26 



Prof. Barton and Miss Browning 



(1) Frequencies. — As solutions of (14), let us try the three 

 following expressions : — 



x = Ee mi \ y = Fe mit , z = Ge mit , 

 Then, substituting (17) in (11), we have 



(m'-flx + ky + kz^O^ 

 kx + (m 2 -K)y + lz = oA. . 

 kw+ly + (m 2 -h)z = Q. ) 



Eliminating x, y, and z from these, we obtain 



(17) 



(18) 





nr—f, k, k 





*, m 2 -h, I 1=0. . . . 





Jc, L m 2 — h\ 



This reduces to 



(m»-A~Z){(m*-/)(m*-ft + Z)-2iP} = s . 



a cubic in m 2 . 



The roots of this cubic may be written 



mr 



h + l, 



or 



r 7 -i(/+A-/)±tv / {(/-/i + /) 2 + cSP}. 



(19) 



(20) 



(21) 



By substitution from equations (15), and using subscripts, 

 these roots become 



??ii = 



2_ 



ms = 



3a + b-±c 



2 



a 



(22) 



Accordingly, the six roots of (21) regarded as an equation 

 in m may be written 



+ m l5 + m 2 , 



+ Wj 



(23) 



And, corresponding to each of these six roots, there will be 

 in the solution an arbitrary constant. 



(2) Amplitudes. — Referring to the equations of motion 

 (14), and the trial solution of these in (17), let us now 

 obtain relations between the amplitudes JE, F, G, which will 

 hold generally, that is, for any of the six possible values of m. 



