TOIISKtN ul (T.l.TAIN |I>];MS OF SHAFTING. 337 



F B t-.ii -fl (c<>flh 2 * + ** 20) (1 * 2 ** 2 ^ 



C062/3 m ,(2+ljV-160' 2/8 



cosh 2 + cos 2/8 " " " . 2ii + lira 

 * Ch - 



C062/3 



.. 



I may remark here that all these differentiations are permissible, provided that 

 be less than a by a finite amount, however small, the series being then uniformly 

 convergent. We may not, therefore, make a actually zero in the last expression. 

 If, however, a is small but finite, then it is easy to see that E must be negative if 

 ft < ir/4. Also if ft < 7T/4, E algebraically increases continuously, until for a certain 

 value of a it reaches the value zero. For all higher values of a it remains steadily 

 positive. 



1 4. Critical Values of a. and ft. 



It follows, therefore, that in all sections for which ft < ir/4, the maximum stress 

 along the sides 17 = ft occurs at the point B, until a certain critical value, a = , 

 is reached, when the stress at B becomes a minimum, and we now have two points of 

 maximum stress on either side of B. 



When ft = ir/4, E is apparently infinite, but it really tends to a finite limit. If 

 we put ft = 7T/4 c where e will ultimately be made very small, and neglect terms in 

 e, ! , &c., we find that the expression becomes 



4 2 cosh 2a 2 sech 2n + 1 2a cosh 2a 2 sech (2/i + 1) 2a + 



* n = n = 1 



/ \ 1 i 1 



- (cosh 2a + 2e) (I + 2e) (-*- - 2ej - sech | 2a ( 1 -f ' 

 = sech 2a + a tunh 2a 3 cosh 2a 2C sech (2i + 1 ) 2*. 



* * Hal 



When a is very great, this will ultimately be positive. Hence, here also we shall have 

 a critical value OQ. This value, however, is easily seen to lie outside the values of a 

 t.ikcM in this paper. 



I have not investigated so carefully the cases when ft > ir/4, as, for the particular 

 sections selected, the maximum stress certainly occurs at B. 



One point, however, is clear. When a is very large, the second and third terms 

 settle the sign of E. The sign of these terms, again, is settled by the sign of the 

 leading terms. E is negative if 



is positive, i.e., if 



cos'/3 ' 1 + 2co8 20 

 * * 



4/8* TT* - 16/3* 



Vol.. CXCIII. A. 2 X 



