in the Spectrum due to Pressure. 789 



distinct normal motion. Under such influence, consisting 

 in a periodic variation in the spring of the coordinate, the 

 free motion changes from the simple oscillation 



A cos {fit + e) 



to the complex oscillation represented bj 



00 



2a r cos \(ji + p + rn)t + e} , 



— 00 



where p and the ratios o£ the a's are determined by the 

 frequency and intensity of the disturbance. With regard to 

 forced oscillations the coordinate absorbs energy from a 

 direct force of frequency equal to that of any element in the 

 series, storing it as a free (complex) oscillation — the property 

 leading to the hypothesis. 



In the phenomena referred to n is small compared with /j,, 

 and it is evident from the general method of analysis that 

 even with a small variation in spring, the elementary ampli- 

 tudes . . . a_!, a , a x . . . are comparable. The general solution 

 may, however, be obtained by a very simple method : — 



x = A (cos ct + a sin nt + e), 



where A and e are arbitrary, is the solution of 



OLTi sin lit 



'x H x -f (<r -f k 2 ft 2 + 2acn cos nt + i* 2 /i 2 cos 2ni)x — 0, 



c + oui cos nt K ' 



an equation determining the motion of a system of natural 

 spring c 2 + ^ct 2 n 2 , subject to positional and motional forces- 

 the strengths of which are periodic functions of the time. 

 n being small the periodic terms in n 2 are negligible com- 

 pared with those in n ; thus 



x + (/ii 2 -f 2afin cos nt)x = 



gives A " f / 1 u 2 r 2 \ . ) 



x = A cos -< I fi - 2 — " )t+otsmnt + € > , 



indicating a reduction in frequency of the second order of 

 small quantities. The square of the amplitude of spring 

 variation is proportional to the energy of the normal motion 

 producing it ; and such motion being subject to dissipation 

 according to the exponential law, for a steady state its 

 energy must be proportional to the pressure — the temperature 

 being assumed constant. Thus the reduction in the frequency 

 of the series is directly proportional to the pressure. 



Although the equation of motion can always be reduced to 



