August 7, 1913] 



NATURE 



;8i 



physical features of the planets and moon. As 

 a geologist the author claims to have formed 

 definite views of his own on these questions, 

 differing in many respects from commonly ac- 

 cepted theories ; but, as he points out, it would be 

 impossible for a writer to substantiate these 

 varied theories unless he had travelled all over 

 the world, besides being, at the same time, a 

 mathematician, a physicist, a chemist, an astro- 

 nomer, and a geologist. Considerable attention 

 is given to theories of the displacement of the 

 earth's axis. 



A collection of theories of this kind, if thus 

 propounded in a proper spirit, is not only interest- 

 ing, but it opens up useful material for future 

 discussion. On the other hand, not the least 

 important feature is the insight which the book 

 affords the general reader of known physical facts 

 and phenomena connected with the earth and 

 planets. 



1 Manual of School Hygiene. By Prof. E. W. 

 Hope, E. A. Browne, and Prof. C. S. Sherring- 

 ton. New and Revised Edition. Pp. xii + 311. 

 (Cambridge University Press, 1913.) Price 

 4s. 6d. 

 The hrst edition of this manual, which was re- 

 \ iewed in our issue for August 15, 1901 (vol. lxiv., 

 P- 373)' was reprinted on three occasions before 

 the appearance of the book in its present form. 

 Six chapters on physiology by Prof. Sherrington 

 have here been added. They aim at emphasising 

 the salient portions of the subject, and deal with 

 the body considered as a mechanism, the blood 

 and its circulation, respiration, food and digestion, 

 the temperature of the body, and muscle and nerve. 

 Library Cataloguing. Bv J. Henry Quinn. Pp. 

 viii + 256. (London: Truslove and Hanson, 

 Ltd., 1913.) 

 Mr. Quinn's book should prove of real service as 

 a guide for young librarians to the various codes 

 Of cataloguing rules. His bright, helpful chap- 

 ters should certainly convince the beginner in 

 library work that the office of librarian is no 

 sinecure ; and the arrangement of his matter, and 

 the subjects chosen for treatment, should enable 

 information on practical cataloguing to be ob- 

 tained with a minimum expenditure of trouble. 



LETTERS TO THE EDITOR. 



1'hc Editor does not hold himself responsible for 

 opinions expressed by his correspondents. Neither 

 can he undertake to return, or to correspond with 

 the writers of, rejected manuscripts intended for 

 this or any other part of Nature. No notice is 

 taken of anonymous communications.] 



Energy in Planetary Motions. 



If a particle of mass m be brought from infinity 

 to distance r by the action of a central attraction 

 varying as the inverse square of the distance, the 

 potential energy exhausted in the process is mp-/r, 

 where ju is the "intensity of the centre." If the 

 particle has experienced no resistance to its motion 

 the kinetic energy is given by the equation 



\mtf=n£. 



NO. 2284. VOL. 91] 



But if the particle be made to move in a circle of 

 radius r about the centre of force, the speed v is 

 given by 



and the kinetic energy \mv 2 represents only half the 

 potential energy exhausted. The other half must 

 have been dissipated or disposed of in some way or 

 other. 



Similarly, if the particle be brought in from motion 

 in a circle of radius r' about t lie centre to motion in 

 the circle of radius r, so 1 hat potential energy of 

 amount m,u( 1 1 1 r') is exhausted, the kinetic energy 

 has been increased by only .Jm,«(i/>— i/r'), so that 

 again only half of the potential energy exhausted is 

 represented by the orbital motion, and the remainder 

 has been expended in doing work against resistance 

 of some sort. The central force has, in fact, done 

 exactlv twice as much work as that represented in 

 the increase of the kinetic energy. 



All this, of course, is perfectly elementary and well 

 known, but it is nevertheless a curious dynamical fact 

 that exactly half of the work done by the attraction 

 must be expended in overcoming resistance. 



I have not seen the corresponding theorem in 

 elliptic motion anywhere explicitly stated. It is as 

 follows : — The time-average of the kinetic energy, 

 taken for one revolution in the orbit, is half of the 

 corresponding time-average of the potential energy 

 exhausted in the passage from infinity to the distance 

 r. A similar theorem holds, of course, for the differ- 

 ences of energy concerned when the particle is trans- 

 ferred from one orbit to another about the same 

 centre. 



Let 2a be the length of the major axis of the 

 elliptic orbit. The speed v at distance r from the 

 centre is then given bv 



\r 2n } 



which, multiplied by nt, is the equation of energy. 

 The potential energy exhausted from infinity to dis- 

 tance r is again mn r , and it can easily be shown 

 that the time-average of the kinetic energy in the 

 orbit is mp. la. 



Parenthetically, it may be remarked that this result 

 is most easily and elegantly established by the follow- 

 ing Newtonian process. If when r is the distance 

 of the particle from the centre of force (one focus of 

 the ellipse) r' be the distance from the other focus, 

 and p, p' be the lengths of the perpendiculars from 

 the foci on the line of motion at the instant, we have 

 r/r' = p/p', and, therefore, since />/>' = b 2 , where b is 

 the length of the semi-minor axis, we have r'lr = f' 2 h- . 

 But the equation for v- can be written 



„ u 2a - >■ u.r' 



Hence integrating for a period of revolution T we get 



where ds is an element of the path, and the integrals 

 with respect to s are taken once round the ellipse. 

 Now, clearly / p'ds is twice the area of the ellipse — 

 that is, 2Trab. Thus 



The period T is 2ir*/a 3 /ft, and so the mean kinetic 

 energy is 



