30 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OP DIFFRACTION. 



It is to be observed that da represents the elementary solid angle subtended at by an 

 element of the riband formed by that portion of the surface of a sphere described round 0, 

 with radius r, which lies between the plane yz and the parallel plane whose abscissa is br. 

 To find the aggregate disturbance at corresponding to a small portion, S, of the plane P 

 lying about O x , we must describe spheres with radii ... r - 26r, r - br, r,r + 6t, r + 26t ..., 

 describing as many as cut S. These spheres cut S into ribands, which are ultimately equal 

 to the corresponding ribands which lie on the spheres. For, conceive a plane drawn through 

 OO x perpendicular to the plane yss. The intersections of this plane by two consecutive 

 spheres and the two parallel planes form a quadrilateral, which is ultimately a rhombus ; so 

 that the breadths of corresponding ribands on a sphere and on the plane are equal, and their 

 lengths are also equal, and therefore their areas are equal. Hence we must replace da by 

 r~ 2 dS, and we get accordingly 



IndS^,^ . mndS „ ,,_ . „ (l-w 2 )dS. , t [ 



£.- T _/'(6*-r), *--— — f(bt-r), £= ' fQ>t-r). . .(42) 



btrT 4u-r *7rr 



Since 1% + mr) + n £= 0, the displacement takes place in a plane through perpendicular 

 to L 0. Again, since £ : i\ :: I : m, it takes place in a plane through Y and the axis of ss. 

 Hence it takes place along a line drawn in the plane last mentioned perpendicular to 00,. 

 The direction of displacement being known, it remains only to determine the magnitude. 

 Let <Ci be the displacement, and <p the angle between 0i0 and the axis of %, so that n = cos0. 

 Then £i s i n <p w ^l ^ e tne displacement in the direction of ar, and equating this to Y in (42) 



we get 



dS 

 &-—**<pf{bt-r) (43) 



The part of the disturbance due to the successive displacements of the films may be got 

 in the same way from (30) and the two other equations of the same system. The only terms 

 which it will be necessary to retain in these equations are those which involve the differential 

 coefficients of £ , j; , £ , and p in the second of the double integrals. We must put as before 

 r for bt, and write r~*dS for da. Moreover we have for the incident vibrations 



£ = 0, ,-0, X,=f(bt'-x), p^-nflbt'-x). 



To find the values of the differential coefficients which have to be used in (30) and the two 



other equations of that system, we must differentiate on the supposition that £, rj, £, p are 



functions of r in consequence of being functions of ,v, and after differentiation we must put 



, r d d 



x = 0, t = t — -. Since — = — I . , we get 

 o dr ax 



®.-^-* (&), 



-lnf(bt-r), 



whence we get, remembering that the signs of I, m, n in (30) have to be changed, 



Z=--7-rf( bi ~ r )> 1' TIT f^ bt ~ r ^ ?■ . r -f'(bt-r) 



iirt 4nrr 47rr 



