November 25, 1922] 



NA TURE 



699 



observers X and Y. The spaces which they use are 

 normal to these axes. Then if v be their mutual 

 relative velocity, 



v = tanh C, 



the velocity of light being unity. 



It may be added that the relation B+C=A' is 

 a particular case of the more general " triangle of 

 relative velocities." Let OP, OQ, OR be a triad of 

 co-directional non-coplanar temporal vectors (Dr. 

 Robb's " inertia lines ") cutting the " open hyper- 

 sphere " (centre O) 



((-- \-- y-- z~= I 



in point-instants P, Q, R, where u isthe time co-ordinate. 

 Let a, b, c be the geodesic arcs QR, RP, PQ within 

 the hyper-sphere. These arcs are minima, not 

 maxima ; their elements in the limit are spatial in 

 character, being normal to time-vectors ; their 

 hyperbolic tangents represent the mutual relative 

 velocities of observers (X, Y, Z) who use OP, OQ, OR, 

 or parallels thereto, as their time-axes. The Euclidean 

 space used by X at any instant is parallel to the 

 tangent space at P to the hyper-sphere, and from 

 the point of view of X the directions of the relative 

 velocities of Y and Z are the tangent-lines at P to 

 the geodesic arcs PQ, PR. The angle between these 

 directions is a circular angle (P), and the metrics 

 of the geodesic, triangle PQR are contained in the 

 formulae 



cosh a = cosh b cosh c— sinh b sinh c cos P, 

 sin P _ sin Q _ sin R 

 sinh a ~ sinh b sinh c° 

 When a, b, c are very small compared with the 

 radius of the hyper-sphere the spaces of the observers 

 are regarded as parallel, and we get the ordinary 

 formulae 



a 2 = b s + c 2 — 26c cos P, etc. 

 When OP, OQ, OR are coplanar we get the 

 relation as before (with change of letters) 

 a = b + c. 



The above remarkable formula for relative velocities 

 was, I believe, first discovered by Dr. Robb, and is set 

 forth by Dr. Weyl (" Space, Time, and Matter," § 22). 

 I am not aware, however, that its direct connexion 

 with the geodesic geometry of the open hyper-sphere 

 has been explicitly noticed. R. A. P. Rogers. 



Trinity College, Dublin, October 31. 



Space-Time Geodesies. 



In Nature of October 28, p. 572, Dr. Robb pointed 

 out the incorrectness of asserting that the length 

 of a " world-line " is a minimum between any two 

 points of it. He gave an example in which the length 

 was neither a minimum nor a maximum. The 

 object of his letter, no doubt, was to remind some 

 reckless relativists that they should be more careful 

 in their language. But there is the danger that 

 some may suppose that he was dealing with a real 

 weakness in Einstein's theory. To dispel this idea 

 we may recall a few well-known facts. 



Treatises on the geometry of surfaces (in ordinary 

 three-dimensional Euclidean space) define geodesies 

 in various ways. Some say that a geodesic is the 

 shortest line that can be drawn on the surface between 

 its two extremities, and they use the calculus of 

 variations to find its equations. This method is 

 open to criticism. The researches of Weierstrass 

 have shaken our faith in the infallibility of the results 

 obtained by an uncritical use of the routine processes 

 of that calculus. But whatever may be said against 

 the process employed, the equations of a geodesic 

 finally obtained agree with t hose obtained by more 



NO. 2/69, VOL. I IO] 



trustworthy methods. For example, we may define a 

 geodesic as a curve such that at every point the 

 osculating plane is perpendicular to the tangent 

 plane to the surface. From this definition we can 

 easily obtain (cf. Eisenhart's " Differential Geometry," 

 p. 204) equations which in the usual abbreviated 

 notation of tensor calculus may be written 



^r+{aB,a } ^— ^r = O t ((7=1,2). 



Einstein's equations (" The Meaning of Relativity," 

 p. 86) are the obvious generalisation of these and 

 differ merely in that the suffixes range over the 

 values i, 2, 3, 4, instead of only 1, 2. His notation 

 is slightly different from the form given above, which 

 is due to Eddington. 



These equations can be obtained by at least two 

 other methods. Einstein uses a " parallel displace- 

 ment " method due to Levi-Civita and Weyl. 

 Eddington (" Report on the Relativity Theory of 

 Gravitation," p. 48) shows that the equations are 

 satisfied (or not) independently of the choice of 

 co-ordinates, and that they reduce to the equations 

 of a straight line for Galilean co-ordinates. This 

 straight line is described with uniform velocity, so 

 Einstein's equations may be regarded as a generalisa- 

 tion of Newton's first law of motion. 



Applying these equations to the example given 

 by Dr. Robb, we find that his space-time curve 

 does not satisfy them unless F"(.r) =0. This means 

 that F(-tr) must be a linear function of x and so it 

 cannot fulfil the required conditions of vanishing for 

 two different values of x, except in the trivial case 

 F(x) =0. Thus the ambiguity seems to lie, not in 

 Einstein's equations of motion, but merely in a 

 particular method of arriving at them. 



As regards the desirability of modifying Einstein's 

 ideas on the nature of time, it is hazardous to give 

 a definite opinion at present. It may be noted that 

 Prof. Whitehead's new book (" The Principle of 

 Relativity ") endeavours to combine all the verifiable 

 results of Einstein's theory with somewhat con- 

 servative ideas concerning space and time. The 

 modified theory leads to some remarkable predictions 

 (p. 129) which should be tested bv experiment. 



H. T. H. Piaggio. 



University College, Nottingham, 

 November 4. 



The Dictionary of Applied Physics. 



The issue of Nature of September 30, p. 439, 

 contained a highly appreciative review of the first 

 volume of the " Dictionary of Applied Physics," and, 

 as editor, I am much indebted to the author for his 

 kind words. One remark, however, has, I gather, led 

 to some misunderstanding ; may I have space for a 

 brief explanation ? 



Dr. Kaye directs attention to some of the " omis- 

 sions," with the view of their future rectification. 

 Most of these " omissions " will be found dealt with 

 in future volumes of the Dictionary. Thus, in an 

 article in vol. hi., on Navigation and Navigational 

 Instruments, by Commander T. Y. Baker, the gyro- 

 compass is treated of very fully, while, in vol. v., 

 Mr. Dobson has a highly interesting article on instru- 

 ments used in aircraft. 



It has been part of my plan to separate the mathe- 

 matical treatment of a subject and its practical 

 applications. In this manner I hoped to increase 

 the utility of the work to various classes of readers, 

 some of whom are interested chiefly in the theory, 

 while others are more closely concerned with the more 

 practical details. R. T. Glazebrook. 



5 Stanley Crescent, Kensington Park Gardens, 

 London, W.n. 



