304 SUPPLEMENT TO CHAP. XIV. [CHAP. V 



Hence the change of length of the axis is 



C* 



As= 1 I Gr + M 



f* 



= J_ I Q 



ETV/ o ~ 



, 

 dt ttt ..... (50) 



The change of the angle between two infinitely near cross-sections, and the 

 actual turning of a cross-section is from (9) and (12), Art. 1, given by 



Ads M G r + M 



Gr + M 



1 /M 1 



** = VJ T d + E 



Finally, from (14) we have 



/rids 

 &<t>d y + I r 



/ 

 Afdx + 



Substitute in these last two equations for A <p and - their values from 



as 



(49) and (52). The double integral thus arising can be resolved by par- 

 tial integration. 



Thus 



Applying this, we obtain 

 Aa?= 



(58) 



We shall assume in the following the axis always circular. 



23. Fundamental Equations General. Upon this assump- 

 tion of a circular axis we have generally 



G = H cos $, N = H sin <f> \ 



M = M + Hr(l-cos$>) I ..... (54) 



Or + M = M, + Hr 

 Hence, from the preceding Art., 



Hr + M. /* 



A = rr-r I dd>rtt<t> ..... (55) 

 E A Jo 



Mor _ Hr 



r* /* 



A = r A <?) (1 cos <J>) H -- / M (1 cos <p) d<t> 



EIJ 



o /* 



I d<j> rttB\n<t>+B ..... (57) 

 J 



H r + Mo 



_ . 

 E A 



