[154] 



and therefore 



Again 

 whence 



PROFESSOR STOKES, ON THE 



i(' + j.-i-i).£(;_i)(i + i-'-). 



2 \c c s c c 3 l n \e cj \c c a) 



(9) 



oc, c 2 + 1 



«, = - — , a 2 - - a, — — - , 

 /n c Ci + t 



c 2 



a a 3 sf c, (c 2 + tf)l _ a * M J \ 



e c 3 c\ (Cj + 1) cj c \c, c 2 / ' *' 



Neglecting t altogether in the formulae (7), we get 



1 1 2/1 1\ 

 c x C 2 fx\C pi 



and therefore 



a a 3 



- + — 

 c c, 



n 

 c 



2tia fl i\ a n l\ 1 



/uVW p/ c\c pj\' ' 



Let AJS+AyR = A.R. The formulas (6), (9), (10), and the corresponding formulas relating 

 to Ay.R, give 



' 1 - + - c --)(^ + y ! ) 



Ai2=- --- 



{- [n --)(o'» + 6'y) -i ( )(aa! + by)\. . . . (n) 



(X. [c \c pi c \c pi J 



The condition of distinctness, as has been already observed, is that AR shall vanish inde- 

 pendently of a? and y, in which case the elementary systems of rings corresponding to the several 

 elements of the dimmed surface will be superimposed on the screen. The coefficient of a/f + y* 

 will vanish when either of the following equations is satisfied : 



c = c, 



or 



1 2 



+ - = -. 

 c p 



(12) 



In order that the coefficients of on and y may vanish independently of particular values of 

 a and 6' we must have 



c =c =p, (13) 



which equations satisfy at the same time both of the equations (12), of which it would have 

 been enough that one should have been satisfied. Hence the rings are formed most distinctly 

 when the luminous point and the screen are both at a distance from the mirror equal to that of 

 coincident conjugate foci. This agrees with observation*. 



Whatever be the position of the luminous point, if a, b', c be the co-ordinates of its 

 image, we have 



a a b' b 



c c c c 



* See Newton's Optics, Book II., Part iv., Obs. 1, for the 

 case in which the curvatures of the two surfaces are alike, and 

 an experiment by the Duke de Chaulnes ( Mem. de VAcademie, 

 1755, p. HI) for the case in which they are unlike. In this 

 experiment a plano-convex lens was employed. Each face in 



succession, after having been tarnished, was turned towards 

 the incident light It appears from a passage at the end of 

 Newton's twelfth Observation that he had himself made ex- 

 periments of a similar nature, the results of which however are 

 not described. 



