Mr. Moon om Fresnel's Theory of Diffraction. 91 

 stant, while s only varies, the latter will take the form 



which shows that the total wave on arriving at P may be re- 

 garded as the resultant of two waves, ^ cos 5 and .r" sin 5, dif- 

 fering in their origin by a quarter undulation, and whose am- 

 plitudes j;' and x" are given by the expression 



the integrals being taken between the limits of n corresponding 

 to s = — AM and s= + oo." 



It would be difficult to conceive of anything purporting to 

 be a mathematical investigation more full of contradictions 

 and absurdities than the foregoing. Recurring to art. 631 of 

 the treatise on Light, we find that when a wave is transmitted 

 through a limited aperture, the disturbance of any vibrating 

 molecule at distance a -i-yfrom the centre of vibration may be 

 represented by 



^^d^sct<pQsin27r(^^-^^, . . . (A.) 



where d^ s is a small element of the front from which the se- 

 condary waves are supposed to diverge, a the radius of that 

 front, "<p{Q) being the function of the angle made by a late- 

 rally divergent vibration with the direct one, which expresses 

 its relative intensity, and which is unity when 9 = 0, and dimi- 

 nishes with great rapidity as 9 increases," the integration being 

 to be effected to the limits of the aperture. 



The accomplished author observes that in the above inves- 

 tigation "we shall have a <$> (9) =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." What is the efficacious part of the surface A M N ? 

 Plainly, if the expression (A.) be correct, it must be that part 



of the surface for which the value of c?"^5«4)(5)sin27r( ™ ^ j 



is significant. Now that expression can only become insigni- 

 ficant by <p (6) becoming very small; but u^{Q) passes through 

 all possible values between 1 and 0, and it is surely impossible 

 to leave out of consideration the whole series of intermediate 

 values. If we take into account the case where «<|)(9) = 1, 



H2 



