The Rev. Dr. Robinson on the Constant of Refraction. 183 



when p is the density at the distance r from the centre, p and a, the same quan- 

 tities at the earth's surface ;* hp the refractive force of air at the density p, and 

 6 the apparent zenith distance. 

 If v?e assume, 



A = ■/!+ V sin e 



V\-\.bp — {l-\-bp')im'e 

 Q = refraction if the earth vpere plane, 



r — a 



s = r, 



r 



Brinkley has shown, page 85, that, 



, — ^ bAdp 

 '' = ^+b^ • 



/ 1 + (2* - s') X A^ 

 and by developing a we find, 



omitting higher powers of b. Developing • (^r we have, 



* These quantifies more strictly relate to the osculating circle, and the constant of a table must 

 be modified accordingly. The quantity — is one of these ; if we assume the mean radius of curva- 

 ture as the standard, and the earth's compression ^^^, then for another latitude, 



I I 



-7 = - X 1 + 0.0004991 X cos 2l. 



Laplace has remarked that this should make the refraction to the north and south unequal. In fact, 

 if we suppose the last rays of twilight to be once reflected, and that refraction ceases with reflection, 

 (in which case I find, taking into account the curvature of the ray, which Delambre has neglected, 

 that the height of the reflecting point is 41.536 miles,) andthe rayis acted on in the case of horizontal 

 refraction, through 8" 43' of latitude. The change of the radius of curvature, and the place of its 

 centre, must make a sensible difference in the two refractions, but the effect of the difference of tem- 

 perature in the two trajectories is perhaps still greater. 



The value of I is also inversely as local gravity, and that of b (or of the density corresponding to 

 a given barometric column) directly as it ; they must therefore be divided and multipUed respectively 

 by 1 — 0.002695 X cos 2l. 



These corrections may seem minute, but are very sensible in low refractions. 



