Wires of varying Elasticity. 95 



whence, following the same method as for the free-free bar, 

 we get as a first approximation, 



cosa £=0 ; 

 and as a second, 



cos cc l= sin <y\j^ + 2~) ' 



7T 



or, putting a Z=(2i-f 1) g in tne sma11 terms, 



7!" 



^ = (2^+1)--^, 

 where 



„ = (2i + l)|{ 



KJ 2HZ 



\ 



! (2i + i) 2 J 

 (2i+lK /M;f 1 _KZ__2H^ ) 



The overtones are thus in this case not strictly harmonics 

 of the fundamental. In the higher overtones this defect 

 becomes very small, but in the lower it is very noticeable. 

 Consequently in a fixed -free bar of the nature here considered 

 the musical character of the note will be decidedly inferior. 

 This fact might prove a good criterion of the isotropic nature 

 of a bar of strictly uniform cross section, especially when taken 

 along with the absence of this defect in the same bar in the 

 free-free or fixed-fixed vibrations. Noticing that, if the suffix 

 I refer to the end z — l of the bar, 



«i=«o(i + KZ), 



we find that if the bar be reversed, the equation for the 

 frequency is 



* 21 V^rl 2^7r\2i+\yy : w 



Po 



Thus, if the fixed end become the free, the pitch of every 

 tone, especially the lower tones, is altered. This, again, would 

 supply a delicate test of the isotropic nature of the bar. From 

 the value of H we see that, if the elastic properties be con- 

 stant throughout the bar, the pitch is highest when the densest 

 end is fixed ; while if the density be constant, the effect of 

 reversing the bar depends on the values of p : q and of m Q : n Q . 

 Since m — n is positive for any substance that can properly 

 be termed an elastic solid, the coefficient of p in H is always 

 positive, and algebraically much greater than that of q, which 

 may be positive or negative. Thus, as a general rule, the 

 pitch will be highest when the most elastic end is fixed. If, 



