302 Lord Rayleigh on the Limit to Interference 



For the grounds of his estimate Ebert refers to an earlier 

 paper *, in which, however, the calculation seems to relate to 

 a problem materially different from the present, that, namely, 

 in which the refrangibility of the light is limited to two dis- 

 tinct values (as approximately in the case of the soda lines), 

 instead of being distributed equally over the same range. In 

 this case (9) is replaced by 



4|l-cos ^^^K-^i) . cos ^^^^:^'+^^^ } ; . (16) 



so that, if a have the same form as in (11), and o! denote the 

 numerical value of cos a, 



as before.. A= (l-^O/d +<«'), (") 



According to (16) the field is first uniform when a = i7r, 

 instead of tt, as from (9). When a = 7r, the bands are again 

 black, and as A further increases there is a strictly periodic 

 alternation between blackness and absolute disappearance of 

 the bands. 



The substitution for a spectral band of uniform brightness 

 of one in which the illumination is all condensed at the edges 

 explains a large part of the discrepancy between (14) and (15) ; 

 but even in the latter problem (15) seems to be a very small 

 estimate of A. According to (15), a = 54°, cos « = *59 ; so 

 that from (17) /i = "26. Bands of which the darkest parts 

 are of only one quarter of the illumination of the brightest 

 parts could hardly be invisible. 



The more nearly correct formula (14) is itself, however, 

 based upon the assumption that all the vibrating molecules 

 move with the same velocity. This is the origin of the law 

 expressed in (9), according to which the bands should re- 

 appear at a retardation greater than that of first disappearance. 

 But the real law of the distribution of velocity is that dis- 

 covered by Maxwell, if there is any truth in the molecular 

 theoryt. That such is the case is recognized by Ebert ; and 

 he argues that the broadening of the spectral band due to 

 velocities higher than the mean, will entail a further diminu- 

 tion in the maximum retardation consistent with visible inter- 

 ference |. I proceed to the actual calculation of the maximum 

 retardation on the basis of Maxwell's law. 



* Wied. Ann. xxxiv. p. 39 (1888). 



t It is here assumed tliat we are dealing with a gas in approximate 

 temperature equilihrium. The case of luminosity under electric discharge 

 may require further consideration. 



X In the earlier memoir (Wied. Ann. xxxiv.) Ebert appears to regard 

 the capability of interference {Interferenz-fdhigkeit) of a spectral line as 



