NATURE 



[January 5, 1922 



Letters to the Editor. 



\The Editor does not hold himself responsible for opinions 

 expressed by liis correspondents. Neither can he undertake 

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Atmospheric Refraction. 



The correspondence on terrestrial refraction 

 from Mr. Mallock and Dr. de Graaff Hunter in 

 Nature of June 9, p. 456, and August 11, p. 745, 

 raises a paradox which I think must have puzzled 

 many readers of Nature besides myself. Mr. Mallock 

 is, of course, quite correct in stating that the diminu- 

 tion of density of the air observable under ordinary 

 conditions is practically linear for moderate increases 

 of height above the earth's surface, and that, con- 

 sequently, the refractive index of the air may for 

 -moderate increases of altitude be taken as diminishing 

 linearly at such a rate that it would reach vacuum 

 value at the height H ot the homogeneous atmosphere. 

 Dr. Hunter is equally correct in pointing out that 

 Mr. Mallock's reasoning, based on the above-men- 

 tioned observational fact, leads to a value of k, the 

 coefficient of terrestrial refraction, which is almost 

 exactly twice as great as that found by observation 

 under ordinary conditions. Dr. Hunter does not, 

 however, point out what I think is the reaf fallacy in 

 Mr. Mallock's argument. 



The difficulty is not to be got over by any considera- 

 tion of temperature-gradient in the air, although it is 

 well known that variations in the temperature-gradient 

 constitute the chief cause of variations in terrestrial 

 refraction. The only way in which temperature- 

 gradient could affect Mr. Mallock's result would be 

 by its requiring a change in the value (4-32 sea-miles) 

 which he adopts for the height H of the homogeneous 

 atmosphere. Whether we calculate H for air at 

 a uniform temperature, as is usually done, or on the 

 assumption of a diminution of temperature with in- 

 crease of height at the rate ordinarily observable (say 

 1° C. for each 200 metres), we obtain a value of H 

 which is nearly the same as that used by Mr. Mallock 

 in his argument. 



May I suggest that the solution of the riddle is to 

 be found in Mr. Mallock's supposition of a ''plane 

 vertical wave-suriace starting from P," whereas the 

 rays of light from a terrestrial point must give rise to an 

 approximately spherical wave-surface ? In the diagram 

 (Fig- i)> which represents a vertical section through 

 the homogeneous atmosphere with the curvature of the 

 earth neglected for simplicity, a plane wave-surface 

 HPO would change its position to BAG in the time t, 

 where HB and OC are proportional to the velocities 

 of light at H and O. But in that time rays from a 

 point P would reach points D and E, such that 

 PD-PA = KPB-PA) and PA-PE = i(PA-PC), be- 

 cause the average velocity along PD would be the 

 mean of the velocities at P and H, and, similarly, the 

 average velocity along PC would be the mean of the 

 velocities at P and O. It is easy to see that this gives 

 a radius of refractional curvature exactly twice as 

 great as that found by Mr. Mallock, and consequently 

 leads to a value for the coefficient of terrestrial refrac- 

 tion which is in agreement with observation and with 

 the tables ordinarily employed by navigators for the 

 dip and distance of the sea horizon. 



It may be woith while to mention here a very 

 likely source of confusion in comparing the values 

 of the coefficient of terrestrial refraction k found by 

 different observers under different conditions, and 

 especially by observers in different countries. There 

 are two definitions of k in use by surveyors, one of 

 NO. 2723, VOL. 109] 



which makes its numerical value double that given 

 by the other. An assistant of mine who read Dr. 

 Hunter's letter was greatly surpris»^d at his statement 

 that fe = o-i33 "is not a value ordinarily met with in 

 practice," because in Egypt we ordinarily use fe = oi3, 

 and our trigonometric levels derived from observations 

 made in the afternoon hours when refraction is at its 

 minimum and steadiest value are found to agree sur- 

 prisingly well over great distances with those found 

 by spirit-levelling. The explanation of the apparent 

 discrepancy between Dr. Hunter's statement and our 

 experience is that we follow the Continental practice 

 in defining k as the ratio of the curvature of the 

 refracted ray to the curvature of the earth, while Dr. 

 Hunter and most English writers define it as half 

 this quantity. 



It does not seem to be very generally known that 

 a rational formula for calculating the coefficient of 

 terrestrial refraction at any point where the baro- 

 metric pressure, air-temperature, and temperature- 

 gradient are known was advanced by Jordan so long 

 ago as 1876. This formula, which is given, together 

 with an account of the theory on which it rests, in 

 Jordan's " Handbuch derVermessungskunde," Band 2, 

 is 



/e = 0-2325 _^- 



760 (I. c.)^^' -^-9-35-)' 



where k is the coefficient of terrestrial refraction, 

 defined as being the ratio of the curvature of the ray 

 to that of the earth, B 

 the barometric pressure 

 in mm., a the coefficient 

 of expansion of air at 

 constant pressure, t the 

 air-temperature in de- 

 grees C, and n the tem- 

 perature - gradient in 

 degrees C. per metre of 

 height. 



Jordan's theory is prob- 

 ably not quite complete, 

 in that it omits any con- 

 sideration of variations 

 in the humidity of the 

 air; but it does take ac- 

 count of variations of 

 pressure, temperature, 

 and temperature- 

 gradient, and these are 

 probably the principal 

 factors affecting the 

 value of k. The ^''^- ■• 



resulting formula is 



very simple and easy of application, and, so far as I 

 have been able to test it in the Egyptian deserts, I 

 have found it to give results which are in good agree- 

 ment with those of observation. It appears also to 

 accord very satisfactorily with Indian experience; 

 for when applied in the two examples given by Dr. 

 Hunter in his letter, one at sea-level and the other 

 at an altitude of 19,000 ft., it yields (allowing for the 

 difference in the definition of k) results identical with 

 those which were found by Dr. Hunter to agree well 

 with numerous observations. John Ball. 



Survey of Egypt, Cairo, December 14. 



In Nature of June 9, p. 456, a letter appeared from 

 Mr. A. Mallock giving a proof that the path of a 

 nearly horizontal ray through the earth's atmosphere 

 is a circle of about 14,900 miles' radius, and later 

 (August II, p. 745) Dr. de Graaff Hunter, of the 

 Indian Survey, wrote controverting Mr. Mallock's 

 statement, and asserting in effect that the radius of 



