Vibrations of a Mod of Varying Cross-Section. 89 



4. If we put 7i = we obtain the case of a wedge-shaped 

 rod free at the thin end and clamped at the other : the 

 direction of vibration being parallel to the planes of tri- 

 angular section *. 



From (11) we have 



u 



= Afc(.*)+B* (jO, (24) 



or M = ^ Il(2 * Al ' )+ v> Jl(2v/A ' ) - 



The equation to determine x 1} and therefore X, is by (23) 

 1 <v 2 # 4 j; 6 



= ITT!~^Ty^^! + 2Tl^^l~3T^m + ■ ,, • (25) 



I find that the first four roots of this equation are 



5-3151, 15*207, 30-020, 49*763. 

 Since x x is the value of x corresponding to z=l, we have 



1 \/{$p) = *" or x= iv(t)' 



giving the frequency corresponding to any root X\ of (25). 

 Also since 



v(lS) = *' 



we have 



* = J* (26) 



From (18) the ratio of the constants A and B is found 

 to be 



A/^fo) = -B/Vr.xfo) = C (say), 



whence u = C{^r_i(o;i)0o(«)-^-i(«i)^oW}' • f27) 

 We notice that 



</>-i = Io(2 v^) and f_! = J (2v/tf). 

 In the first mode we have #! = 5*3151, and therefore 

 u = -C{19-2773^ (.r) + '293327</)o(.r)}. 



* The case of a vibration perpendicular to the planes of triangular 

 section is treated by F. Meyer zur Capellen in Wiedemann, Annalen der 

 Physik, vol. xxxiii. 1888, p. 661. 



