[168] PROFESSOR STOKES, ON THE 



24. The notation being the same as in Art. 10, we have merely to employ the equations 

 (17) and (18), without making any approximation depending on the smallness of (f> and 0\ 

 The thickness t may still be supposed small compared with c and h. Neglecting t for a first 

 approximation, and then substituting in the small terms the values of tan <p' and sec d> got 

 from the first approximation, we find 



sr + c- 

 Interchanging c and h, s and u, and subtracting, we get finally 



*-'(^--^-v / '-^) 



(33) 



25. For the achromatic line R = 0, and therefore s : c :: u : h. Hence s is to u in 

 the constant ratio of c to h, and therefore, by a well-known geometrical theorem, the achromatic 

 curve is a circle, having its centre in the line L E g produced, and cutting this line in the two 

 points in which it is divided internally and externally in the given ratio. The latter of these 

 points may be formed by producing LE to meet L E produced, and the former by producing 

 LL till the produced part is equal to the line itself, and then joining the extremity of the 

 produced part with E. Hence the construction given in Art. 11 for determining the bright 

 band of the order zero continues to hold good whatever be the angle of incidence. 



26. In the neighbourhood of the image the bands are sensibly straight, being arcs of 



circles of very large radius. To find the mean breadth of a band, it will be sufficient to 



suppose the point P to lie in the line L g E , to differentiate equation (33) making R, s, and u 



dJi X 



vary together, while s + u remains constant, replace — — by -s , and after differentiation take 



u and s to refer to the small pencil, regarded as a ray, by which the image is seen. If i be 

 the angle of incidence, we may put after differentiation s = c tan i, u = h tan i. We thus 



find 



_ Xch v u 2 — sin 2 i , „ 



j3 = _ t (34) 



2 1 (c + h) sin i cos 3 i 



On account of the largeness of the angle of incidence, the breadth of the bands is sensibly 

 uniform, and therefore /3 may be regarded as the breadth of any one band. It is to be 

 remembered that /3 denotes the linear breadth of a band as seen projected on the mirror. If 

 we denote the angular breadth by •&, we have on account of the smallness of ar 



/3cosi Xc %/u* -dn*i , , 



ar = t ; = Ti TT ~ Jt ~- : ( 35 ) 



h sec i t (c + h) sin 2 i 



