The Rev. Dr. Robinson on the Constant of Refraction. 185 



P = l-^ 



as 



21 



}/P'^P=-a"^(n + l)in+2r 



The term ^sdp, is the same on either hypothesis, being a result of the atmo- 

 sphere's equilibrium ; the coefficients of the higher terms differ, those on the hy- 

 pothesis u T Increasing much more rapidly, ifrdp is that which Brinkley added to 



5 P 

 Laplace's expression, using the arithmetical mean, which gives - X —5. I have pre- 



o a 



ferred the geometric mean of the separate terms, as giving less weight to u t, 



which is especially necessary near the limit of convergence.* If we develope q, 



pass from sines to arcs, and put u for . , ,. — , we shall have, 



^ ^ sin 1 



r" = ^ X tang e 



V? sin 1" u'sin^ \" 



+ ^^-f^ X tang' e + t^^ X tang^ 6 (q'. q") 



— -X^, X - . ^ e [1 .00000 + 6 X tangle (1.06698)1 (a. a ') 

 sm 2 a cos'' "- ' o \ /^ \ j 



+ shr2^ ><^^-^^t2.44949 +6 X tangle (5.04119)] (/3. /3') 



- si;^ ^ S • S" ^ ^8.65117 + h X tang^ e (26.92202)] (7. y') 

 + ^ ^ ^* ^ • ^ [38.43867 + h X tang^ 6 (160.08103)] (8 . I') 



A xi^,.^. [199.22000 &c.]. 



sm 2 a* cos'' 



* The original intention was to have assumed the terms zz ^01 X a'l' ; a and a' being arbitrary 

 factors determined by observation ; but as the simple -v/i X i' was found to satisfy my observations, 

 VOL. XIX. 2 B 



