424 Determination of Young's Modulus for Glass. 



Mr. Bell's formula 



81= \ ^-cos — j-dz (6) 



Jo fe o ^ 



This gives the correction U necessary to obtain the length 

 of the equivalent rod of uniform section. 



The proof does not assume S to be the mean area of the 

 cross section. But the assumptions that SS/S is everywhere 

 very small, and that the squares of the integrals in (3) are 

 negligible, will be in general most satisfactorily fulfilled when 

 S is the mean section. 



When S is more than usually variable, a slight increase in 

 accuracy would probably be obtained by determining k 

 directly from (3). In this event it would be convenient that 

 S () should be the mean section, as we should then have 



( cos2 T 



Jo V L 



. 9 7TZ\ SS j ' 



sm- —=- )-^-dz = (J, 

 I J fe 



and so 



C l 9 7TZ SS . C l . 9 TTZ 8S, If* 27TZ SS _ 



I cos- -y- . -~-dz=—\ snr r . a-dz= 1 \ cos -=— . ~— dz. 



Jo l fe o Jo l b o ^J / fe 



We should thus have still really only one integral to 

 evaluate. 



If the tube were fixed-fixed instead of free-free, the origin 

 being still at an end, we should obtain for the correction to 

 the length 



8Z=-j o '(8S/So)cos(29r*/Z)<fe. ... (7) 



This is numerically equal but opposite in sign to the cor- 

 rection obtained for the free-free rod ; thus by combining 

 results from the two species of vibrations we could eliminate 

 the effects of variation of section, assuming them of course 

 to be small. 



In a fixed-free bar, z = being the fixed end, we find 



Bl= -ft (B$/&o) COS (wz/Qdz. ... (8) 



In all these cases an increase of S near a fixed end or node 

 tends to make 81 negative or raises the pitch, while an increase 

 of S near a free end tends to make 81 positive. In other 

 words, an increase of material stiffens the bar elastically when 

 near a node, but acts mainly as a load when occurring near a 

 section of maximum amplitude of motion. 



