766 On Induced Stability. 



to that of the first, or lower, and a is half the amplitude ot 

 the applied motion. Writing these equations 



[aW-{g + *n\J nt + e- int )}^e + bW$ = (), 

 [ p B 2 -{ 9 + un\e int + e- int )}]cj ) + qD 2 d = 0, 

 we have two particular solutions, each o£ the form 



= f A r e (c+rin) \ = 2 B r e {c+rin) \ 



— 00 — 00 



where 



{-g + a(c + ri>i) 2 }A r -aii 2 (A r _ 1 + A r+1 ) + 6 (c + Wn) 2 B r = 



(r) 



The set of conditional equations, (r), determines c and the 

 relative values of the coefficients. When a is small the terms 

 diminish rapidly from A and B , and when a approaches the 

 limit zero, an remaining finite, we obtain 



(ap-bq) 2 c* + {2(an) 2 (a 2 +p 2 + 2bq)-g(a+p)(ap-bq)}c 2 

 + k(ctnY—Z(anYg{a+p)+g 2 {ap — bq) = Q. 



The roots of this quadratic in c 2 are real for all values of an. 

 For stability the quantities 



2{an) 2 {a 2 +p 2 + 2bq) —g(a + p) (ap — bq) 

 and 4:(an)±-2(a ) i) 2 g(a + p)+g%ap-bq) 



must be positive. Thus stability is always ensured by making 

 the frequency of the applied motion sufficiently large. For 

 two equal rods, each of length /, the condition is 



(an) 2 >0-683Zy. 



It may be noted for the sake of comparison that for a single 

 rod of length 21, for stability (any 2 > \lg. 



2. In the case of a chain of three uniform rods, each of 

 length Z, we obtain the c equation by a method similar to the 

 preceding ; — 



26(cZ) 6 + 9(259^-4%)(cZ) 4 + 9{154156/* 2 - 11572filg + 112(^) 2 }(cZ) 2 

 -+-81{27040//, 3 -12480//% + 570/*(/a) 2 - 5(^) 3 } = 0, 

 where p—fy(*nfi and a is small. 



The roots of this equation in c 2 are real for all values of an. 

 They are negative if 



(«?i) 2 >l-7%, 



the condition for stability. 

 March, 1909. 



