COLOURS OF THICK PLATES. [163] 



lest any one in repeating the experiment should observe them, and mistake them for some- 

 thing relating to the colours of thick plates. 



17. The formula (21) determines the breadths of the several fringes, which are unequal, 

 except in the case in which the eye and the luminous point are at the same distance from 

 the mirror. It will be convenient, however, to investigate a simple formula to express what 

 may be regarded as a sort of mean breadth. Let the mean breadth be defined to be that 

 which would be the breadth of one fringe if the rate of variation of the order of a fringe, 

 for variation of position in a direction perpendicular to the length of a fringe, were constant, 

 and equal to the rate in the neighbourhood of the projection of the image, and let /3 be this 

 mean breadth. 



Putting y = in (21), differentiating on the supposition that R and w vary together, and 

 after differentiation putting ah (h + c) -1 for m, we find 



,_ Zta 



dR = — - dx; 

 fj.cn 



and since, according to the definition of /3, X _1 diZ = fi~ l dx, we have 



£t& •: « 



When c = h the bands are straight, and of uniform breadth, that breadth being equal to /3 ; 

 and when the bands are not very much curved /3 may still be taken as a convenient measure 

 of the scale of the system ; but the formula (25) is not meant to be applied to cases in which 

 the projection of the image of the luminous point falls at all near the centre of the circles. 



Section III. 



Rings formed by a curved mirror, and viewed directly by the eye, when the luminous 

 point and its image are not in the same plane perpendicular to the axis. 



18. The rings and bands of which the theory has been considered in the two preceding 

 sections may be regarded as forming the two extreme cases of the general system. In the 

 first case, the rings appear to have a definite position in space ; in the second case, every 

 thing depends upon the position of the eye. These are the cases of most interest, but there 

 are some properties of the general system which deserve notice. 



In order that rings may be thrown on a screen, it is necessary that the retardation of 

 one of the interfering streams relatively to the other should be sensibly constant over the 

 whole of the dimmed surface, or at least over a large portion of it. But when the rings are 

 viewed directly by the eye, we are concerned with so small a portion of the dimmed surface, 

 in viewing a given point of a ring, that the rings may be seen very well in cases in which 

 they could not be thrown on a screen. Moreover, we have seen that even independently of 



45—2 



