of the Interference of Homogeneous Light. 89 



Differentiating the general equation sin <p = ft sin <p' 9 we have 

 cos p d <p = ft cos p'flfp' 



J 7 / 7 cos 



and dp' = as — ,. 



r r jot cos f 



Substituting this vajue of d <p', and the value oW—v - — —-- 



we have 



, x ft cos 2 $>' COS <fl , , , , „ ju. 2 — 1 



cos 3 <p ju. cos <p r v r i"' 



, x 7 cos<p' , , ,, N a 3 — 1 



or, d r = — rd<p r — (tan <p' + tan \1/) — ; 



cos <{> v ]"- 



r in this equation is still a variable quantity, but we may eli- 

 minate it by considering a perpendicular let fall from the 

 image of the luminous point upon the first surface of the prism 

 produced, as ae, fig. 3. Calling this perpendicular distance e, 

 we have e 



cos <p % 

 and substituting this value 



d>> m e — d<p £m£ (tan <?' + tan J/) ^=- ! , 



cos <p cos <j> x <■* 



7 v t cos 0' , # , N a 2 — 1 

 dr = — ed(p ~ (tan <p'+ tan ty) ~ . 



This equation is most probably not integrable in the ge- 

 neral form we now have; but by supposing the pencils very 

 small, as they really are in cases of interference, we may sub- 

 stitute for (tan <p'+ tan \J/) a term containing only tan <p' and 

 a constant; and we shall find, on recurring to numbers, that 

 we may make this approximation, as well as the former one, 

 without introducing any material error. 



Thus at the angle of minimum deviation, we have 



tan $' + tan V = 2 tan <p' ; 

 introducing this value, therefore, our equation becomes 



r COS" <$ ft 



'J* cos$' Ci sin <p' ft 9 — 1 

 = — ed§ 5 — 2. 



cos- $ cos £ 

 Tliird Series. Vol. 2. No. 8. Fcfi. 1833. N 



cos 2 <p ft r /tcos*^ ft 



