98 Mr. C. Chree on Bars and 



same as for a uniform bar whose material is identical with 

 that at the central cross section of the bar. In both cases the 

 pitch is higher or lower than that calculated from the material 

 at one end, according as the ratio of the rigidity to the density 

 increases or diminishes with the distance from that end. 



In the fixed-free bar the overtones are not in general har- 

 monics of the fundamental, though the defect becomes very 

 small in the higher overtones. The actual pitch of any tone 

 is higher or lower than that calculated from the material at 

 the fixed end, according as 



q{ir*(Zi + 1) 2 -4} -cr f {ir\2i+ 1) 2 + 4} 

 is positive or negative, i denoting the number of the overtone. 

 Thus the pitch is always higher if the density diminish and 

 the rigidity do not decrease ; it is always lower if the rigidity 

 diminish and the density do not also diminish. If both den- 

 sity and rigidity increase, the pitch is higher or lower than 

 the calculated, according as the proportional variation in 



rigidity to that in density is greater or less than ^i|i^l!±i ; 

 the reverse being the case if both diminish. For the funda- 

 mental tone the determining ratio is * 2 + ♦ 

 If the bar were reversed, 



*- u "vfy ~^r l+ ^(27+17/ ! • (66) 



so the pitch is altered by reversion. 



To get the highest pitch, G.x the end for whose material the 

 product of the density into the rigidity is the greater. 



There are several practical applications which might be 

 made of the preceding results. Several of these have been 

 suggested in passing, so we shall only briefly consider them. 

 It is obvious that from (25) and (35) we can deduce the 

 mean 'values of Young's modulus, and of the rigidity between 

 the fixed end and any determinate cross section of a wire 

 by observing the displacements w or v at that cross section. 

 This may be done at a whole series of cross sections, while the 

 wire is subjected to the same forces of tension or of torsion 

 throughout its entire length, applied as in the ordinary 

 experiments. We should thus be able to determine the law 

 of variation of the modulus of elasticity, or of the rigidity to 

 any required degree of accuracy throughout a considerable 

 length of one and the same wire. The results might be em- 

 ployed to determine the degree of uniformity in the structure 

 of the wire, or they might be used to determine the effect on 



