Uniform and Varying Sectional Area. 119 



and the last four of these give 



-d 5 r -u 5 / 3 ' 5 J 4 ' U " 



Henee the type to be assumed as a first approximation is 



Ik V\ (12 ctl a , 1\ 



-fflSi{ *=*§'™ (*.-&+*)*, 



and (7) becomes 



dx l ~ EI 



-^' {?*-*& + J){z-*)dz} 



_ 16 pcoi/'// : \c ZV /ar' .r G \ 



~ T El? \io " 6 + io ~~ 3u )' 



Integrating and keeping the end conditions in view, 



y= 1 Elr ( ' 01 ^ /V -' 008 ^ ^ 5 + -002,380,95> 7 



—000,595,24 ^--010,119,051^), . . (8) 

 which is the approximate vibration-curve, and 



_£ =97 . 10 98^. 



For the second approximation, using (8) in (7) \ and 

 simplifying as before, 



-tl = -310,751 ^ff ( - 1-02513 l°x + 1-68650 ZV 

 ate Jiil 6 



-•83 /V + -19841 ZV -•03307 lx 9 + -00662a- 10 ) 



... - y =-310751^ (10386 Z u a— 17086 ZV 



+ -084331/ V- -01984 ZV 



+ -00276 ZV- -00030 lx xx + -00005 .r 12 ) . 



This is a close approximation to the equation of the centre- 

 line, and gives 



-y =-0102725^^; 

 u\ til yi 



AT 9-866 HJ 



