Uniform and Varying Sectional Area. 117 



and the equation is 



Substituting for y z in equation (4) 



d 2 y = p<0 ■; 

 dx 2 EI yi 



rMH?)<->- 



fow(- s '*--*7 + ' 0l ?> 



and performing the integrations, evaluating the constants by 

 the end conditions, 



"" y = BlM" 0819 ^ ¥ ~~ * li2 ' 698 > 41 ^+'0416 x 4 



-'Oiy +'000,198,4l| 8 ), . . (6) 



which is approximately the curve of vibration of the bar 

 and gives 



-y 1= -08194^V, 



or from (3) 



M 3-493 JcV 

 2tt Z 2 ' 



where A: = radius of gyration of cross-section, 



U = \ /—= velocity of transmission of longitudinal 

 P vibrations in the rod. 



The expression given by Rayleigh may be put in the form 

 AT _ 3-5160 HJ 

 ~ 2tt Z 2 ' 

 showing that the ratio between the two values of N is 

 0*9935. 



The next approximation gives a still closer value and a 

 ratio of 0*9998. It is obtained by inserting y z from equa- 

 tion (6) in (4), and using the value just obtained for y u 

 whence 



-•01 j + -000,198,41^ J(a?--s)<fe 

 = U t ^r d -040972 ZV- '018,783,07 ZV + -00138a 6 



Hil 6 \ 



-•000,264,55^ +-000,002,21-^ J, 



