COLOURS OF THICK PLATES. [153] 



Although this formula was obtained on the supposition that the points L, L ly L 2 , L 3 , 

 M, M lt M 2 , M 3i lay on the positive side of the plane of wy, it is true independently of that 

 restriction. For when one of the foci L, L lt L 2 , L 3 , from having been real becomes virtual, 

 or from having been virtual becomes real, the corresponding ordinate c, c x , c 2 + t, or c 3 changes 

 sign. At the same time, in the expression for the retardation distance passed over is converted 

 into distance saved, and vice versa. Hence in any such expression as (2) the sign of one or 

 more of the lines is changed. But in the expansion of the radical by which the length of such 

 line is expressed, the sign of c, c 1} c 2 + t, or c 3 must be changed at the same time, and there- 

 fore no change is required in the expanded expressions. 



To eliminate c x and e 2 from (3), we may observe that we have very nearly 



e /x c e, Ci + 1 



and similar expressions hold good for e', &c. Hence 



R ^ e'H c\ f | c\ c\ + t \ (?t <?, / t c, c 2 +t\ 



We might, if required, express c x and c 2 in terms of c by the formulae of common optics, 

 without making any supposition as to the magnitude of t. In practice, however, t is usually 

 small compared with c, e u &c., so that we may simplify the above expression by retaining only 

 the first power of t. We thus get 



_ t le % <?\ t id* + b' 2 a? + ft 8 \ 



Rm *\?-?r;r7i c^) < 5 > 



4. Before proceeding to apply this expression, let us investigate the conditions of distinct- 

 ness. Denoting by A X E, A V R the additions to R on account of the terms involving a?, y, we 

 get from (l) 



\c c z l \c c 3 J 



-GC-9*-<HW' «> 



and A y R may be obtained by interchanging a and b, so and y. 

 We have by the formulae of common optics 



V H-l 1 1 2 1 ua-11 



- + ~ . , - = + - ; ... (7) 



Ci r c c 2 + t s Cx + t c 2 r c s v ' 



whence, supposing * small, expanding as far as the first power of t, and putting for shortness 



---(fi-l)( )--, (8) 



s r s \r si p v ' 



so that p is the radius of a speculum having the same focal length as the mirror, we obtain 



„ + - ="- + — ■J-j + + - — i + -s> ; 



c c 3 p n \p" pr r 8 pc c ) 



Vol. IX. Part II. 44 



