Wires of varying Elasticity. 93 



Introducing the proper constants, we find 



A-A -f A (? sin 2 "° z \ ft (l-cos2a o 0) I 

 A=A - 2 (A ^- ^__j-J3 ^ / 



-|b « ^ • • (54) 



B=B . + H|V<i^ + B.( s -=|a)} 



+ 2-A « s 2 . . . (55) 



For a free-free bar, R=0 when 2 = and when z — l\ 

 therefore 



B =0, and — AsinaZ + Bcos al — 0. 



Neglecting smaller terms as usual and putting B = 0, the 

 latter equation may be wTitten 



A sin ct l + A K« Z 2 cos <* l — B cos a /. 

 Introducing the above values of A and B, putting B = 0, we 

 get 



9in ^ { 1 _ H (, _ sjn|^ J + w cog ^ 



, fH/l-cos2«J\ K „\ 



Remembering that the terms in H and K are very small, we 

 get as a first approximation, 



sina £ = 0, or a Z = wr. 



ITT 



As a second approximation, substituting -j for a in the 

 smaller terms, we get 



sina /+ 2 wr( — 1)*=0. 



If we suppose oc l = i7r — e, where e is very small, we thus have 

 . KZ 



,...= !f(l-¥> >■■:■■ («) 



and 



The frequency of the fundamental tone of the bar corresponds 

 to t=l, and the overtones to the consecutive integral values 

 of i. Thus, to the present degree of approximation, the over- 



