Wires of varying Elasticity. 97 



Let /32 = _P then for the vibration-frequency ^- in the 

 n ^ 7r 



following cases we have 



Bar of two materials. 

 Free -free, 



n^ x tan (falj + n 2 (2 2 tan(/3 2 Z 2 ) = 0; " 

 Fixed-free, 7h a 



tan(/3 i / 1 )tan(/3 2 y-^=0; f. ■ 



Fixed-fixed, 2P2 | 



r< 2 /3 2 tan (fiJJ + n^ { tan (fi 2 l 2 ) = 0. J 



(62) 



J5ar of three materials. 

 ree, 



^Jcot (fcy cot flyy + ~4 2cot WA) «>t/W) 



Free-free, 



«2 

 »1 



Fixed-free, 



+ ^|5-cot(M)cot(^ 3 )=l ; 



^|tan (fti,) tan (££) + ^|tan (M) tan (/S,/,) 



Fixed-fixed, 



+ j|tan(M)tan(|3 3 y = l; 



Jl 1 cot (M) cot (ft*,) + jf cot (ftZ,) cot (ftZ 3 ) 



^2P2 W 2P2 



+ ! g^ 3 cot(ft? 1 )cot(ftz 3 )=i. 



Bar varying continuously but slowly after the law 



P^oO- + *'*)> n=n .(l + gaf) J 

 where terms of order (cr'l) 2 , {ql) 2 may be neglected. 



[Here we have merely to write a' + q for 2H and & ' — q for 

 2K in the corresponding previous problems.] 



Free-free, 



•-Wjft-T*')' ( 64 > 



Fixed-free, 



*" 21 'Vftl 1 4 ' «»(2t + l)»i* tW) 

 Fixed-fixed, same as fixed-free. 



Thus in the fixed-fixed and free-free bars the overtones 

 are harmonics of the fundamental, and the frequency is the 



Phil. Mag. S. 5. "Vol. 21. No. 129, Feb. 1886. H 



(63) 



