232 Lord Rayleigh on the 



An application o£ Lagrange's method gives as the differential 

 equations of motion, 



c x dt 2 +a ' S " ~ U 

 1^ 2 CX^-X! X^-Xa| 



c 2 dt 2 +a 1 s + s 7- "^ 



(3) 



ld 2 X d 2 X 3 — X 2 _ 

 c z dt* + a S' " 



By addition and integration 



.A.1 JLo -X-O _ , , . 



~+— + — = 0, (4) 



c l c 2 C Z 



since in the case o£ free vibrations all the quantities X 

 may be supposed proportional to e pt , so that d/dt may be 

 replaced by p. 



From (3) and (4) by elimination of X 3 , 



(&+h)*-S-°. • • • • • w 



whence as the equation for p 2 



S + ${nr +£ ^} + ^t^+^)+«i} = o. (6) 



In the use of double resonance to secure an exalted effect, 

 as in the experiments of Boys and of Callendar, we may 

 suppress the direct communication between the second 

 resonator S' and the external air. Then c 3 = 0, and (6) 

 becomes 



To interpret the c's suppose first the passage between 

 S and S' abolished, so that c 2 = 0. The first resonator then 

 acts as a simple resonator, and if pi be the corresponding^, 

 we have pi 2 /a 2 = - c^S, as usual. Again, if S be infinite, we 

 have for the second resonator acting alone, p 2 2 /a 2 = —c 2 /$' ; 

 ajid (7) may be written 



P * 



■p 2 (pi 2 +jV+J/> 2 2 )+j?iV = 0. . . (8) 



