278 Resolution of Light into Undulations of Flat Wavelets. 



All that remains to be done is to start the rigid system 

 forward in the direction OX with the speed of light. It then 

 represents what occurs in nature, and furnishes the following 

 theorem : 



Theorem XI. 



If ufw's of all the wave-frequencies that lie within the 

 band mn in the spectrum travel in the direction OX ; if they 

 are of equal intensities and polarized alike; and if their 

 phases are such that their crests coincide at some one instant 

 of time with any one plane YZ 'parallel to their wave-fronts : 

 then this light produces at each station in space an illumination 

 which lasts for a period \/8v, ivhere 8 is half the range of 

 wave-frequencies within the band: with perhaps glimmers of 

 illumination before and after that period. 



These glimmers would require for their extinction the 

 simultaneous presence of certain appendage ufw's which, 

 however, in many practical problems represent so small a 

 part of the light that they need not be taken into account. 



20. We may also infer the following theorem, which 

 is the converse partly of Theorem VIII. and partly of 

 Theorem XI. 



Theorem XII. 



Light that at our station extends over a limited space and 

 lasts for a limited time is theoretically susceptible — i. e. would 

 be susceptible if the physical conditions prevailing in nature 

 justified our pursuing a resolution of the kind to its mathe- 

 matical limit — of being resolved into v t ic's infinite in number 

 and each occupying the whole of space for all time. 



In dealing with the actual problems of nature, a resolution 

 is found to be sufficient which falls short of this extreme, and 

 which is well within what the physical constitution of matter 

 tolerates. 



21. We are now in a position to effect the second of the 

 two useful substitutions spoken of in § 4. If we have only to 

 consider what happens at our station within a limited duration 

 which we may call r. and which may be a second, a minute, 

 an hour, a week, or any other : then we may proceed Jb 

 follows : — The light with which we are concerned may be of 

 various wave-frequencies extending over either the whole or 

 a part of the spectrum, and the rays of which the spectrum 

 consists may be of any (either the same or different) inten- 

 sities and states of polarization. They need not be in any 

 way related to one another. We may conceive the spectrum 

 to form a ma}) of wave-frequencies. Then we may divide 

 the length of this spectrum into equal degrees, each of which 

 corresponds to a fixed difference of wave-frequency, which 

 we may call S, and we may make these decrees large or 



