1908-9.] On G-ronp- Velocity and Propagation of Waves. 409 
individual waves and wave-groups travel inwards — an interesting result in 
connection with the dynamical theory of group-velocity. 
§ 27. Lord Kelvin’s investigation of 1887 therefore shows that the 
mathematical theory of group-velocity explains the modus operandi of 
dispersion. In relation to sequences of light-waves, — perhaps one of the 
most important applications, — where we are entirely concerned with the 
wave-length, and not with the individual waves, the theory emphasises the 
importance of group-velocity in all cases of refraction. Lord Kelvin’s 
diagrams not only illustrate the manner in which group-velocity is con- 
cerned in the formation of a “ front ” and “ rear,” and in the propagation 
of an initially finite disturbance, but also show how the velocity of the 
individual waves, according as it exceeds or falls short of the corresponding 
group-velocity, influences the distribution of the energy of the initial 
disturbance among the various wave-lengths of the disturbance throughout 
the medium. This is of importance whether we are concerned with the 
maintenance of distinct marks by which a group of waves may for a time 
be recognised, or merely with the energy contained in the part of a dis- 
turbance having a particular wave-length, which is of interest in connection 
with the theory of radiation. The extreme smallness of the amount of 
energy contained in the very short waves is a feature of all the diagrams. 
Thus in fig. 34 the energy of the initial disturbance would need to be very 
great if we are to regard the energy of even the largest imperceptible wave 
in diagram 6 as sensible. 
§ 28. It is not intended in this paper to enter into a discussion of 
particular applications of the theory of group-velocity presented above, 
but it may be useful to point out in conclusion one application of importance 
in the theory of radiation, which is suggested by § 27. This is to the case 
of a black body, say a plate of metal, which is slowly heated on one side. 
At first only long heat-waves are observed, but shorter and shorter heat- 
waves are emitted as the temperature rises. At about temperature 525° C. 
the first visible rays appear, and the plate becomes red, owing to the long 
light-waves it sends out. As the temperature is further raised, shorter and 
shorter wave-lengths appear in appreciable amount, and the colour of the 
plate changes from red to yellow, and finally to white. 
The points of importance in relation to the diagrams are that the 
various wave-lengths are emitted in the order corresponding to their group- 
velocities, and that the energy of the disturbance required to make each 
wave-length perceptible is greater the shorter the wave-length; exactly as 
in the continuous water-spectra exhibited in the diagrams of fig. 34. It is 
interesting to find the idea underlying this application expressed in the 
