60 Proceed'uKjs of the Royal Irish Acadvinij. 



The corresponding stresses are then 



iVj = _ iV^ = 2/>t ^ sin 7iz, N-, = 0, 

 (I 



Ti = - fx —B,, Ki {no) cos m, T. = — B„ {na) cos nz, 

 d (I 



— 



= fx O sin nz. 



a 



It appears, therefore, that there acts on each element plane of the 

 curved surface a force whose type is QaB^smiiz. The B„ will be 

 known from the Fourier expression for actual force distribution. 



A strip da- in breadth of the curved surface bounded by two near 

 parallels to the axis is therefore acted on by a torque whose axis is 

 normal to cylinder 



. I 



= a^lB^do- 2 sin nzdz = aUB^dcr - eos 7izdz 

 }-h J-hn 



= aQB„d(r 2 sin nh = a^lB,,da- X 2 sin 



2 



the total torque being therefore 



2a2^\id(j — Bn — sm — - • 



TT- S- 2 



For a corresponding internal strip, the torque 



^ I . STT 



= 2r^n,d(r — B,, — sin ^ , 



wliere = ~(- Ki(nr) 



dr\r 



If the thickness of the cylinder be small compared with its breadth, 

 we have / , 1 



^o{nr) = -= 6- jjnr, K^inr) = — ^ 

 J27r J27rj«r 



n,= ,_ -( 1 + — • 



It appears, therefore, that the torque diminishes rapidly from the 

 surface according to the exponential law. 



This corresponds for a circular cylinder to the result given in 

 Tait and Thomson (Art. 728, p. 563, vol. i., 1st ed.). 



