Rotatory Inertia on the Vibrations of Bars. 59 



Now, at a free end, 



54=0. 



d%* 



or 7i 2 ^> = 72 2 ^- 



Also 4 ^, ^_f> 



^ 4 (Yi</>' + 7 2 t') + (-7i 3 *' + 72 8 f ') = ; 

 *'(^-7i 2 )7i+t'(^ + 7 2 2 )7o-0 ; 



r^'=7i^'- 



Using these relations and those in § XL 1, 



Ji^ + ('Jf) 2 } <?f=37 '^' +2(1+7i2) ^ + ^ (3+V) ^' 



becoming, when rotatory inertia is neglected, 



thus agreeing with Rayleigh when the necessary trans- 

 formations have been made. 



3. Clamped end. — At a clamped end 



dfd?f ±dfdf 

 dX dp + dX dg dp 



Also, y=0 + i|r = O 



df .. ^at a clamped end. 



^j=7i9 +7sT =0) 



Using these relations and those in § XL 2, we find 



2 J { f + (%1 } rff = * Sf (2 + xi) + 7i< ^' + 7i ^" 2 - 



4. Pivoted end. — At a pivoted end, 



J V + WfJ ^"aiSp'Sf Sv2f»- X 5f3x' 



Now /= 4> + yfr, 



and at a pivoted end ^r = 0, always. 

 .-. /=<£, 



ft /" 9 (tat , , 9 / c. . / 



^F = -^' ^p = - 2 ™ 1 *-* * ?<p * 



Lata 



^p--7i0. 



