TORSION OF CERTAIN FORMS OF SHAFTING. 



Clearly, \\lien we look at the second of expressions (22), we see that the second 

 term must ultimately become vanishingly small, provided irct/2ft > 2a, or ft < w/4. 

 For such values of ft, then, S B tends to the definite limit p.rc tan 2ft cos ft, i.e., for 

 ft = 7T/6, S B //ITC tends to the limit 1 P 5 for large values of a. 



For values of ft > ir/4 it would seem that S B increases numerically to an indefinite 

 extent. 



Consider now the second expression for S A . Clearly, if a exceed a certain value, 



tanh " - approximates so closely to unity that we may replace it in the series by 

 unity, and the error will only be a small fraction of the series itself. We then find 



S A . , 8/8 (cosh 2 + cos 20) (- 1)" 



- - 



* tends to sinh (sec 2ft- I) - " - ; 2 



cosh 



when a is large. 



Substituting ft = ir/6, and using the Euler-Maclaurin sum-formula to calculate the 

 series, I find 



= sinh a - 

 c/,, =ir/8 TT cosh 



brg 



and remembering that 



coeh 2 + 4 1 -PI 



- = 2 cosh a .- T = 2 cosh a, if a large, 

 cosh 2 cosh 



sinh a = cosh a - = cosh a, if a large, 



cosh a. + sinh a 



we see that, when a is large, 



= -cosh (-452147), 



and, therefore, increases numerically indefinitely. 



On the other hand, if a be very small, we get a flat section, A being now the point 

 nearest to the centre. Looking at expressions (21) and (22) we see easily that 



f^M tends to - 2 



\ftTCJ 



and 



ft 

 or, neglecting squares of a compared with the first power, 



Now 2 - ( " ^r, lies between 1 and 8/9, hence l^\ < ( cos ft] a, 



