SoS 



NATURE 



[August 26, 1920 



using a formula which permits of a ready checlc. 

 This only takes about five minutes for each sight, 

 and is, no doubt, the best way ; in fact, it is the 

 only safe way where a very considerable degree of 

 precision is aimed at. But most navigators prefer to 

 avoid computation so far as possible by the use of 

 tables, and in ordinary circumstances the altitude 

 tables used in the Royal Navy will give sufificiently 

 accurate results. The great defect of the tabular 

 method is that one has to round off the dead- 

 reckoning latitude to the nearest degree for the 

 -assumed position in order to enter the tables, and 

 consequently the position-lines may extend over so 

 grept a distance on the chart that their curvature 

 cannot properly be neglected. With logarithmic 

 calculation, on' the other hand, the actual dead- 

 reckoning position can be taken as the assumed 

 position, and the position-lines will then be so short 

 that their curvature can be neglected without any per- 

 ceptible loss of accuracy. 



It may not be out of place to remark in conclusion 

 that the' utility of the Sumner line or position-line 

 principle is not confined to position-fixing with a 

 secant at sea. I have shown in two recently pub- 

 lished papers ("Notes on the Working of the New 

 Navigation," Cairo, 1918, and "The Prismatic Astro- 

 labe," Geographical Journal, July, 19 19, p. 37) that 

 the "new navigation" is capable of useful applica- 

 tion on land in conjunction with theodolite observa- 

 tions and wireless time-signals, and that determina- 

 tions of geographical position of very considerable 

 accuracy may be made in this way. The method has 

 since been put into practice by Dr. Hamilton Rice on 

 exploratory land surveys in South America (see the 

 Geograplucal Journal for July, p. 59) with satisfac- 

 tory results. " John Ball. 

 Survey of Egypt, Cairo, July 24. 



Relativity and Hyperbolic Space. 



Observation tells us that while gravitation 

 dominates the history of a lump of matter rnoving in 

 the vast ocean of free aether, it has practically no 

 effect on the history of a pulse of light in similar 

 circumstances. Since last mail I have investigated 

 the bearings of space being hyperbolic on light-rays. 



The central-projection map of the space, used 

 before, in which r = tanh ©, where r is the radius 

 vector of the map and Re the radius vector in the 

 space, will be called a gnomonic map ; planes are 

 mapped as planes. If the projection used be given 

 by r = 2tanh ^9, the map will be called stereographic ; 

 small regions are mapped in correct shape, spheres 

 and planes as spheres, and the two sheets of 

 a pseudo-sphere as two spheres intersecting and 

 making equal angles with the sphere repre- 

 senting the median plane, in a circle lying on the 

 absolute, r = 2. (A pseudo-sphere is the locus of a 

 point at a given distance from a given plane, called 

 its median plane. The characteristic of the map- 

 sphere which represents a plane is that it cuts the 

 absolute r = 2 orthogonally.) A point (x, y, z) on the 

 gnomonic map becomes [x/(i+^'r^), y/(i + ir'), 

 2;/(i + 5'r^)] on the stereographic map. 



The behaviour of a ray of light is fully described 

 by saying that its path on the gnomonic map may 

 be put in the form x^/a'^+y^=i, where a is less 

 than I, and that the eccentric angle is t/R, where t is 

 co-ordinate time. This ellipse really represents the 

 two branches of a pseudo-circle ; the ray goes out 

 •to infinity (in the space) along one branch and returns' 

 along the other, the complete circuit having the 

 period 27tR. The median line of the pseudo-circle 

 NO. 2652, VOL. 105] 



passes through the origin — that is, through the 

 observer. 



If from a given point rays start in all directions 

 there will be a definite wave-front. For a finite time 

 before t attains the value of a quarter-period, gn-R, 

 this front will form the single sheet of a true sphere 

 the centre of which recedes to infinity, whereupon the 

 front develops the two sheets of a pseudo-sphere, the 

 one proceeding in the same direction as before, and 

 the other, together with the median plane, returning 

 from infinity, having been reflected back by the 

 absolute. By the time t = ^nR the median plane has 

 just reached the origin, and the reflected sheet is 

 chasing both the other sheet and the median plane 

 back on the way to infinity. In the next quarter- 

 period these motions are reversed in order of time, in 

 direction of motion, and in position relative to the 

 origin. At the time < = 7rR the front has contracted 

 down to a point focus situated on the opposite side 

 of the origin from the radiant point at a distance 

 equal to that of the point. At the time t-ZnR the 

 original circumstances recur, and everything is about 

 to be repeated. A ray always moves normal to the 

 front, although the centre of the true sphere and 

 the median plane of the pseudo-sphere themselves 

 move from and to infinity in a finite time. 



All these motions can be exactly imitated in 

 Euclidean space. Let, at a given point in such a 

 space, the velocity of light be i-|-r74R', the same in 

 all directions, and let the sphere r = 2R be a perfect 

 reflector. Then light will in this medium behave 

 exactly as does the light in the stereographic map 

 (when the scale of that map is increased in the ratio 

 of R to i). Indeed, this seems the easiest method to 

 get the differential equation of the path of a point 

 in the hyperbolic space, for which fdt is stationary. 

 I may remark, however, that when the equation is 

 obtained, later work is much simplified by changing 

 the dependent to a form corresponding to the 

 gnomonic map. 



In the stereographic map the rays after an even 

 number of reflections, by the absolute, form a system 

 of coaxial circles through the radiant point and that 

 point on the opposite side of the origin which is 

 inverse to the sphere r = 2. (For radiant point let 

 o = x~a = y = z. Then for the second point mentioned 

 it is meant that o = x+^/a = y = z. Ordinary inverse 

 point would be o = x-^/a = y = z.) After an odd 

 number of reflections they are similarly related to the 

 focus mentioned above. The fronts are the spheres 

 cutting these coaxial circles orthogonally. 



Alex. McAulay.' 

 University of Tasmania, June 10. 



The Antarctic Anticyclone. 



In Nature for August 5 Mr. R. F. T. Granger 

 remarks : "The same conditions, i.e. the surface out- 

 flow and the central descent of air, exist in Prof. 

 Hobbs's polar ice-cap anticyclone; the only difference 

 is the physical origin." 



In the case of the ice-cap there are other differences 

 as well ; the temperature is lower in the case of an 

 ice-cap than in an anticyclone. The ice-cap conditions 

 which resemble those of an anticyclone are, as Mr. 

 Granger says, "surface outflow and the central 

 descent of air." The differences are low temperature, 

 low pressure, and different physical origin. My 

 suggestion was that these differences made it in- 

 advisable to call them both anticyclones. 



R. M. Deeley. 



Tintagel, Kew Gardens Road, Surrey, 

 August iS. 



