Variably Coupled Vibrations. 331 



This oives 



D" 



•' /= ttp e ^ 



Put (G) in (o); then we obtain the auxiliary equation in x 

 x A + 2/,u 3 + (b 4- in 1 )* 3 + 2M.i' + w 8 6 - a 2 p = 0. . (7) 



This is a complete equation of the fourth degree, but since k 

 is small compared with the coefficients of x 2 and x, we may 

 assume a solution in the form 



a;==— r+ip or —s + ig, . , . . (8) 



where r and 5 are small quantities whose squares are 

 negligible. 



Thus we have the equivalent equation 



(x 4 r—ip) (.« 4- r + ip) (x + s— iq) (x + s + iq) = 0, 



<or a? 4 + 2(r 4 s> 3 + Q? 2 + g 2 4- r 2 + s 2 + <krs)x 2 



4 2(/> 2 s 4- gV + r 2 s 4 ?'5 2 ) * + (p 2 4- r 2 ) (? 2 4 s 2 ) = 0. 



.... (9) 



This, on omitting the negligible quantities, becomes the 

 approximate equation sufficiently accurate for the present 

 purpose 



<r 4 4- 2(r 4- s)x* + (p 2 4- q 2 ),v 2 4- 2(fs + q 2 r)x +p 2 q 2 = 0. (10) 



The comparison of coefficients in (7) and (10) yields 



r + s = k, (11) 



jfi + if=b + m*, (12) 



ph + gr=kb, (13) 



pp-q- — m 2 b — a 2 p (14) 



From (12) and (14) we may eliminate q 2 and obtain a 

 biquadratic for jt> 2 , whose roots may be called p 2 and q 2 . 

 We thus find 



2p 2 = b + m 2 + v / {{b-??i 2 ) 2 + 4:a 2 p\ | 



2q 2 = b + m 2 - x /{(b-m 2 ) 2 + ±a 2 p} }' ' 



(11) and (13) give 



pt-q 2 ' S p 2 -q 2 ' ' 



Z2 



