[158] PROFESSOR STOKES, ON THE 



only of the dimmed surface as lies within a cone having M for vertex and the pupil of the eye 

 for base ; and the bands will be seen distinctly if R do not alter by more than a small fraction 

 of X when x, y alter from one point to another of the portion of the dimmed surface which 

 lies within this cone. Now we have seen already that the bands are in all cases seen dis- 

 tinctly in the neighbourhood of the image when the image itself is seen distinctly, so that 

 when the image is real the bands may even be thrown on a screen, in which case a com- 

 paratively large portion of the dimmed surface is concerned in their formation. We may 

 conclude that in the present case the bands will be seen with sufficient distinctness throughout, 

 provided the image of the luminous point be seen distinctly. 



9. In considering the distinctness or indistinctness of the bands, we are concerned with 

 the finite size of the pupil of the eye ; but in investigating only their form and magnitude 

 we may suppose the pupil a point, and reduce each pencil entering it to a single rav, 

 which forms the axis of the pencil. Let E be the eye, or rather the centre of the pupil, 

 h its perpendicular distance from the mirror, and suppose the axis of % to pass through E. 

 The ray by which a portion of a band is seen as if at M cuts the mirror in a point whose 

 co-ordinates a?, y are equal to a, b' altered in the ratio of k to h — c', so that 



Substituting in (15), we get 



R = ^[^ + y i )-}A(*- a y + (y-i>Y\]> • • • 06) 



from which it may be observed that c has disappeared, as evidently ought to be the case. 

 The expression (16) might have been at once deduced from (15) by putting the co-ordinates of 

 the eye in place of a, b', c ' . The reason of this is evident, because the retardation is con- 

 stant for the same ray, and a ray may be defined by the positions of any two points through 

 which it passes. We may therefore employ the points E and P, instead of M and P, to 

 define the ray, and may therefore at once substitute the co-ordinates of E for those of M in 

 the expression for the retardation. 



10. To determine the forms, &c. of the bands, nothing more will be requisite than to 

 discuss the formula (l6). As however this formula was obtained as a particular case from 

 a very general, and consequently rather complicated investigation, in which the curvatures of 

 the surfaces were supposed to have any values, and as the bands to which it relates are of 

 great interest, the reader may be pleased to see a special investigation of the formula for the 

 case of a plane mirror. 



Retaining the same notation as before, except where the contrary is specified, let L , E 

 be the feet of the perpendiculars let fall from L, E on the plane of the dimmed surface, and 

 let L P = 8, E P= u. Let R x be the retardation of a ray regularly refracted and reflected, 

 scattered at emergence at P, and so reaching E ; R 2 the retardation of a ray reaching E after 

 having been scattered at P on entering into the glass, and let R x - R 2 = R. Let LSTPE 

 be the course of the first ray, which emanates from L, is regularly refracted at S, regularly 



