85 



Jipproximate integration of the Fluxional Equation. 



3. Let Q represent the refraction that would take place if 

 the surface of the earth were a plane, and the different strata 

 of air parallel thereto, in which case the ratio of a to r 

 would be the ratio of equality. Therefore equation (2) 



— pi sin. e ^ I ^b (p) 



feecomes Q = ^Trb7T7T^f^^^^^T(^)^^7T=^ ^ (3) 



Hence R ^ J^ V^/-izr_(JL±^'^) ) ^"lll- 



r ■v/ I -f i p _ /i ^- i (p) \ ^ sin. « 9^ 



Let-^=1~. W 



r 



:w ^, • _ Qji— «) 



Then JR — -j====== -— ——— . -— -^ rr— (5) 



'1 ^bf~n-\-b{f)\ sin. ^9 



or jR = Q (1 — s) (1 — s tan. ■= ^) = Q — -^^ neglecting the 



second and higher powers of s, also §, (§) and their powers* 

 It is obvious that for the part of the atmosphere which 

 makes the refraction sensible, 5 must be very small. 



By equat. (3) 



Q = — -J /> p. tan. neglecting §>, (§) and their powers. 



• Hence ii = Q + f^rh~ nearly. (6) 



Now/^s= gs— /gs = is—f~~ <by equat. 4.) 



