550 



NA TURE 



[April 29, 1922 



Gradient. Degrees 

 Centigrade per metre. 



o-ooo (isothermal) 

 o-oo6 (average) 

 o-oio (adiabatic) 

 0-03414 



^=0° 

 16,980 

 20,600 

 24,000 



infinite 



18,220 

 22,100 

 25,760 



i=o" 

 0-117 

 0-096 

 0-082 



/ = io° 

 0-109 

 0-090 

 0-077 



Mr. Mallock's value, 14,900 geographical miles = 

 1 7, 150 miles, agrees nearly with the isothermal value for 

 < = o°. (In my former letter I had not recognised that 

 Mr. Mallock's result was in nautical miles.) The result 

 is too small as a usual value, because he takes the 

 temperature gradient as zero and the surface tempera- 

 ture to be freezing point. Dr. Ball's explanation is 

 incorrect as pointed out by Commander Baker 

 (January 26) ; and further in that he states in his 

 second paragraph that the difhculty is not to be 

 got over by any consideration of temperature gradient. 



Commander Baker, in his letter (January 5) has 

 arrived at a similar result, for a horizontal ray, as I 

 have. The temperature gradient, however, of 1° C. 

 per 200 feet, which he says will give my results, is in 

 error ; it should be per 600 feet. 



In the second paragraph of this letter. Commander 

 Baker says that neither Mr. Mallock nor I give an 

 adequate presentation of the facts, in that the 

 assumption is made that the ray is circular. I do 

 not think that this deduction can be made rightly 

 from my former letter of August 11. I may say at 

 once that I entirely agree that the ray is not in 

 general circular, especially when the ray is close 

 to the earth or sea surface. However, in cases 

 met with in land surveys (excepting rays which 

 continue very close to the ground) one may compute 

 the refraction practically by the use of a coefficient of 

 refraction which represents the curvature at height 

 {2ha + h,,)l-^, as stated in my letter. The use of 

 different coefficients of refraction for different heights 

 essentially involves the idea of a ray of varying 

 curvature except in the case of a truly horizontal ray. 



Now work on the diurnal change of refraction on 

 inclined rays shows up the importance of the varying 

 conditions of temperature gradient in the layers near 

 the earth. I have not yet been able to reduce the 

 case of rays, which lie mostly or largely in these lower 

 layers, to a formula, though I think there is fair hope 

 of doing so in some cases. Extreme cases, in which 

 there is obvious and varying mirage, will not be 

 amenable to treatment : but I think a ray, 20 feet 

 above the surface, probably will. But I gather that 

 Commander Baker is chiefly interested in rays over 

 the sea, at a height of 30 feet or less. In the Survey 

 of India such cases naturally do not arise, and I 

 have not had any observations of this kind to 

 consider. However, in my Professional Paper No. 14 

 (Survey of India) I have given some deductions as 

 regards dip of the horizon (vide pp. 96-100), arriving 

 at the formula 

 Dip in seconds from point at height h = 



56-33(;i'- 15-13 An* 

 where At'^FAt, h' = h{i-o-2204'F)lo-'/7g6, F = 5i9-4//, 

 t + At and t being the absolute temperatures at levels 

 of observer and sea respectively. This formula is based 

 on cos(dip) = (i-l-/2/y)"Vo/M which involves only the 

 terminal values of m. 



I have tabulated the corresponding dip in Tab. 

 LIII. loc. cit. for various values of h' and At\ and I 

 should be very interested to hear from Commander 

 leaker or others to what extent my formula represents 

 the facts of observations. J. de Graaff Hunter. 



Survey of India, Dehra-Dun, U.P., India, March 2. 

 NO. 2739, VOL. 109] 



I AGREE with Dr. Hunter that my letter of January 

 5 contained a numerical error when I stated that a 

 temperature gradient of 1° C. per 200 feet would give 

 a ray curvature corresponding to the refraction co- 

 efficient given in his letter of August 11, 1921. 



Dr. Hunter also takes me to task for my comment 

 that both he and Mr. Mallock assume the refracted 

 ray to be circular. I think I have a certain amount 

 of justification for this, as in his letter of August 11 

 he speaks of the curvature of the ray " tacitly assumed 

 to be circular," although later it is true that he states 

 that the coefficient of refraction has different values 

 at different heights. 



It was rather in connection with the formulae upon 

 which the nautical tables for the dip of the sea horizon 

 are based that I take exception to any assumption 

 that adequate results can be obtained unless varia- 

 tions of curvature are considered. 



As stated in my letter of January 5, it is impossible 

 to draw a circle which touches the surface of the sea 

 and also becomes horizontal at a height of say 30 feet 

 above the sea, and unless consideration is given to a 

 form of ray path that can satisfy these conditions it 

 is impossible to get a zero value for the dip. 



Dr. Hunter quotes from his Professional Paper 

 No. 14 (Survey of India) a formula which he has set 

 out there from which the dip is to be evaluated, and 

 asks to what extent this formula represents the facts 

 of observations. I have, unfortunately, no data of 

 measurement of the dip made in connection with the 

 temperatures of the sea level and at the bridge, but 

 on theoretical grounds I cannot admit that this 

 formula is correct. It will be seen that the dip 

 becomes zero whenever A'=i5-i3A^', which is equi- 

 valent to saying that, if the temperature rises uniformly 

 1° F. per 15 feet, the dip is zero at all heights. Con- 

 sider now what will happen to a ray of light which 

 starts off from the surface of the sea tangentially. 

 In an atmosphere of uniform refractive index that 

 ray would proceed in a straight line and ultimately 

 depart from the earth entirely. With a refractive 

 index that diminishes with height the ray will be 

 bent towards the earth, and if the rate of diminution 

 is great enough that ray will at some point become 

 horizontal and the dip will be zero. Let us say that 

 this point is at a height of 30 feet above the sea. 

 Dr. Hunter's formula requires that the temperature 

 should be 2° more at 30 feet than at sea level, and if 

 the rise of temperature is uniform in this 30 feet his 

 formula also requires that the dip should be zero, 

 and therefore the ray horizontal, at all heights below 

 30 feet. This is obviously a fallacy, for if the ray 

 was always horizontal it would never reach the height 

 of 30 feet at all. 



The fact is that in an atmosphere where the layers 

 of uniform refractive index are spheres concentric with 

 the earth, the dip can only be zero, if at all, at one 

 height. Below that height the dip will be positive 

 with a maximum value at some lower level ; above 

 that height no ray tangential to the earth's surface 

 can be seen at all, and the depression or elevation of the 

 sea horizon requires an entirely different explanation. 



The equation upon which Dr. Hunter's formula is 

 based brings out this point quite clearly. This 

 equation is ^os (dip) =fi,rlfji{r + h) . 



In an atmosphere in which ix{r +h) is, at some height, 

 less than fi^r the dip becomes imaginary, for its cosine 

 is greater than unity. The dip could only be zero 

 for all heights for an atmosphere in which ix{r +h) is 

 constant, and in such case a ray once horizontal 

 would remain horizontal for a complete circuit of the 

 earth. Thos. Y. Baker. 



Admiralty Research Laboratory, 

 Teddington, Middlesex, April 6. 



