112 



PKOCEEDINGS OF THE AMERICAN ACADEMY. 



m 



d ll + k 2 r \ _2t d ^-Q 



df- +kX 3c 3 tf* 3-U 



or 



m 



dx 

 dt 



— + k 2 x 



2e 2 



+ 3c 



(PxV 



dt 2 ) 



0. 



As numerical values we may assume 



e = 4.7 10- 10 , m = 9 lO" 28 , 

 Hence 



c= 3 10 



10 



2 e 2 



= 6 10 



-'.21 



3 mc 3 

 Substitute t = ar and choose a 2 F = 1 



d 2 x 



dr 2 



+ x-6k 10- 24 



Then 



d?x 

 d? 







and 





~d 2 x , 



+ 6M0-^Y=0. 



Now ft = 2tc/\, where X is the wave length. The observed range of X 

 in the spectrum is from 10~ 5 to 3 10~ 2 with X about 5.5 10~ 5 in the 

 center of the visible spectrum. The range for the coefficient of the 

 last term in the equations is therefore from about 10~ 7 in the ultra- 

 violet to about 4 10 -u in the infra-red with 2 10 -8 for the value near 

 the middle of the range of visibility. 



The numerical forms of the equations are therefore 



d 2 x 



5 + x 



d?x 



d? 







and 



with 



dx 

 dr 



'd?x 

 dr 2 



+ x 



+ K 



r/-.r 

 dr 2 



= 



(5') 



(6') 



10- 7 > k > 4 lO- 



ii 



or 



2 10- 8 



the latter being taken as a sort of standard value. 



(It is not necessary to make the special assumptions involved in 

 the electron model of the oscillator if we desire merely to determine 

 the order of magnitude of k. If (5') be replaced by 



d 2 x dx 

 -d? +K Jr +X 



o, 



(7) 



