THE ABSOLUTE INTENSITY OF INTERFERING LIGHT. 



325 



J - 



or, that part of 



beginning with r= - 1 . 



1 1 m 



which does not contain y is 2 '/,-'+ i'/r+2 



,r+l 1.3...(2r+l) 



= 2(-l)" -J 



l>eginning with r= 1. 



.-. that part of the product 



(y + L) 



which contains neither ^ nor 2, is 



2 C- X ) 





be g innin g with r=-l,OT (since the first term 



is 1) ; that part of the product which does not contain y or z in 



.... 

 ^ 4 ^ 2.4...(2r + 



beginning with r = 0. 



But this is precisely one of the factors of the expression for the total intensity. 



TT a 2 c 2 \ a b~ 



.-. total intensity = - ^ * 



r ( -M 

 z in 1+ y - v 



(that term which contains neither y nor 



Now, the actual value of m is infinity, since the integral has to be taken between 

 the limits and oo , and has been expanded in such a form as to vanish for the 

 former limit. But in order that the expression which we have just determined 

 may also vanish when m=0, which it must do if it be the proper formula for ex- 

 pansion, we must add to it some function which does not contain a term inde- 

 pendent of y or z. We may add any such term we please. For our present 

 object it is not necessary to add any at all ; but it may be thought fit to do so. 



We observe, then, that - 



~ y 



-, which always involves y, will suffice. 



