278 SUPPLEMENT TO CHAP. XIV. [CHAP. I. 



/ da Ci * C j * rv*da I 



v = J da--J vda+ -J -^- 9 = A + - because since 



9 is measured from the centre of gravity, I vda = Q. 



vda r . rv'da I 



Again, 



Therefore the insertion of these values of the integrals in the equations 



for _ and _ above gives 

 E 



G_ _ bds rAd(f 

 E ~" ds ~ \_~d~T 



d ~ r d s 7 

 M 



Q A<Z M 



Inserting the second in the first of the above equations, ==- = 5 A = , 



and hence 



a M 



dt ~EA 

 Inserting this in the second equation above, 



Ad<f> _ M_ M Q 



ds ~El 



(c) Change of length and position of axis. 



From (8) we have at once for the elongation of axis, 





or, when c is very small compared to r, 



From (9) we have for the change of direction of the tangent to the axis 

 1 /VM M Q \ _ 



A * = EJ (T + AH + AT) ^^ or from (10) for r 



constant, that is, for a circle, 



When is very small compared to r, we have 



