TOUSION (.| c I;|;T.M\ FORMS OF SHAFTING. 



The equation giving the maxima or minima is found as before. It is 



|~ <Pw I dJ /(ho . . 



2 , . . W ginh 2 =0, 



ill}- .1 1/7] ifrj /Jf- 



or, using the first of expressions (17) for /' 



341 



sin 2j} tanh 2a + 8 (cosh 2 + cos 



2* 



(2n 







. 



vtf (2n -f 



2n + \irn 



Ida sin L'rj (rush 2a + C<)B 2/3) "^" L'a 



cosh 2 + cos 2i; H 7o (-'+ l)V-f- 1C 



-4_-=0. 



, -'/i + lw/3 

 com 



2 



The left-hand side is always positive. Hence there is no root except 17 = 0, and 

 tlutt corresponds to a maximum. 



That no absolute maximum can exist inside the section can l>e proved as follows: 

 We have 



xz/p. = dw/dx ry, yz/fj. = dw/dy + ix. 



Suppose at any point P inside the section S 2 = xz 2 + yz' 1 is a maximum. The 



above forms for xz + yz being independent of axes, let us take for axes of x and y the 

 direction of the resultant stress across the section at P and the perpendicular to it. 



So that yz = at P. 



Consider a near point P'. Let xz, yz l)e the stresses at P'. Then, since S* is a 

 maximum at P, 



.'-.- > xz" 1 + i/z", 



hut 

 Therefore 



M 



.^s 



or xz is a numerical maximum. 

 But since 



therefore also 



yz" > 0. 

 arz 2 > xz", 



d'wjdx* + d- w/dy* = 0, 

 *() rf() 



and it is well known that no function can have an alsolute maximum or an absolute 

 minimum inside a region where it satisfies LAPLACE'S equation. Tin's is in fact a 

 particular case of the general theorem that a potential cannot have a maximum in 



tree space. 



