66 Mr Macdonald, On the torsional D^^y 1> 



putting p = e~^+% q — e~^, 



ds___ ^Tcq , o ; /-, _ 2\ [ ^ P'^ 



dv~ (1 + qy'^ ^^^ ^)[(l + qy + (i+pqy^ 



'^{l+qff^-- 



Now q and p are less than unity ; therefore -^ is positive when 

 ?; = /3 and 5 diminishes with 77. 

 When r] =a, 



hence when 7^ = a, — is negative, and s diminishes as 77 increases ; 



therefore s is greatest when 77 = /3 or 7; = a, ^ being zero. 



When ^ = TT, 



41 "^ , iu a . o 7 / 1, 1 \ ^ we-"'* cosh n (ri —8) 



s = — TC coth ii + TC coth yS + 2to (cosh 77 — 1) Z .— j , ^ ' "^ ; 



2 1 ^i^h n{/3 — a) 



ds 



-y- and s are positive when 77 = /3, therefore 5 diminishes with 77, 



and when 77 = a, s diminishes as 77 increases. 



When 1 = 0, 77 = /S, s = 2t6 + ra' + 2x6 -^y-^,+ ... , 



1 = 0, 77 = a, 5 = ra + 2x6 7 rr-g + . . . , 



(a — by 



J. n 07/, ^rha^a'^ 



| = 7r, '77 = A 5=2x6-xa + . ,_^,, + ... , 



2x6a'^ 

 I = TT, 77 = a, s = -xa+ / _^V2 + • • • • 



The shear at the surface differs from that at the surface of 

 a solid cylinder by a quantity depending on the square and 

 higher powers of the radius of the cavity, and if a > 26 + d , that 

 is, if the cavity is at a distance from the axis of the outer 

 surface less than half the radius of the cylinder, the shear is 

 greatest at the surface and the strength of the cylinder is practi- 

 cally unaltered if the cavity is small. If a < 26 + a', that is, 

 if the distance of the cavity from the axis of the cylinder exceeds 

 half its radius, the shear is greatest at that point of the surface 

 of the cavity which is most distant from the axis and is greater 



