Wires of varying Elasticity. 91 



By the same method, we find for the frequency of vibration 

 in three materials forming portions of a bar of lengths Z 1? l 2 , 1%, 

 writing (m + 7z)a = N for shortness, 



Free-free, 



^cotOi/i) cot (a 2 Z 2 )+ ^ 



cot (a^i) cot (ct 2 l 2 ) + ir cot (a 2 Z 2 ) cot (a 3 Z 3 ) 



N 2 2 



Fixed-free, + N^ Cot Wl) Cot <" 3 ^ = X ; 



^- tan (a^i) tan ( 2 / 2 ) -f ^ tan (a^) tan (a 3 Z 3 ) 



Fixed-fixed, + % tan (a ^ tan <«■« = ^ 



-^ cot (ai^) cot (u 2 h) + ~ co t (a 2 / 2 ) co t ( a3 / 3 ) 



+ -j^r- cot ( ai ^) cot (a 3 Z 3 ) = l. 



There is no difficulty, except the length of the expressions, 

 in extending the method to four or more materials. The 

 resulting frequency-equations obviously will follow each a law 

 which might be established inductively, so that the form for 

 each of the three types for any number i of materials might 

 be at once written down. Any difficulty would be purely 

 trigonometrical, and in the free-free and fixed-fixed types 

 would be much lessened by the interchangeability of the ends. 



The relations (43) or (44) may clearly be regarded as rela- 

 tions between the constants of any two successive materials. 

 They were established independently of any assumption as to 

 the difference in constitution of consecutive layers. We may 

 therefore, remembering our previous hypothesis, regard them 

 as in the limit applicable to a continuously varying material. 



Thus, writing z for I &c, and putting a 2 = ot 1 + ~rdz &c, 



when we proceed to the limit we easily find 



N clA r A M v , B . d "I d (m + n 



(m + n) a — - = — — (1 — cos 2etz) + ^ sin 2ccz K - 



J dz L 2 . * J dz 



-*(m + n)Bz~, . . (49) 



and 



( '" +n)a § = [f sin 2az+ I (1 - cos 2a ^] c ^r 



+ et(m + n)Az^. . . (50) 



