COLOURS OF THICK PLATES. [159] 



reflected at T, and scattered on emergence at P. The lines LS, ST, TP will evidently lie 

 in the plane LL P. Let d), <p' be the angles of incidence and refraction at S. We have 



R 1 = LS + ZpST + PE, 



= csec(p + 2fit sec <p' + -y^A 2 + w 2 ), (17) 



and 



c tan <p + Zt tan (f> = «, sin cp = /jt sin 0' (18) 



The obliquity being supposed small, we may expand, and retain only the squares of small 

 quantities, the terms thus neglected involving only fourth and higher powers. We get in the 

 first place from (17) 



R 1 = c+2nt + h + %(c<p* + 2 f it<t>'* + ^-\ (19) 



But equations (18) give 



A. A.' M * 



</>= fid) = 



and substituting in (19) we get 



Ri = c + 2nt + h + 



To obtain i? 2 , it will be sufficient to interchange c and A, s and u, since if we supposed the 

 course of the ray reversed it would emanate from E, be regularly refracted and reflected, then 

 scattered on emergence at P, and so would reach L. Interchanging, subtracting, and re- 

 ducing, we obtain 



f « 2 « 2 1 



R = tl \ (20) 



\h(nh + 2t) c(fiC+2t)\ V ' 



This formula is more general than (16), since no approximation has yet been made depending 

 on the magnitude of t. In practice, however, t is actually small compared with c and h, so 

 that we may simplify the formula by retaining only the first power of t, which reduces (20) to 

 (16), inasmuch as w 2 - x 2 + y 2 , and s z = (w - a) 8 + (y - b) 2 . 



11. Let us now proceed to apply the formula (16) to the explanation of the phenomena. 

 In discussing this formula, it is to be remembered that <v, y are the co-ordinates of the point 

 of the mirror on which a fringe is seen projected. Since the direction of the axis of y is 

 disposable, we may make the plane of y, x pass through the luminous point, in which 

 case 6=0, and 



t((l 1\ . . 2ax a 2 ] 



For a given fringe R is constant. Hence the fringes form a system of concentric circles, 

 the centre of the system lying in the axis of do. If a be the abscissa of the centre 



ah 2 , / ah ah 



ah' . fan ah \ , . 



Now ah(h-c) ' and ah (h + c) -1 are the abscissae of the points in which the plane 

 of the mirror is cut by two lines drawn through the eye, one to the luminous point, and the 



