296 Mr. L. C. Jackson on 



These may be re-written : 



M^ + Mm 2 ?/ = Mm*uz, . . . (1) 



^ji+ (Nn 2 -4- M?nV> = Mm*«y, . . . (2) 



where ??i and n are derived from the free isolated vibrations 

 of M and N respectively, viz. : 



y = a sin mt and z = b sin nt, 



and 9 # 



m 2 = ~. t 



Following the analogy of electrical practice, we may write 

 the coefficient of coupling 7 as given by 



2 _ _MmV_ 

 'y-N^ + MmV * ' ' 3) 



Solution and Frequencies. 

 To solve (1) and (2;, try in (1) 



y = <* • • • • i£) 



This gives ; = ^ + ^ ^ (5) 



Then (4) and (5) in (2) give the auxiliary equation in x : 

 NV + x 2 (l$m 2 + N;i 2 + Mm 2 * 2 ) + Nn 2 m 2 = 0. . . (6) 

 This may be re-written in the form 



x± + w 2 (p 2 + q 2 )+p 2 q 2 = 0. .... (7) 

 Hence x = ±pi or +g-i. . . .' . . . (8) 



From this point we shall treat the case in which M = N, 

 m = n. 



Thus the equation (3) for the coupling now reduces to 

 2 _ * 2 

 7 ~"l-f« 2 * 



Hence, on inserting the usual constants, we may write for 

 the general solution and its first derivatives, 



y = E sin (pt + e) + F sin (qt + </>), (9) 



9 5 9 2 



r — ra* . . m z — c 

 — 5 — iii sin (pt + e) + 5-= 



T^-Brin ( ? i + e) + V-J sin (?« + <#»), (10) 



