1893.] strength of a hollow shaft. 63 



throughout the matter of the prism, and 



f=\{^' + y')'rC .....(3') 



at the boundary, where G is an arbitrary constant which will 

 not be the same for the two bounding circles. 



Transforming to ^, rj where 



a; + ^2/ = c tan 1(1 + i?7), 

 ilr satisfies 



from ?7 = a to r]= ^, and 



re" cosh V , 1. /K\ 



•\|r= = ^4- const (5) 



^ cosh 7] + cos ^ 



when ■?; = a and when t] = /3, 



77 = a being the outer bounding circle, and 

 7]= ^ the inner. 

 Equation (5) may be written 



^Ir = 2tc- coth 7} 2 (-T e-"" cos 72^ + C (5 ) 



1 



when ?; = a or /3. 



Assume t = ^ (-)'^ (^^e-^" + ^^e'^") cos w^ (6), 



1 



g-2n^ cothyS - e~2*^" coth a 

 then ^ „ = 2 (-)" TC^ g-2np _ g-gTia 



J^n = 2 (— )"■ TC^ g2n^ _ g2}ia 



Therefore 



^, , e-»^ coth/3 sinh ?i(T7-a)-e"""eo th asinh n (y-^) 

 f = 2tc^ 2 (-r .inh^r^W^) 



cos 71^... (8), 



and 



" , , e-^^coth)gcoshyi(77-a)-e-"°cot hoccoshw(?;- /3) 



^v = </>=2tc^ 2 (-)" sinh^/8^) ^ 



sin ?i|^...(9) ; 

 hence the displacements are known. 



(7). 



