[152] PROFESSOR STOKES, ON THE 



PM - V { (a' - xf + (6' - yy + (c' - *)*} i ^3 - \/ H«3 - *Y + (p 3 - yf + (c 3 -*)>}, 

 and % = — (a?* + j/ 2 ), nearly. Expanding, we get 



Let V = c - c 3 + Vj + Vyt where Vj is the sum of the terms containing a and «, and F y ' 

 the sum of the terms containing 6 and y. Then 



2c' 2c 3 



There is no occasion to write down VJ, since it may be deduced from Vj by writing 6, y 

 for o, a?. Taking V x to denote for F what VJ denotes for V', we get by interchanging a and 

 a', a 3 and Og', and subtracting, 



•~ 2c' " 2c 3 2c + 2c' 3 ' * " * () 



In order that the rings may be formed on a screen with perfect distinctness, it is necessary 

 that the difference of phase of the several pairs of streams which come from the several points 

 of the dimmed surface should be the same ; in other words, that the retardation R should be 

 independent of oc and y. Deferring for the present the investigation of the conditions of 

 distinctness, we may observe that when these conditions are satisfied the expression for R must 

 be the same as if x and y were each equal to zero. We have therefore 



R = \L, 3T\ - \M, Z], 

 where 



[L, M] = (L to L 3 ) + c'-c 3 + ^-,-^-, 



*&C £ C3 



and [M, L] is formed from [Z,, M] by interchanging the co-ordinates of L and M. In the 

 above expression e 2 has been written for shortness, in place of a 2 + b 2 . Now supposing 

 c, Ci, c 2 , c 3 , to be all positive, and denoting by A, B the points in which the first and second 

 surfaces respectively are cut by the axis of the mirror, we have 



(L to L 3 ) m AL - ixAL^ + fxBL^ + nBL„ - \kAL % + AI^ ; . . (2) 



which gives, on expanding, 



& H e \ /KCi 2 t*-V* 



(Z, to 1^) = c + fiCi - - — + n (ci + t) + — + fi (c 2 + t) + ■ 



2c r ' 2c t v ' 2(c L +t) v ' 2(c 2 + J) 



ne.? e 3 2 



- |UC 2 - — + c 3 + — 



2c 2 2c 3 



C + C 3 + 2nt + — + 



fite^ fi.te : 



1 



2c 2c 3 2C!(ci + 2c 2 (c 2 + £) 

 We have therefore 



m eg e? e? \ 



' 2 Vi (c', + c' 2 (c' 2 + t) c, (c, + t) c, (c 2 + 01 ' 



