576 Dr. J. Larmor on the 



be so, the prism is very badly suited for the analysis of this 

 type of radiation, and no amount of adaptation of the result 

 will bring the prism into conformity with the ideal grating. 



The Fourier mathematical process * 3 as also the ideal 

 grating which reflects back the disturbance in echelon so to 

 say, operates by simply selecting and piecing together elements 

 existing in the original radiant disturbance, so as to isolate 

 periodic wave-trains that on superposition would reproduce 

 the form of the original vibration-curve. 



On the other hand, what the prism may do to a given 

 isolated pulse would seem to depend on its own constitution. 

 The customary mode of investigation would be to replace the 

 pulse by the equivalent infinite system of component Fourier 

 wave-trains, to find the effect produced on each of them by 

 substituting its expression in the differential dynamical 

 equations of the dispersive medium, and to add the results 

 thus found. Though the original Fourier expansion of the 

 pulse is always analytically legitimate and definite, it is not 

 always allowable, without scrutiny as to convergency, thus to 

 operate on its separate terms and add. Indeed, the particular 

 component waves whose period is a free period of the medium 

 would increase infinitely in importance in the result : thus it 

 must be ascertained whether this infinity of intensity is more 

 than compensated by infinite smallness of the element of 

 period over which it ranges, before the procedure which 

 includes it can be accepted as mathematically legitimate. 

 The insertion of a very small frictional term in the dynamical 

 equation will, however, secure that the component vibration 

 remains finite though great at this critical period, and the 

 analysis then becomes entirely valid. But the problem is 

 only shifted ; it has now to be ascertained whether the limit 

 which this solution approaches as the friction is reduced 

 indefinitely is the same as the solution previously arrived at 

 in the absolute absence of friction, — whether in fact there 

 exists a definite limit. That there is in many cases no definite 

 limit is merely another way of expressing the theory of 

 anomalous or selective dispersion, in which the final steady 

 result depends essentially on the magnitude of the small 

 viscous term that must be introduced in order to evade 

 infinities, while the mode of the gradual establishment of 

 that result is likewise undetermined. The question thus 

 arises, whether the proportion of the energy of an incident 

 isolated pulse that goes into this selective vibration is capable 

 of determination by operating anatytically upon its Fourier 

 analysis 'in this way, — whether, in fact, a different line of 

 * Cf. < ^ther and Matter/ § 162. 



