October 31, 1901] 



NA TURF. 



64 



of three dimensions. In chapter iii. the author settles 

 down to the composition, resolution, and equilibrium of 

 a general system of coplanar forces, and gives a very 

 good exposition of the subject ; but in this chapter, of 

 nearly fifty pages, no use of either the calculus or three- 

 dimensional geometry is made, except in three pages 

 devoted to the common catenary. Now the understanding 

 of this very important and extensive section of dynamics 

 is well within the power of any student even if he is 

 quite ignorant of these branches of pure mathematics, 

 so that it seems a pity that he should be kept back in his 

 dynamical studies until he has passed through " an 

 adequate course in the differential and integral calculus." 

 Passing over chapters devoted to the determination of 

 centres of gravity and the composition of forces which 

 are not coplanar, we come to chapter vii., which treats 

 of the principle of work. This chapter is somewhat 

 meagre, consisting mainly of what is known as " book- 

 work," and not containing sulScient illustration of the 

 applications of the principle to concrete cases. Until 

 the student comes to chapter viii. he will experience no 

 difficulty in the author's treatment of the subject ; but 

 when he reaches this chapter, on " motion produced by 

 constant force," he will find a good deal about the nature 

 of ''inertia regarded as a force" which will be very per- 

 plexing. His main difficulty will be to decide whether 

 the author means the " force of inertia " to be one 

 exerted by a body or upon it by some agent or medium. 

 Thus, at the beginning of art. 288 it would appear to be 

 a force exerted by the body : — 



" The property of matter through which it resists any 

 change of motion, in accordance with the First Law of 

 IMotion, is called Inertia." 



But a few lines farther on we have the sentence : — 



" Now, just as the resistance of a fixed body in contact 

 with .that upon which the force acts, and preventing its 

 motion, is regarded as a force equal and opposite to the 

 force which would otherwise produce motion, so the re- 

 sistance to motion in the body when free is regarded as 

 a force equal and opposite to the active force which pro- 

 duces the motion." 



Let us suppose a particle M acted upon by forces 

 whose resultant is P and kept from moving by the resist- 

 ance, N, of a fixed surface B ; then the force N is exactly 

 equal and opposite to the force P. Again, imagine the 

 body M acted upon by the same force P and unresisted 

 by any fixed surface ; M will have an acceleration a, and 

 the statement is that there is acting on .M a force resisting 

 the acceleration a — this force being clearly produced by 

 something which in our thoughts replaces the above fixed 

 surface B — that this force is equal and opposite to " the 

 active force which produces the motion." So far, what 

 this "active force " is is not clear ; but the next sentence 

 defines it : — 



" Thus the force of inertia acts upon a particle of mass 

 m only when there is an acceleration a, and its value is 

 >na, while its direction is opposite to that of the 

 acceleration." 



Now observe that if the particle had no acceleration, 

 this force would be zero, while in the first part of the 

 analogy (where also a = o) the supposed analogous force, 

 N (the resistance of the surface B) is not zero. 

 NO. 1670, VOL, 64] 



However, from this and from subsequent statements 

 it is clear that, in the author's view, a force of inertia 

 really acts on a particle m which has an acceleration n, 

 and that this force is scalarly and vectorly equal to - ma ; 

 that is to say, it is D'Alembert's fictitious " reversed 

 effectiv-e moving force." But this is not in accordance 

 with the statement at the top of p. 288 : — 



"And the inertia which acts upwards is, at that point, 

 simply the resistance of the body to being moved away 

 from the tangent at o." 



It is certainly strange that a force acting on a body 

 should be the resistance of the body to being moved. 

 The author, however, clearly defines his conception, 

 which he calls that of "kinetic equilibrium," at the top 

 of p. 244 :— 



" For example, suppose a man whose weight is W to 

 be standing on the floor of an elevator which begins 

 to descend with the known acceleration a. The forces 

 acting on the man are his weight, W = ;«;,'', acting down- 

 ward, his mertia, oth, acting upward because the accele- 

 ration is downward, and the resistance R of the floor 

 of the elevator acting upward. Since the forces are all 

 vertical, there is but one condition of equilibrium, namely, 

 \V = R-f-;«a." 



The objection which a student will raise to this is that 

 if the man is really acted upon by the upward force ma, 

 the man is really at rest and not in motion at all. 



D'Alembert never attributed anything but a fictitious 

 existence to his " reversed effective forces," and he was 

 right and consistent all through. The real objection to 

 his principle is that it teaches us to be dissatisfied with 

 the actuality (viz. motion), and to seek refuge in a fiction 

 (viz. rest). The teaching of Newton's second axiom is 

 quite different : it accepts motion as a fact and deals 

 with it. 



The remainder of the book gives somewhat short and 

 easily readable discussions of central orbits, motion 

 (especially uniplanar) of rigid bodies, moments of inertia, 

 and impulses. 



OUR BOOK SHELF. 

 The Earliest Inhabitants of Abydos ; a Cranial ogical 

 Study. By D. Randall-Maciver. Plates viii 4- tables 

 16. (Oxford : Clarendon Press, 1901.) Price lar. bd. 

 net. 

 In the present work Mr. Randall-Maciver presents to 

 the public the craniological material which he obtained 

 in Upper Egypt in the winter of 1S99-1900, and the 

 results which he has deduced from it. In a series of 

 eight plates he gives us photographs of a large number 

 of skulls which he obtained from two cemeteries at 

 Abydos, which, he says, belonged to the earliest and the 

 latest stages of the pre-dynastic period, and to these 

 he adds some sixteen tables of mmute craniological 

 measurements. The first cemetery contained only pottery 

 of the earliest forms, black-topped, polished red, and 

 white ornamented red, and the second degraded wavy- 

 handled vases and other pottery of well-defined classes. 

 The remarks which Mr. Randall-Maciver makes in his 

 short preface may be regarded as a continuation of those 

 expressed in his " Libyan Notes," and we observe that he 

 still holds the view that the theory of the Libyan origin 

 of the pre-dynastic or proto-dynastic Egyptians is " based 

 on wholly inadequate evidence." The pre-dynastic 

 Egyptians were, he thinks, a mixed race, but as a whole 

 that race was not Berber ; on the other hand, he does 

 not deny the existence of an original Berber substrnumi, 



