Mr. Moon on Fresnel's TJieory of Diffraction. 93 



finite integrals id r cos — r-^ and Id r sin — /^ cannot be ac- 

 curately expressed ? — I reply that it is unnecessary that the 

 integrals should be accurately expressed, but they may be ap- 

 proximately. In the above investigation, as in every other 

 where the theory of secondary waves is employed, in making 

 the approximation, Fresnel stops at the point where it suits 

 him ; and after neglecting all powers of s higher than the 

 second up to a certain point, immediately begins to give them 

 significance. This will readily appear, for we have without 

 approximation, 



/=PN = '/N02+P02-2PO.NOcosNOP 



v/' 



+ (a + rf ~2a(a -i-r) cos — , 



(where P O = a + ^) an expression which is unmanageable 

 unless s be small, in which case we have 



f=K/ 



r^ + a{a + rf-^=.r + ^j% 



and 



a: = /dscos2'jr (7^ — ^) = / dscos— {vt —f) 



Now since we have agreed that all powers of s above the 

 square may be neglected, we are authorized to put 1 for the 



cosine of the arc — - — . s% and the arc itself for its sine in 

 \ 2ar 



the above expression. Making these substitutions and inte- 

 grating, we have 



27r, ^ . 27r a + r s^ . 27r, , . 



and if Sj and s^ be the limits of s, 



27r 

 = (53— Sj)secdcos— -(«5if — r + 6), 



A 



where 



tan — J = — - . ^ ^ LJ 



\ A lar 3 



