1893.] strength of a hoUoiu shaft. 65 



If there should be any number of such cavities placed so that 

 the areas over which their effect in changing the shear is signifi- 

 cant do not overlap, the twist is given by 



T = .,/(l-|w^)-./(l-2^' 



where I is the moment of inertia of the area of the cross-section 

 of the cylinder, and /' of the cavities about the axis of the cylinder 

 supposed solid ; and if /' is small compared with / 



T = To (1+2 



-yir = TC- COth yS ( T ■ z. 



^ \cosn 77 + cos ^ 



3. To find where the shear is greatest. 



It is stationary in value when f = or vr and is then given by 



77 cosh t] + l dyjr 



S — — 7C tanh ^ -\ ~- when t = 0, 



2 c drj ^ 



and s = — TC coth ^ ~ when ^ = tt. 



z c drj 



Writing 



Vcosh 77 + cos ^ / 



^ , Z. (— )" e""" sinh n (97 — B) cos nP 



- 2t&c Z '-^ ^-r — 7^5^^^ — P , 



J Sinn n {p — a) 



we have, when ^ = 0, 



5 = — TC tanh ^ + TC coth 8 

 2 



o 7> / 1 , 1 ^ V '^ (-)" ^""" cosh ;i (77 - /9) 



- 2t6 (cosh 77 + 1) Z ' ■ , — -^ V ^' 



^ 1 smh 71 (p — a) 



^ = - — sech- ^ - 2r6 Z ^ . , — 7^5 ^-^^ — — (cosh 77 + 1) 



c?77 2 2 1 smh n (13 -a) ^ ' ' 



„ * 71 i-Y e-^" cosh niri- B) . , 



- 2t& S ^ ■ , — -5 ~ — — smh 77, 



J smh 72 (p — a) 



When 77 = yS, 



cZs TC ,„/3 ^ , f,7t(-)'*e-««sinh/S 

 ti77 2 2 smh n (p — a) 



