Vibrations of a Rod of Varying Cross-Section, 

 In the third mode x x — 38*453, 



u= 0(4306-80^! (w) + -0290930 ^0)}« 

 We then have as corresponding values : 



95 



2*/x. 



cc. 



&(#). 



^iO). 



w/C. 



T5-12 

 15-14 



J8-40 

 \8-42 



6-5536 

 6-6049 



1764 

 17-7241 



27-3792 



27-7838 



- -000265556 

 4- '000042188 



+3-57988 



- -880662 



- -347188 

 + -989993 



The nodes 

 Again : 



are thereto 

 x = 

 z/l = 



re given bj 

 6-59477, 

 •171502, 



r 



17-66184, 

 •459298. 





2Vz. 



X. 



dx' 



dty x 

 dx' 



%/°- 



T6-36 

 16-38 



/9-80 

 19-82 



101124 

 101761 



2401 

 241081 



1-366545 



12-2416 

 12-4124 



- -000187291 



- -0000824478 

 - -000123829 



- -766870 

 + -0338437 



+ -001051 

 - -172201 



Hence the points of maximum excursion are given by 



*•= 10-17341, 24-01059, 



z\l= -264567, -624414. 



Returning to the equation (27) 



M=C{^r_i(ar 1 )0 o (a?)— ^-iC^)^^)}, 



we notice that when x is small and x x relatively great, the 

 chief: term in the expression is the second. Thus the posi- 

 tions of the nodes, loops, and points of inflexion are given 

 approximately by the roots of the equations 



that is by 



*»-«■ f°= ' ^=°> 



J 1 (2 v /a;)=0, J 2 {2y/x) = 0, J 3 (2 s /x)=Q, 



respectively ; the approximation being closer the higher the 

 mode and the nearer the points in question to the thin end. 



