76 



NATURE 



[November 28, 1895 



himself says in the preface that " in teaching, I have found 

 it advantageous to supplement the work at each stage by 

 additional examples of the processes given in the text ; as 

 well as by references to special books in which particular 

 questions are examined with greater detail." Considered 

 as a synopsis of the principles of statics, particle and 

 rigid dynamics, hydrodynamics, and even portions of 

 elasticity, the book may safely be recommended either to 

 mathematical students, or to such students of physics 

 or engineering as have undergone the necessary pre- 

 liminary grounding in higher analytical methods. 



In its new form. Prof Tait's " Dynamics " might be not 

 inaptly described as a " Thomson and Tait for Beginners," 

 and in scope and plan it is not very different from Ziwet's 

 American treatise on " Theoretical Mechanics." It 

 opens with a short introductory chapter on. Newton's 

 laws, followed in chapter ii. by over 80 pages devoted 

 to pure kinematics, under which heading the analysis of 

 strain receives a fair share of attention. Chapter iii. 

 opens with a few further definitions relating to the laws 

 of motion, chiefly extracted from " Thomson and Tait," 

 followed by an outline of statics of a particle and the 

 theory of attractions, including Green's theorem and the 

 method of electrical images ; the whole being condensed 

 into 36 pages. Chapters iv. and v. deal with particle 

 kinetics, and include 14 pages on the principle of least 

 and varying action, and 6 pages on Lagrange's general- 

 ised equations. Statics and kinetics of a rigid body 

 follow next, then a short chapter on the dynamics of a 

 chain, in which vibrations of strings occupy only 7 pages. 

 After this comes a still shorter chapter containing a few 

 of the simpler applications of the theorj' of elasticity. In 

 chapter x. the fundamental equations of hydrodynamics 

 are disposed of in 25 pages, and are followed by a chapter 

 on waves, based on Prof Tait's " Encyclopasdja " article on 

 the subject. The author concludes with a short philo- 

 sophical discussion on the " objectivity of force." 



Unlike the ordinary run of text-books, this one con- 

 tains no collections of examples, but in most of the 

 worked-out problems, only the main results are stated, 

 so that the reader will find plenty of work to do in filling 

 up the gaps. Perhaps this plan will afford as good or 

 better training than the usual routine of " bookwork and 

 riders." There is a further advantage in the elimination 

 of unnecessary formulas from the text. The judicious use 

 of small type has also done much in making the mathe- 

 matical portion appear less formidable, and the use of 

 dark type has been avoided by adopting the method of 

 indention. 



A reviewer naturally turns his attention sooner or later 

 to the sections which deal with the laws of motion, and 

 these cannot be said to be above criticism. The recog- 

 nition of Newton's scholium to the third law as a 

 separate dynamical principle is good, but we should have 

 been glad to have seen greater definiteness in dealing 

 with the second law. We all know that "quantity of 

 motion " is to be interpreted as momentum, but Newton 

 makes no mention of time in his statement, and modern 

 text-book writers have remedied the deficiency by re- 

 stating the second law in the following three different 

 ways : 



(i) Rate of change of momentum is proportional to 

 the force. 



NO. 1361, VOL. 53] 



(2) Change of momentum in a given time is pro- 

 portional to the force, (In other words, change of 

 momentum varies as the force when the time is kept 

 constant ; the fact that it varies as the time when the 

 force is kept constant, being regarded as too obvious to 

 require stating as a law.) 



(3) Change of momentum is proportional to the 

 impulse of the force, impulse being defined as the time- 

 integral of a force. 



Now, Prof. Tait appears to regard the law from the 

 second standpoint on page 8, and from the first on 

 page 100, and yet reference to the " Principia " suggests 

 that the third interpretation is the most strictly New- 

 tonian, since Newton evidently regarded /<7rr^ as capable 

 either of being impressed gradually, or of being con- 

 centrated about a single instant of time. A statement 

 of Prof Tait's real views would have been far more 

 acceptable than the series of definitions of " change of 

 momentum," " rate of change of momentum," " rate of 

 change of kinetic energy," " space-rate of change of it," 

 with which chapter iii. opens. If a rate had been defined 

 generally, these definitions would be superfluous. 



In § 118 we are told that "the second law gives us 

 the means of measuring force, and also of measuring 

 the mass of a body." We had always thought (and still 

 think) that the third law is required before both force and 

 mass can be measured. For in comparing two forces. 

 Prof Tait supposes them applied in turn to the same 

 body. But this test affords no criterion of the equality 

 of two forces when applied to different bodies, as they 

 have to be in comparing the masses of the bodies. If 

 we assume an independent definition of equal forces (as,, 

 for instance, in Prof Tait's illustration on p. 1 11, where 

 he assumes that the same locomotive exerts approximately 

 the same pull on two different trains), the second law, of 

 course, affords a means of comparing two masses, but 

 not otherwise. 



We are sorry that Prof Tait has not altogether 

 banished from his book that highly misleading and 

 artificial phraseology of "a point having several simul- 

 taneous velocities " in connection with the parallelogram 

 law (§ 117). Combined with the definition of velocity as 

 " rate of change of position," this always seems to suggest 

 the (absurd) notion that the point moves into, and therefore 

 occupies, different positions at the same instant. All that 

 the parallelogram of velocities really proves, and what 

 Prof Tait proves in §§ 30, 31, is that if the velocity of 

 A relative to B and the velocity of B relative to C are 

 represented by two sides of a parallelogram, the diagonal 

 represents the velocity of A relative to C. In passing 

 from this to the parallelogram of forces, the discontinuity 

 of reasoning could perhaps be overcome by a reference 

 to Newton's commentary on his second law, which states 

 that the effect of several forces is the same whether they 

 be impressed simultaneously or successively. 



In § 108 the author defines as " the ' component ' of 

 a force in any direction " what is now, with great ad- 

 vantage, usually called the resolved part of the force in 

 that direction. 



We should also like to see the use of the word " stress " 

 avoided in connection with Newton's third law (§§ 174, 

 352), until some definite understanding has been arrived 

 at as to whether the term is to signify a force, or (like: 



