480 Prof. 0. W. Richardson on the Theory of 



for every value of v. For it is easy to show that the only 

 regular function of v not involving T which satisfies the 

 equation 



f 





X(v)e wdv = 0, (12) 



if v is also independent of T, is ^(v) = 0. For let 

 X( v ) = a o+ 2 a s v s +b s v~ s 



s—l 



over the range v <j/<qo . B}^ repeated integration of (12) 

 with respect to dT from to T or repeated differentia- 

 tion with respect to T, we see that for every integer p 



J v ±p x( v ) e * T ^ = (13) 



Multiplying the integrals (13) by the corresponding con>~ 

 stants a a x a 2 .... &i 6 2 ... . and adding we get 



[ x (v)] 2 e~^dv = 0. 



_ *? 

 Since e rt and v are always positive it follows that ^(i/) = 



for v < v< go . Applying this result to (11) it follows that 



<f>(v) = hv (14) 



Thus the quantity of energy which an atom abstracts from 

 the radiation before it is liberated, under the influence of 

 light of frequency v, is Itv. 



This result has only been shown to be valid at low tem- 

 peratures. It appears to follow from the following 

 assumptions : — 



(1) That the distribution of energy in the radiation is- 



given by Planck's formula. 



(2) That such a function as e¥{y) exists and the effects of 

 the spectral components of mixed light are additive. 



(3) That photochemical actions are fundamentally inde- 



pendent of temperature at low temperatures and that 

 they do not contravene the second law of thermo- 

 dynamics. 



(4) That the part of f which comes from the radiation is- 

 negligible. 



This demonstration shows that the considerations about 

 the specific heat of electricity and electron reflexion which 

 entered into proofs * I have given of this and related formula? 



* Pliys. Rev. vol. xxxiv. p. 146 (1912) ; Phi]. Mag. vol. xxiii, p. 624 

 (1912), vol. xxiv. p. 570 (1912). 



