PuKSER — Applicatiom of BeHHcVa Functiona to Physics. 87 

 We may therefore uritc 



TT J ^ ^ 



where e = — the equation determining a. 

 127r 



II. ^ compared ivith JR. 

 Here we may write 6*"^= 1, = tn^. AVe now find 



^P,,R\{nR) Y,{nR) sin = ^ 2Q„r, 



^»^-\ ;^,sin^^ir,(^)r,W^^. 



Jo ^ +a„/ A 



This is seen to be of the order ^J\^^^{^^ j ' ^^5" therefore bo . 



neglected in terms of — • "We may then, in this case, write with 

 great accuracy 



V= — [ dx. sin2 ^-4(1- "iKiix) Y,x). 



TT J R X- ^ 



If we divide this into two ranges, x = to x ^ \^ x = \ to 



^X X 



^= oc, we may write in the former sin— = ^77, this part of V 



A R 



becoming thus 



t' 2aR n r 

 j^,~^^{l--2lu{x)Y,{x))dx = f.^^. 



In the second part we may write 



and the second part is seen by reduction to assume the form 



a^^i^^. + ?^-J^ COSX-, where e = 

 p, q being certain numerical coefficients. 



