Variably Coupled Vibrations. 333 



differentiations of these with respect to the time, we find 

 = E sine+F sin <f> | 

 = G sin e + H sin (/> J ' 

 v = Ey> cos e + Fq cos <£ 

 = Qp cos e + H^7 cos </> 



Equations (24) and (25) are satisfied by 



e = 0, </> = 0. . . . 

 Then from (18), (19), and (21), we have 



_ -p 2 + b n^b-m*- S{Q>-m*)* + 4a*p} 



(24) 

 (25) 



(26) 



G: 



H = 



.E 



ap 



2ap 



q 2 j¥b F _b-m *+ y/ \ ( b-m 2 ) 2 + ±a 2 p\. 

 ap 2a p 



\(27) 



(26) and (27) in (25) give 



= Gp + E.q 



whence 



p(-q*+f) 



-p 2 + l> 



1 

 v | 



!> 



(28) 



(29) 



F = 



?(-<f+i> 2 ) J 



Thus we see that the ratios of the amplitudes of the quick 

 and slow motions for the y and z vibrations respectively are 

 given by 



V 



lb + m 2 W { (& - m 2 ) 2 + la 2 p \ J 



E _ (b -m 2 + V {(b-m 2 ) 2 + ±a 2 p})(b + m 2 - </ {(b - m 2 ) 2 + 4 a 2 p] ) 



h 0*0) 



(b 



; i 



- A /{(6-m 2 ) 2 + 4a 2 /3 })(6 + m 2 + N /{(/>-m 2 ) 2 4-4a 2 /3 ])^j 



or writing 8 for %/{(& — m 2 ) 2 + 4a 2 />} 



G_ rM-m 2 — 8~i» 1 



H " U+m 2 + SJ 



E _ (6 -m 2 + g)(& + m»-S)* f 

 F ~(&-m 2 -S)(6 + >» 2 + 5)*J 



(31) 



