90 Mr. Moon o)i Fresnel's Theory of Diffraction. 



A M F, according to the theory laid 

 down in art. 628 et seq. For sim- 

 plicity, let us consider only the pro- 

 pagation of undulations in one 

 plane. Put A O = a, A B = i, and 

 suppose A = the length of an undu- 

 lation ; and drawing P N any line 

 from P to a point near M, put 

 PN=/, NM=5, PB = ^; then, 

 supposing P very near to B, and 

 with centre P radius describing tlie 

 circle Q M, we shall have 



^=PQ + QN= V(^a-vbf + x^-a + Q.^ = b-{- 



+ QN. 



2{a + h) 



Now Q N is the sum of the versed sines of the arc s to radii 

 O M and P M, and is therefore equal to 



+ 



20M ' 2PM 

 so that, finally, 



f=h + 



2\a ^ b ) 



a+b 

 2ab 



+ - — -s^ 



2 (a + 6) ' 2ab 

 Now if we recur to the general expression in art. 632 for the 

 motion propagated to P from any limited portion of a wave, 

 we shall have in this case a. (p{&) = 1, because we may regard 

 the obliquity of all the undulations from the whole of the effi- 

 cacious part of the surface A M N as very trifling, when P is 

 very distant from A in comparison with the length of an un- 

 dulation; and as we are now considering undulations propa- 

 gated in one plane, that expression becomes merely 



y =■/ d s s'm 2 TT -i fp — 4" p 



and the corresponding expression for the excursions of a vi- 

 brating molecule at P will be 



X=//5C0S^(f-{). 



If then we put foryits value, and take 



^ /t b ^^ \ , /2{a+b) 



^Ht - A - 2M^T^) =^' '• V -T^r- = ''' 



and consider that in those expressions t and x remain con- 



