AGGREGATE EFFECT OF INTERFERENCE. 169 



The principle assumed as the basis of adulation is this*: " The effect of 

 any wave in disturbing any given point, may be found by taking the front 

 of the wave at any given time, dividing it into an indefinite number of 

 small parts, considering the agitation of each of these small parts as the 

 cause of a small wave, which will disturb the given point, and finding, by 

 summation or integration, the aggregate of all the disturbances of the 

 given point, produced by the small waves coming from all points of the 

 great wave." 



I took, then, the simplest case which can be conceived, viz. that of an 

 infinite plane wave. There can be no doubt that the result in this case 

 ought to be the following : that the disturbance produced is the same in 

 intensity as that corresponding to one of the points in the disturbing 

 wave. 



Let b be the perpendicular distance of the given point from the wave, 



then it is evident that if c sin — (vt — x) be the disturbance of any point in 



the wave, the effect produced, according to the above principle, will be re- 

 presented by 



2tt J crdr sin — (vt — y/r 2 + b 2 ) x some quantity. 

 Nor is it less evident that the result actually is c sin — (.vt — b). 



A 



What therefore is the multiplier in question? If it is not constant, it 

 must be some function of r. 



Denote r 2 by %, and let the multiplier bey (ss) : 

 then our equation assumes the form 



7T . f* d% sin — - (vt — \A + U)f (as) = sin — (vt — b). 

 But by the very elegant theorem of M. Liouvillef 



/o" <f> (x + a) a*- 1 da= (- l)^/^ (x) dx^, 



* Airy's Tracts p. 267- Traite de la Lumiere, par C. H. D.Z. (Huygens), p. 17. A. Leide, 

 1690. t Journal de l'Ecole Polytechnique, 21« Cahier, p. 8. 



Vol. VII. Part II. Y 



