342 



SCIENCE 



[N. S. Vol. XXXIX. No. 1001 



parenoy by any gradual progression. The index 

 of refraction of air is so near unity tliat there will 

 be no sensible regular reflection in passing from 

 one layer to the next, and the transmission of each 

 layer may be expressed exponentially by Bouguer's 

 formula, but with different coefficients of trans- 

 mission for the several layers. 



Thus, suppose E„ to be the original intensitj' of 

 a beam of light incident upon the outermost layer 

 at the angle whose secant is m. 



Then after passing successive layers the remain- 

 ing intensities become 



El = S„<2i'"i, E. = Eoai™! • a..'":, 



En = E^d^m^a^m., . . . a„m„. (1) 



The value of the secant of the angle of incidence 

 will change slightly in passing from layer to 

 layer from two causes: first, the curvature of the 

 earth; second, the refraction of the beam in air. 

 These causes produce opposite effects, the first 

 tending to increase the angle of incidence, the sec- 

 ond tending to diminish it as the beam approaches 

 the earth's surface. Their combined effect is de- 

 pendent on the height to which the temperature 

 exercises absorption and on the distribution of 

 density with the height. But it is generally sup- 

 posed that the absorption of the air above 40 

 miles from the earth 's surface is negligible, and, 

 remembering that the atmospheric density dimin- 

 ishes with the height, it appears that for zenith 

 distances less than 70° the effect of change of the 

 secant of the angle of the incident beam from the 

 outermost to the innermost layer of the atmosphere 

 will not introduce error greater than 1 per cent. 

 Accordingly for zenith distances less than 70° we 

 may write approximately 



£„=E„(o^a„ . . . a,,)™. (2) 



The symbols a-^, a. . . . a„ denote constants 

 (providing no change of transparency occurs dur- 

 ing the interval of time in question), and their 

 values are slightly less than unity. We may substi- 

 tute for their product a single constant, a, itself a 

 proper fraction, and remembering that En is the 

 intensity at 'the earth's surface, above denoted 

 simply by E, we have 



E — EoQ^n. (3) 



LIMITATIONS OF FORMULA 



No mention is made in this expression of the 

 barometric pressure, but it is easy to see that an 

 alteration of barometric pressure would signify, 

 under the conventions adopted in deriving the 



formula, a change in the number of layers, n. 

 This would cause an alteration of the quantity a, 

 which is the continued product of the transmis- 

 sion coefficients of the layers, by introducing addi- 

 tional multipliers Cti+i, 07,.+ ... or by the 

 withdrawal of some Oa— 1, an— 1. . . . Since we 

 have no means of determining the value of the 

 terms so introduced or taken away, there is no 

 means of correcting for change of barometer in 

 the use of the expression (3) and it would, for in- 

 stance, be impossible to compute, from knowledge 

 of the values of E, E„, a and m for one station, 

 what would be the value of E at some station of 

 different barometric pressure. 



From this we see that the unit of air 

 mass to be taken for each station is the air 

 mass traversed by beams from zenith celes- 

 tial objects between the station itself and 

 the outer limit of the atmosphere, and that 

 the barometric pressure can not be em- 

 ployed in the computation to reduce ob- 

 servations at different stations to a common 

 unit of air mass. 



The determination of the solar constant 

 of radiation, based upon the demonstration 

 which has just been given, depends upon 

 the following assumptions : 



1. In a homogeneous medium, a homo- 

 geneous ray loses a fixed proportion of its 

 intensity in every equal length of its path. 



2. The earth's atmosphere may be con- 

 sidered as made up of a great number of 

 layers concentric with the earth, each ap- 

 proximately homogeneous in itself over the 

 area swept through by the solar beam be- 

 tween zenith distances of 70° and 30° 

 during the time required for this sweep 

 of the beam. 



3. Surface reflection of the outer bound- 

 ary of the atmosphere, or the boundaries of 

 its internal layers, is negligible. 



4. Except in the known red and infra- 

 red atmospheric bands, the transparency 

 varies gradually from wave-length to wave- 

 length, or if atmospheric absorption lines 

 exist, the energy they absorb is inconsider- 

 able. 



