192 M. Cauchy's View of the Undidatory Theory of Light, 



tions will propagate themselves in space with velocities equal, 

 two and two, but proceeding in opposite directions, and re- 

 presented by /2', /2", /2'". 



We have already observed that from the form of the func- 

 tions (36. 37.)> Sq ^^^ »i ^^ve recurring values when q is 



increased by —j- ; and the similar remark made with respect 



to the function (43.), it will also be seen, is not restricted to 

 that particular case, but applies equally to the general formula 



(38. )j when t is increased by t7>. Thus, then, adopting the 



notation of (45. 46.) for the intervals of recurrence in space 

 and in time, we have directly from those expressions 



/2 = 4" (55.) 



Or we find in general that there is always a constant relation 

 between the length of a wave and the velocity of its propaga- 

 tion ; or, in other words, that the velocity of the propagation 

 is directly as the lengths of the waves, and inversely as the 

 times of the oscillations of the individual molecules of the 

 aetherial fluid ; or, what is the same thing, the interval of the 

 time of the recurrence or arrival of two successive waves at 

 the same point. 



It must be recollected that, in order to simplify the investi- 

 gation, we have proceeded solely with reference to a single 

 displacement in the direction of each of the axes ; or, more 

 precisely, it has been conducted on the assumption made at 

 first, that we might consider each of the expressions (17.) as 

 reduced to a single term : those expressions, however, really 

 involve the sum of a number of similar terms. In the ex- 

 pressions (23.), therefore, which represent the initial values of 

 f »j ^ and of their differentials, as well as in the equations 

 (33.), the same consideration must be attended to, that is, we 

 must take 



f = -^ [^0 cos A: g + go sin Jc q] {5(5.) 



&c. 

 f 1 = JS* [di cos ^ ^ + gi sin k q] (57.) 



&c. 

 f = 5* [A' «' + A" 8" + A'" «'"] (58.) 



&c. 

 We shall then have only to introduce the values of a' «" «"' as 

 above found, and the motion of the system may be considered 

 as produced by the combination of many, or even an infinity 

 of similar motions, each the same as those represented by the 

 equations (43.) and (53.). 



Finally, in order to complete the analytical view of the sub- 



