when Light is radiated from Moving Molecules. 299 



and that for the most part each molecule radiates independ- 

 ently, there seems no escape from the conclusion that the 

 character of the aggregate radiation must be governed by 

 Doppler's principle. 



If V be the velocity of a molecule, 6 the inclination of its 

 motion to the line of sight, the natural wave-frequency N is 

 changed by the motion into n, where 



„=^I±^, (1) 



and V is the velocity of hght. If A, X be the original and 

 altered wave-lengths, so that 



A = y/N, X=V//i; (2) 



then 



V 



\ +v cos V 



= A 1 1 — ^ cos ^ I approximately, . . (3) 



when v/V is small. 



As a first approximation, Ebert supposes that the velocity v 

 of every molecule is the same. In this case the spectral band, 

 into which what would otherwise be a mathematical line is 

 dilated, has the Hmiting wave-frequencies 



N(H-^), n(i-^) (4) 



and between these limits is of uniform brightness. For 

 the number of molecules whose lines of motion lie between 6 

 and d + d6 is proportional to sin^cZ^, and this again by (l)is 

 proportional to dn. It is here assumed that the spectrum is 

 formed upon a scale of wave-frequencies ; but for the present 

 purpose the range concerned is so small that it becomes a 

 matter of indifference upon what principle the spectrum is 

 disposed. 



The typical case of interference arises when two streams of 

 homogeneous light are superposed, which differ in nothing 

 but phase. If B denote this difference of phase, the vibrations 

 may be represented by 



cos -^/r -j- cos (i/r + 6), 



or by 



2cosiScos(^/r + iS); (5) 



and the intensity is 



l=4:Cos^S (6) 



