478 Mr. G. H. Bryan on the 



therefore 



£ = -fM{e + ^t); 



whence 



W=EA C ( e + ^ T )^+ | nli % j a r 2 dx* 



= 2Eha(e 2 + | a^re + g a A ^A + | ntfar 2 . 



Let w be the axial tension, G the couple about the axis 

 required to produce the deformation. Then evidently, 



dW dW n 



de dr 



In the experiments described by Prof. Perry, G = 0, and 

 t is the quantity observed. We therefore have 



= ~ = 2® ah (J a 2 <f>e + | a^r\ + ~nh*ar, 



dW / 2 \ 



w=-~ = 2Eah ( 2e + | a'^r J. 



Eliminating e, we find 



w=-tx 16Aa 2 ( 7-=: Ea6 + w -, — r )• 

 \15 ^ a 2 a<j)J 



Writing 0=—r, so that 6 measures the untwisting per unit 

 length, this gives 



e= 



16ha 2 (^Ea<j> + n~-~) 

 \15 r a 2 acf)/ 



the required formula for 6. 



When a$> is exceedingly small the formula gives 



.» Wd(f) 



which agrees with Prof. Perry's result. 



The value of 6 soon, however, attains a maximum as <f> is 

 increased. This will be the case when 



* The first term of this expression might have been written down at 

 once by considering the elongation of the fibres of the strip when pulled 

 out, while the second term follows immediately by taking the same ex- 

 pression for the torsional rigidity as that assumed by Perry. On the 

 whole I think the method here given is preferable, as it involves fewer 

 assumptions, and shows more fully within what limits we may consider 

 the results as approximately correct. 



