resistance to Torsion of Certain Forms of Sha/tiny, with Special Reference 



to the Effect of Key ways. 



i L. N. G. FILON, M.A., Research Student of Kings College, Cambridge, Fellow of 

 University College, London, 1851 Exhibition Science Research Scholni: 



Communicated by Professor M. J. M. HILL, F.R.S. 

 Received June 1, -Head June 15, 1899. 



1. Object and Methods of the Investigation. 



THE object of the present paper is to obtain solutions of the problem of torsion for 

 cylinders whose cross-sections are bounded by confocal conies. It is mainly an 

 extension of DE SAINT- VENANT'S investigations, and is based upon his general 

 equations of torsion. 



The method employed depends upon the use of conjugate functions and i), such 

 that = const, represents coufocal ellipses and 17 = const, confocal hyperbolas. 



The use of conjugate functions for the torsion problem has been suggested by 

 THOMSON and TAIT ('Natural Philosophy'), by CLEBSCH ('Theorie der Elasticitiit 

 fester Korper,' 33-35), and by BOUSSINESQ ('Journal de Mathe'matiques,' 

 pp. 177-186, Se*rie III., vol. 6). CLEBSCH has used such elliptic coordinates to 

 solve the torsion problem for hollow cylinders bounded by confocal ellipses, and 

 DE SAINT- VKNANT has applied conjugate functions to the same problem for shafts 

 whose sections are sectors of circles ; curvilinear coordinates have also been employed 

 by Mr. H. M. MACDONALD (" On the Torsional Strength of Hollow Shafts," ' Proc. 

 Camb. Phil. Soc.,' vol. 8, 1893, p. 62, et seq.), but I am not aware that the actual 

 solution has yet been obtained for sections bounded by both ellipses and hyperbolas. 



The work proceeds on lines analogous to those developed by SAINT- VENANT 

 himself, in his solution of the problem of torsion for the cylinder of rectangular 

 cross-section. The strains and stresses are expressible in terms of infinite series 

 involving circular and hyperbolic functions. 



The boundaries of the section are given by constant values of and 17. The values 

 "f f aiv taken to be a. 



The conditions from which the unknown quantity w (the shift parallel to the axis) 

 is determined are 



tPw/daf + tfw/dy 1 = 



29.12.99. 



