Rotatory Inertia on the Vibrations of Bars. 53 



equation for the periods. By this means we not only see 

 exactly what we are neglecting but avoid the risk of casting- 

 out a term from the differential equation. 



The effect is greatest in a free-free bar, when it may become 

 considerable in short thick bars. The difficulty of realizing 

 in practice such a bar, however, renders this of small 

 account. 



It appears that if X and N be the frequencies when 

 rotatory inertia is retained and rejected, respectively, we may 

 write 



2 4 6 



JST : N=l + a-^- +b^j +c~ + 



a: 4 . 



where -^- and higher powers are vanishingly small. The 



coefficients a are negative and have been calculated for bars 

 under various terminal conditions and for the fundamental 

 and first five overtones. 



I Funda- 



1st 2nd 



3rd 



4th 



5th 



! mental. 



overtone. overtone. 



overtone. 



overtone. 



overtone. 



Clamped-free ... 2'3238S9 



16-207656 38-649553 



71-45090 



114-06690 



166-55652 



Clamped-clamped 6151311 



23-025059 49-453258 



85-792832 



131-999001 



188-075039 



Free-free ' 24740415 



54-462303 93433364 



142341572 



201-114041 



269-756448 



Pivoted-free 13-615576 



35 585395 67-439112 



109-162518 



160-755527 



222-218142 



Pivoted-pi voted. J 4834802 



19-739209 44-413220 



78-956835 



123-370055 



177-652879 



Pivoted-clamped .' 5'756265 



21-448204 4701S760 



82-458981 



127-768804 



182-948233 



IX. Xormal Functions. 

 In (5) take Y = and put, 



v = %f p {x) (A p sinjrt 4- B^ cos pt) 



p 



= %f p cj)p. 

 v 

 .'. (5) becomes 



V h — e j>4 ^P 



whei 



:e 



. EA 



