DIFFRACTION GRIMALDI'S FRINGES. 173 



originatiiif^ between h and c will bo in complete conflict with tlioso between a 

 and /). Tlius tlic power of ab to interfere with A« will be nearly neutralized, 

 and the point B will receive, once more, nearly all the illumination which Aa 

 is capable of sending to it. And in like matuier, if the screen be s^nccessively 

 raised to the points d, e, &c., similar alternations of diminished and increased 

 brightness may be inferred. After passing the fifth or seventh division, how- 

 ever, these successive maxima and minima cease to be perceptible in white light ; 

 a consequence partly due to the unequal lengths of the undukitions of the diifer- 

 ent colors, and partly to the diminishing length and increasing obliquity of the 

 successive divisions of the wave front. 



Dr. Lloyd has illustrated this ease in the following felicitous manner. Let 

 the light received at B from the half wave PA be r(;presented by 1, and that 

 from the total unobstructed Avave by 2. Represent the effect of Aa by -j-m, 

 that of ab by — ;^^', that of be by +7?i", and so on. Then we have, 



2=1 4- w — m'-\-7)i"—7n"', &c. 



Now as each of the successive literal terms is greater than that Avhich follows 

 it, if we cut the series at any point the value of all the terms which succeed on 

 the right Avill have tin; same sign as the first of them; and the sum of the remain- 

 ing terms on the left will be less than 2 if the value cut off is positive, and 

 greater than 2 if the value cut off is negative. 



Should these popular illustrations of a somewhat difficult subject appear un- 

 satisfactory, it may be observed that analysis leads to the same results, although 

 the processes arc complicated. Without going into details, we may remark 

 that the intercept aa' is evidently a function of the angle ARa. Put h for the 

 intercept, w for the angle, and V for the resultant molecular velocity at B. 

 Then, if all the derivative waves begin simultaneously in the arc AQ, the com- 

 ponent molecular velocity at B, due to any elementary wave will be expressed by 



?;sin2- ' .dco ; v being the maximum molecular velocity of the derivative wave, 

 and ?. the length of an undulation. Hence — 



dY . ^ 7i . f(oj) 

 -,— =rsni2---=rsni2-— V^, 

 den / / 



since k is a function of co which may be represented hyj[(jj) . 

 Ify(<u) be the differential coefficient of y\w), we shall have — 



f'{aj)dY=VSm2K^^^p-./'{a>).daj. 



And ff'{w)d\=Y.T{o>)=-l-cos2-^p- + C. 



OrV.FH=:|^(l-cos2.4-)- 



AsJ'iw), which is the intercept, is always increasing with w, and its differ- 

 ential coefficient also, this expression makes it evident that the value of V 

 must pass through a series of maxima and minima, since the expression 



1 — cos2- -^' undulates between the values and 2. These maxima and minima 



become, moreovei*, less marked as w increases, since F(w), whatever it may be, 

 must increase also. 



This illustration, however, excludes an important consideration, which is, 

 that, owing to the constantly increasing obliquity of the arc m to the direction 



of the intercept, the value of -— - as above given should be multiplied by the 



du) 



cosine of the sum of the angles ARa and ABa; which sum is a function of w 



