96 



Mr. P. F. Ward on the Transverse 



The degree of approximation is illustrated by the following 

 tables. 



Nod 



es of wedge-shaped bar. 





Second mode 



3-64662 

 3-67160 

 3-67055 



12-25816 

 12-30668 



25-80624 



— 



Third mode 



Fourth mode 





Eootsof J x (2 V^J^O . 



3-67049 



12-30461 



25-87487 



44-38026 



Points of Maximum Amplitude. 



First mode 



6-69001 

 6-58993 

 6-59392 



17-91733 

 1770437 



3407126 



— 





Third mode 



Fourth mode 





Boots of J 2 (2 Va .) = o. 



6-59366 



17-71249 



33-75520 



54-73142 



Points of Inflexion. 





9-91236 

 10-18901 

 10-17609 



23-21586 

 23-84908 



41-37971 



Second mode 



Third mode 



Fourth mode 





Eoots of J 3 (2 V*)=0- 



1017661 



2381939 



42-34886 



6. Free-Free Bar. 



Since both ends of the rod are now free the conditions to 

 be fulfilled at z = and z = l are 



cok z ^ 2 = 0, and ^- 1 cok 2 =~ ) = 0, 

 0? oz \ Oz" J 



which in our case reduce to 



a » +8 ^=0, and -f-(,.. + .^) =0 . 



ax* dM\ da;-/ 



