CONTEMPORARY ADVANCES IN PHYSICS 175 



and solving for g, 



.?=^'-X^, (19) 



or going over to the differentia! notation, which will not only look more 

 natural but will signify that the result which we have just attained is 

 strictly valid in the limit for infinitesimal differences of wave-length: 



g^ V - X(dvi'd\). (20) 



This is the formula for the group-speed; for the term "group-speed" 

 is the usual one for what I have been calling "speed of beats." Like- 

 wise phase-speed is commonly used to denote the speed of the individual 

 sine-wave trains. 



The term "group-speed" is in one respect unfortunate; for it implies 

 that any "group," that is to say any sequence of uneven and irregular 

 wave-crests and troughs, is propagated with a perfectly definite speed. 

 However this is true only for the simplified group which we have been 

 considering, the beat formed of no more than two wave trains; and 

 even for this it is exactly true only in the limit, where the wave-length- 

 difference between the trains approaches zero. All other groups 

 change in form as they advance. Now there is always something 

 arbitrary in defining "speed" for something which changes as it goes, 

 like a puff of smoke or a cloud. The arbitrariness is nil in only the 

 limiting case which I have just been formulating. However, it must 

 not be exaggerated. A bunch of irregular crests and troughs may 

 retain enough of its form and compactness, as it travels over a distance 

 many times as great as its width, to justify the statement that it has a 

 speed of its own. And if such a group turns out, on being analyzed in 

 Fourier's way, to consist mainly of sine-waves clustered in a small 

 range of wave-lengths, then its speed will not be far from the value of g 

 computed by equation (20) for a wave-length in that range. 



Now these deductions explain a very remarkable experiment by 

 Michelson, which otherwise might have disproved — indeed I do not 

 see how it could have been interpreted otherwise than as destroying — 

 both the wave and the corpuscle theory of light. I will preface the 

 account of this experiment by saying that for light in empty space the 

 speed of all wave-lengths is the same,** so that there never is any dif- 



* The chief evidence for this statement is astronomical. If light of one color 

 traveled faster than light of another, a luminous star emerging from behind a dark 

 one would be seen first in the faster-travelling hue; in fact there would be a .se- 

 quence of colors, the same for every emergence of every such star, and spread out over 

 a time-interval proportional to the distance of the stars. Nothing of the sort has 

 ever been observed, although there are plent>- of luminous stars revolving around 

 dark ones which regularly occult them. 



