586 



NA TURE 



[October i6, 1890 



It is a pity that the term vis viva has been allowed to 

 remain in this last edition, and that it was not entirely re- 

 placed by kinetic energy : the contrasted term vis inortua 

 has been dead for a long time, and vis viva should have 

 followed long ago. 



The transformations of the dynamical equations into 

 the Lagrangian and Hamiltonian forms are introduced at 

 rather an early stage ; and the subject is resumed in the 

 last chapter x., on "Theoretical Dynamics," written by 

 the late Prof. W. F. Donkin. These transformations are 

 merely analytical illustrations of the change of in- 

 dependent variables, the form of the equations depend- 

 ing on whether we express the kinetic energy in terms of 

 the generalized velocities or the generalized momenta. 



A clear and expressive notation, somewhat in the style 

 of that found necessary in Thermodynamics, would make 

 these equations more intelligible and convincing ; but in 

 any case, the application to definite problems, especially 

 where the geometrical constraints present any peculiarity^ 

 is so difficult and refined, that these equations are 

 dangerous weapons to put into the hands of any but 

 advanced students. 



The principles of Least Action and of Least Constraint 

 'are also introduced here by the author ; interesting 

 verifications are thus afforded of well-known problems ; 

 but these principles again would not be employed for 

 choice ; and although the author pleads in their favour, 

 we think it should not be forgotten that the principle of 

 Least Action was employed to bolster up the Corpuscular 

 Theory of Light. 



Newton's principle of mechanical similitude, in the next 

 section, is, however, of great practical importance, and we 

 see its application in the constantly increasing size of our 

 bridges, ships, and guns. In its particular application 

 to naval architecture, a corollary goes by the name of 

 Froude's law (also enunciated by Reech), which asserts 

 that in similar vessels run at speeds proportional to the 

 square root of the length or the sixth root of the dis- 

 placements, the resistances are as the displacements ; and 

 thus the naval architect is able to infer, from the known 

 performance of a ship or a model, what to expect on a 

 different scale. When we make, in any two similar 

 machines of the same material, the velocities in the ratio 

 of the square root of the linear dimensions, we ensure in 

 this manner that the stress per unit area in the material 

 is the same, and thus the two machines are equally 

 strong ; so that this law of corresponding speed is 

 most useful in the practical application of Newton's 

 law of similitude. 



A valuable section on Units, No. 9, points out that there 

 are only two systems which need be considered : the 

 British foot-pound-second (F.P.S.) system, and the metric 

 centimetre-gramme-second (C.G.S.) system. The author's 

 numbers for the conversion of one system into the other 

 are not exactly according to the latest determinations ; 

 thus it is more accurate to make i metre = 39'37o79 

 inches, and i foot = 30'4794 centimetres. The metre 

 was originally designed so that the kilometre should be 

 the centesimal minute of latitude, for use in navigation ; 

 but taking the sexagesimal minute of latitude as 60S0 

 feet, the Admiralty standard, then the above figures make 

 the length of the earth's quadrant 10,007 kilometres, in- 

 stead of 10,000, as designed. It has been decided, hovv- 

 NO. 1094, VOL. 42] 



ever, for electrical purposes that lo'' centimetres should 

 be called a quadrant, although about o'07 per cent. out. 



Recent redeterminations of the weight of a metre cube 

 of water, and of the volume of 10 gallons or 100 pounds 

 of water, made with the greatest care, have revealed per- 

 ceptible discrepancies with former estimates ; so that the 

 definition of the kilogramme as a decimetre cube of pure 

 water at its maximum density must be considered a 

 purely academic definition, and not sufficiently precise for 

 legal purposes ; the ultimate appeal being to the lump 

 of platinum preserved in the Conservatoire des Arts et 

 Mdtiers. 



A Committee of the British Association is at present 

 engaged in attempting to fill up the gaps in our dynamical 

 terminology : the author introduces the dyne and erg, due 

 to a former Committee, but not the kine, spoud, bole, and 

 barad, recently settled upon as names for the C.G.S. units 

 of velocity, acceleration, momentum, and pressure. The 

 C.G.S. units are too minute for practical purposes, so that 

 electricians now employ the joule, of 10'' ergs, and the 

 watt as the volt-ampere, or power doing one joule per 

 second — units based really upon the commercial units of 

 the metre and kilogramme, instead of the centimetre and 

 the gramme. These microscopic units were adopted by 

 the original Committee apparently merely to gratify the 

 fad of making W := sN , instead of looo^'V. 



The astronomical unit of mass is defined in § 143 ; but 

 if it is difficult to measure the volume of a kilogramme of 

 water, the probable error in the determination of this 

 astronomical unit of mass is immensely greater ; so that 

 to our mind this unit had better be discarded, and the 

 gravitation constant introduced into the equations, using 

 its provisional value, lo-^ X 6'48 C.G.S. units (Everett,. 

 " Units and Physical Constants," § 72). 1 



Chapter iv. discusses the equations of motion of a ' ^ 

 rigid body expressed in terms of angular velocities and 

 their increments, &c. The author adopts various illustra- 

 tive methods, but to our mind the simplest procedure is 

 to establish the general equations, with the usual notation 

 hi - /'!2^3 + /';A= L, . . . ; and then Euler's equations, 

 &c., follow as particular cases. By adding the terms due 

 to the employment of a movable origin we obtain the 

 form of the Hamiltonian equations required in the dis- 

 cussion of the motion of a body moving in a liquid ; and 

 here is a good opportunity for the introduction of Dr. 

 Routh's principle of the Ignoration of Co-ordinates, re- 

 quired to complete the theory of the generalized equations 

 of motion. 



Prof. Price could make a very useful book for students 

 of elementary mathematics by taking out and printing 

 separately the part on uniplanar motion and its illustrative 

 examples (chapter v.) : the complication of the subject 

 of rigid dynamics is more than doubled when we consider 

 motion in three dimensions ; but in two dimensions the 

 subject is within the grasp of most students, who will 

 thus acquire a good working knowledge sufficient for 

 most purposes. At the outset the determination of simple 

 moments of inertia is required, and this involves a know- 

 ledge of integration ; so that a student, untrained in the 

 Calculus, can make very little headway. It is a pity that 

 the lack of the slight knowledge of integration required 

 for this purpose prevents most of our students from going 

 on to the real study of the pendulum, the motion of thf 



