and 



nhni) = (n 

 ,sh m)=0) 



(21) 



Vibration of Bars treated simply. 589 



Proceeding as before we find P = and R = 0, and that 

 Q(cos m + cosh iri) + S (sin m + sinh m) = 



Q(— sin ?7i + smh m) + S(cos m-f-cori 



From the second of these, omitting a constant multiplier, 

 we find 



u = (cos m + cosh m) (cos oc' — cosh a?') 



-f (sin m — sinh m)(sin a/— sinh x). . (22) 



Further, by equating the two expressions for Q/S from (21) 

 we derive an equation to determine m, viz. : — 



cos m cosh m + 1 = 0, ") 



or ( ' • ' (23) 



sec m = — cosh m, ) 



The second of these forms is convenient for the graphical 

 exhibition of the relations of the roots as shown in the lower 

 part of fig. 4. From this it is seen that the higher values of 

 m which satisfy (23) are approximately as the odd values 

 of 7r/2, the approximation being very roughly fulfilled for 

 even the second value and showing a serious departure at 

 the first value alone. The frequencies of the possible tones 

 are as the squares of these values for m. But as regards the 

 relative frequencies of the overtones to the prime, this de- 

 parture of the first note is vital, and in a statement of these 

 relative frequencies quite obscures the simple relation sub- 

 sisting among the higher overtones which is so clearly seen 

 from the diagram (lower part of fig. 4). The tones for a 

 fixed-free bar are shown in column 2 of Table IV.; these 



Table IV. — Tones for Fixed-Free Bar. 



Odd values 

 of tt/2. 



1-571.. 



4-712.. 



7-854.. 

 10996.. 

 14-137.. 

 17-279.. 



Relative 



Intervals from Approximate 



Yalues of m. ; Frequencies, i p ^ ^ "V 1 " 



N arm 2 -Prime to Overtones 



m 1 = l - 875 

 m 2 = 4-694 

 m 2 — 7 '855 

 ^=10996 

 ^3 = 14-137 

 w B =17-279 



1 



6-267 

 17'55 

 34-39 

 5685 

 84-93 



Octaves. Equal 

 Semitones. 



and 

 and 

 and 



777 

 1-60 

 1-24 



and 9-95 

 and 4-90 



Notes. 





Phil. May. S. 6. Vol. 1-4. So. 83. Nov, 1907. 2 B, 



