Vibrations of a Rod of Varying Cross-Section. 103 



11. The case of the conical bar is obtained by making 

 n=l. From (40) we then have 



1 x 2 . x* x 6 



= 



+ 



3!5! 1 ! 4 ! 7 ! ' 2 ! 5 ! 9 ! 3 ! 6 ! 11 ! 



the lower roots of which are found to be 



15*406, 31-750, 52,858. 



The positions of the nodes and points of maximum 

 excursion are found to be as follows : — 







X. 



*/L 



First Mode. 



Node 



667792 



•433462 





9-94075 



•645252 



Second Mode. 



Nodes 



6-59220 

 17*79314 



•207628 

 •560413 





10-18177 

 23-55458 



•320687 

 •741876 



Third Mode. 



Nodes 



6-59380 

 1771046 

 33-83296 



•124746 

 •335057 

 •640073 





1017648 

 2382701 

 4207101 



•192525 

 •450774 

 •795925 



Appendix. 



Wave Motion in a Canal of Variable Section. 



Take the #-axis parallel to the length of the canal, and let 

 S denote the area of a cross-section, b the breadth at the 

 surface, and rj the difference between the ordinates of the 

 free surface in the disturbed and undisturbed states. The 

 equation of motion is then 





~dt 2 b $x\~ txp 

 orifice cos (*+«), |^(sg) + A 



* Lamb, Hydrodynamics, Art. 183 (4). 



2 M 



= 0. 



(1) 



(2)* 



