94 Mr. C. Chree on Bars and 



tones of a free-free bar of the kind here considered are strictly 

 harmonics of the fundamental tone. 



Referring to (53) for the law of variation of a or of a / £- 



we see that 



where M 7 and p f are the values of these quantities when z= j?, 



i. e. at the middle point of the bar. Thus, the whole series of 

 tones is the same as for a uniform bar whose material is the 

 same as that of the given bar at its central cross section. 



If we suppose the material of the bar at the end z = 

 known, and from this knowledge the pitch calculated, then 

 the real pitch will be higher or lower than the calculated 

 according as K is negative or positive, i. e. according as 



— — w — y — <r is positive or negative. 



om —n Q r fe 



Thus, if the density increase from that end while the elastic 

 properties remain constant, the calculated pitch is certainly 

 too high. Since m —n () is positive for any known material, 

 if we suppose p and g to be of one sign, the calculated pitch 

 is too low if the density be uniform, while the elasticity 

 increases from the known end. If the density and elasticity 

 both increase or both diminish from the known end, the result 

 depends on the relative amounts of variation, and also on the 

 relation between p and q. If we suppose g=p (i. e. all elastic 

 properties to alter at the same rate and in the same direction), 

 the calculated pitch is too low \ip —g' be positive. In this 

 case we may say : — 



" If the elasticity and the density of a free-free bar vary 

 both directly with the distance from one end, the tones pro- 

 duced by the bar are harmonics of the fundamental, and the 

 pitch of each is higher or lower than the pitch as calculated 

 from the material at that end according as the ratio of the 

 elasticity to the density increases or decreases." 



For a fixed-fixed bar we get, pursuing similar methods, 

 exactly the same result as above in (56) for the free-free bar. 

 Thus the overtones in this case also are harmonics of the 

 fundamental, and are identically the same as those of the free- 

 free bar ; and the results as to the variation of density and 

 elasticity obtained above apply without modification. 



For a fixed-free bar, 



A o = 0, and — A sin a/+ B cos al — i) ; 



