Lord Rayleigh's Investigations in Optics. 45 



so that 



l = u L+ll = V (22) 



V U V u v 



Accordingly (18) becomes 



From (17) and (23), by multiplication, 



/*V 4 = c± 

 u'{u' + /) uv 

 or 



cV(?/ + o 



Again, from (17), 



"~ C 1 \ + 2/*V 2 M J 



^{^ ^D - • • • (25) 



From (24) and (25), 



,2_1N / „2y 





^C /2 /LtV 2 V 2fjbC 



-z-u is negligible 

 longitudinal aberration is simply 



In most cases l-r-u is negligible in the small term, and the 



8 3yBin»y-l) 



The linear width of the image is Sv . 6, where 9 denotes the 



angular semi-aperture ; and the angular width of the image is 



hv n . 11 cos 6 A 



— c/, or, since =u, 



u ; 1 u 



'fr'-l)™**. ..... (28) 



fJb 2 CC /2 v J 



To apply this result to the spectroscope, it is convenient to 

 compare (28) with the angular deviation due to an alteration 

 of refractive index from /ju to \x + 8/jl. This is equal to 



2S/ub sin <//_ 2S/JL sin <fr , Q . 



cos cj> /a cos </> ^~ ' 



If (28) and (29) be equal 



36 2 (ji-y ') 



