74 



NATURE 



[November 25, 1897 



into contact with water-saturated rocks. The association 

 of volcanoes with ocean-margins he considers to be due 

 to the fact that both classes of phenomena mark planes 

 of weakness in the earth's crust. This chapter is well de- 

 serving of careful study, and the final one presents an 

 interesting sketch of the chief events which mark the 

 life-history of a volcano. 



The book is carefully written, and well illustrated by 

 maps and process-blocks from photographs. It would 

 be a convenience if some simple mark had been used to 

 indicate on the large map those volcanoes which have 

 been active within recent times. W. 



THE PRINCIPLE OF CONSERVATION OF 



EN ERG V. 



Das Pr/na'p der Erhaltung der Energie und seine 



Anwendunginder Naturlehre. Von Hans Januschke. 



Pp. X -I- 456. Medium 8vo. (Leipzig : B. G. Teubner, 



1897.) 

 ' I ""HOSE who are engaged in teaching applied m.athe- 

 -*- matics cannot fail to appreciate the wide advan- 

 tages arising from according greater prominence to the 

 principle of conservation of energy than it obtained in 

 the text -books of the last generation. Unfortunately, 

 however, this principle, when stated in the restricted 

 form in which it is most easily understood — viz. the mere 

 assertion of constancy of the total energy, kinetic and 

 potential, of a material system — is insufficient of itself 

 to determine the actual motion of systems with more 

 than one degree of freedom, and, moreover, cannot be 

 applied to find the passive reactions arising from con- 

 straints. This particular point has been brought out 

 forcibly m the recent controversy on "energetics" in 

 which Boltzmann, Planck, Helm and others have taken 

 part. Some further assumption or generalisation is 

 necessary ; either the principle of physical independence 

 of force, or the extension of the principle of energy to 

 virtual displacements {i.e. the principle of virtual work), 

 or the hypothesis that the equation of energy holds good 

 separately for every particle of a material system for the 

 components of motion in every direction, or the assump- 

 tion of the variational equation, or the principle of least 

 action ; all these alternatives are practically equivalent, 

 and enable us to construct an energy theory of dynamics. 

 Only quite recently Prof. Boltzmann, writing in Wiede- 

 mann's Annalen, suggested the possibility of building up 

 the equations of motion, first of rigid bodies, and then 

 of fluids and elastic solids, from the principle of energy 

 aided by suitable subsidiary hypotheses ; and the present 

 volume is interesting as showing how this method works 

 out when applied to a somewhat elementary text-book. 



We fear that Herr Januschke hardly emphasises 

 sufficiently the subtle difference between the restricted 

 form of the principle and these necessary generalisations. 

 At any rate, his deduction of d'Alembert's principle 

 (p. 42) strongly reminds us of Clerk Maxwell's proof of 

 the Lagrangian equations (" Electricity and Magnetism," 

 vol. ii. § 561), the fallacy in which has been pointed out 

 by Prof. J. J. Thomson. If we differentiate the equation 

 of energy 



W^ + 2(F/4 + Imv"^) = constant 



NO. 1465. VOL. 57] 



with respect to the only independent variable, the time 

 /, where dAjd^ = v, we obtam 



_/ „dk , dv\ 

 2 F , - + mv-- 1 = 0, 

 \ dt dt ) 



whence, by taking account of the fact that (according to 

 Herr Januschke) dh and dv are of opposite signs, it is 

 possible to write down the equation 



iY-J-^yh^o. 



But here dh = vdt, and so the variations dh are not all 

 independent, but their ratios are connected by relations 

 of the form 



dh]^ __ dh^ a%3 _ 



Z/j Z/g v^ 



There is here no justification for the inference that 

 such an equation will hold for variations other than 

 such as are connected by this relation, i.e. for displace- 

 ments other than those the body really undergoes. As 

 regards the change of sign mentioned above, comment 

 is superfluous. 



In treating projectiles, the principle of independence 

 of motions is assumed, and with this aid no difficulty 

 occurs. But we naturally pass on to the treatment of 

 " centrifugal force " as a more crucial test of the energy 

 method, and here we find the result obtained either by 

 a wholesale disregard of algebraic signs, or at any rate by 

 what appears to an ordinary reader as such. Taking two 

 particles ;«iand m^ connected by a string of length r^-\-r., 

 and revolving in circles of radii r-^ and rg, the kinetic 

 energy is 



For equilibrium the author writes down from this : 



a?W = vi^v^dv^^ — m^^v^ = o 



(why the sign of the second term should be changed is 

 not obvious). Hence he infers that the tension in the 

 string is 



dv^ 

 '~dr 



dVn 



dr 



Putting V = rw where %v is the angular velocity, he gets 

 for the centrifrugal force 



_ WiZ// 



;«2V 



This result is obtained on the supposition of 71 being, 

 directly proportional to r. But in the theory of central 

 orbits, it is known that unless work is done on the 

 particle by tangential forces, the angular momentum, 

 and not the angular velocity, is constant, and hence v 

 ought to be taken inversely proportional to r, which 

 w^ould reverse the sign of the result. Hence Herr 

 Januschke's method really makes the normal accelera- 

 tion tend in the wrong direction. 



After deahng with rigid bodies, the equilibrium and 

 motion of fluids are considered ; but this portion-'does not 

 extend to the general equations of hydrodynamics (where 

 we should have most liked to see how the "energetic" 

 method works out), Torricelli's theorem and the hypo- 

 thesis of parallel sections being alone considered. Gases 

 follow next, then a chapter on " molecular forces " deahng 

 with elasticity and capillarity. Chapter v. deals with 

 heat, and includes a very fair exposition of the first and 

 second laws of thermodynamics, the subject being openedj 



