January 20, 192 1] 



NATURE 



655 



General Dynamics. 

 Higher Mechanics. By Prof. H. Lamb. 

 Pp. x + 272. (Cambridge: At the University 

 Press, 1920.) Price 255. net. 



WE have here, as was to be expected from its 

 author, an excellent statement and ex- 

 planation of the principal theorems of what may 

 properly be called higher dynamics. Prof. Lamb's 

 title is "Higher Mechanics," which is in accord- 

 ance with the usage which obtained before the 

 appearance of Thomson and Tail's "Natural 

 Philosophy." In his preface to the "Principia," 

 Newton says : " Mechanicam vero duplicem 

 veteres constituerunt, rationalem, quae per 

 demonstration's accurate procedit, et practicam." 

 Thus Newton uses the word "mechanica" quali- 

 fied by the adjective " rationalis " in the sense of 

 " scientia motuum qui ex viribus quibuscunque 

 resultant," the science of the motions of bodies. 

 The point is not one of great importance, but 

 we prefer with Thomson and Tait the name 

 "dynamics" for the whole science of rational 

 mechanics. 



The book begins with chapters on the kine- 

 matics of a rigid body, in which degrees of free- 

 dom, displacements of a rigid body, and theorems 

 regarding these are very clearly explained and 

 illustrated. Then follows a chapter on statics, in 

 which the usual theorems are set forth, with a 

 short but adequate account of the theory of 

 screws. A treatment of moments of inertia is 

 given which, perhaps, might with advantage 

 have been fuller. Binet's theory of plane quad- 

 ratic moments is included, while his theorem 

 that the principal axes of inertia at any 

 point P are the normals to the three quadrics 

 which are confocal with the central ellipsoid of 

 gyration, and which intersect at P, is not attri- 

 buted to its author. The subject of plane quad- 

 ratic moments does not seem of any but a purely 

 mathematical interest, and one rather grudges the 

 page devoted to it in this brief chapter. 



After a chapter on instantaneous motions, we 

 are led on to the equations of motion of a rigid 

 body, with reference to a system of moving axes, 

 which finally, when the body turns about a fixed 

 point, is identified with the principal axes of 

 moment of inertia passing through the fixed point. 

 In connection with the equations of motion for 

 this case — Euler's equations— which are of the 

 form 



\p-{B-C)qr L, 



the usual statement is made that the equations 

 may be written : 



A^=L + {B-C)qr, 

 wn. 2fi72. vol.. To61 



and that then the quantities typified by (B-C)qr 

 are referred to as the components of a "centri- 

 fugal couple." This couple is quite rightly de- 

 signated " fictitious " ; but, apart from the lack 

 of reality which afflicts it, its introduction 

 seems undesirable. To speak of these quantities 

 as "centrifugal couples" seems a perfectly arti- 

 ficial, unphysical, and almost unintelligible mode 

 of regarding the matter. Centrifugal couples are 

 worse than centrifugal forces, those unrealities 

 against which Tait used to fulminate vehemently. 

 What one wants to convince the student here 

 is that A^ is not the whole rate of growth of 

 angular momentum about the fixed axis, with 

 which the principal axis at the instant coincides, 

 and must be supplemented by the rate of growth 

 -(B-C)qr which exists in consequence of the 

 motion of the axes. The whole rate of growth 

 is Ap-{B-C)qr, and there is no ray of light 

 thrown on the subject by carrying the second part 

 from its proper place to the other side of the 

 equation and calling it a couple. The use of 

 " centrifugal couples " is justifiable only when the 

 action of, or reaction on, an axis is in question. 



We hope that we shall not be misunderstood. 

 Of course, Prof. Lamb is perfectly aware of all 

 this ; but our point is that, as the question is one 

 of getting correct ideas into the minds of students, 

 this venerable and " Through the Looking-Glass " 

 mode of referring to these terms would be 

 better forgotten. Students, and even some 

 writers, are iall too prone to suppose that the 

 component growths of angular momentum are 

 \p, Bq, Cr, and the equations are naively written 



as A^ = L, etc. 



It is, by the way, very important, and very easy, 

 to take the terms Ap, - Bqr, Cqr, and show how 

 each of them arises. 



We miss here what is certainly very instructive : 

 the application of the hodograph to the motion 

 of a rigid body— for example, a top. A sequence 

 of vectors all drawn from a point O is erected in 

 space to represent the successive values of the re- 

 sultant angular momentum. The curve in space on 

 which lie their extremities is the hodograph for the 

 rigid body motion, and the resultant couple at 

 each instant is represented by the velocity of a 

 point which moves so that it is always at the ex- 

 tremity of the vector which at the instant repre- 

 sents the angular momentum. 



In connection with holonomous systems, the 

 only reference is to Hertz, though the fact that 

 there are systems which are not holonomous for 

 which Lagrange's dynamical method "fails," and 

 which require special treatment, was pointed out 

 by Ferrers long before Hertz's treatise appeared. 



