Makoh 8, 1907] 



SCIENCE 



385 



dynamical principles of d'Alembert, Hamilton 

 and Jacobi, expressed in the familiar way; 

 and concludes by discussing certain well- 

 known propositions relating to holonomic sys- 

 tems. 



Chapter II. — Ein allgemeiner Satz, die Be- 

 wegung einer reibenden Fliissigkeit betref- 

 fend, nebst einigen Anwendungen desselben. 

 Proceeding from the fundamental equations 

 of motion and the equation of continuity of 

 a viscous incompressible fluid, the author de- 

 rives a general reciprocal theorem expressing 

 a relation between any two states of the fluid 

 satisfying the fundamental equations of mo- 

 tion. This theorem is shown to be capable of 

 very extensive application, and is shown in 

 operation through the consideration of several 

 problems. 



Chapter III. — Ueber die Entstehung turbu- 

 lenter Eliissigkeitsbewegungen und iiber den 

 Einfluss dieser Bewegungen bei der Stromung 

 durch Eohren. In this chapter is found a 

 discussion of the motion of a viscous fluid 

 when the velocity has reached a value which 

 has been called by Osborne Reynolds the ' crit- 

 ical velocity.' After this velocity has been 

 reached turbulent motion may arise. The 

 general equations of motion for an incompress- 

 ible viscous fluid furnish the starting point in 

 this investigation. The question as to whether 

 or not turbulent motion arises is referred to 

 that of the stability or instability of the regu- 

 lar motion. Upon the regular motion small 

 variations are imagined superimposed, the 

 kinetic energy of which is found, and the time 

 derivative of this quantity is used as a cri- 

 terion of stability of the regular motion. 

 Several particular cases which submit readily 

 to extended discussion are then taken up, 

 among them being the case of stationary mo- 

 tion in a cylindrical tube. 



Chapter IV. — Les equations du movement 

 des gaz et la propagation du son suivant la 

 theorie cinetique des gaz. The discussion 

 given in this chapter was apparently insti- 

 gated by a criticism of Jochmann of the first 

 paper of Clausius on the molecular theory of 

 gases. In his criticism Jochmann remarked 

 that it appeared difficult on the views of 

 Clausius to account for the propagation of 



sound waves. Professor Lorentz points out 

 that Jochmann's difficulty arose from his fail- 

 ing to note that the element of volume with 

 which the mathematical physicist deals is not 

 infinitesimal in the rigorous sense of the word. 

 He then proceeds to a general discussion of 

 the subject, following in the main the methods 

 of Boltzmann. 



Chapter V. — Ueber die Anwendung des 

 Satzes vom Virial in der kinetischen Theorie 

 der Gase. The equation of van der Waals, 

 {p-{-a/v'){v — h) '=B(l-'rat), wherein p, v, 

 t are, respectively, the pressure, volume and 

 temperature of a gas and B, a, h are con- 

 stants depending on the nature of the gas, 

 is here the subject of discussion. This 

 equation is derived from the so-called Prin- 

 ciple of Virial first employed by Clausius. 

 The principle is expressed by the equation 

 2(Xa; -f- Ty -j- Zz) = — "Siimu', for the case 

 of equilibrium, wherein X, Y, Z are the com- 

 ponents of the total force acting on a molecule 

 of mass m whose center of mass is given by 

 the coordinates x, y, z, it is the velocity of the 

 center of mass of the molecule, and the sum- 

 mations are extended to all the molecules. 



Restricting himself to the case where the 

 density is not too great, the author is led to 

 the equation given by van der Waals. A sim- 

 ilar restriction was tacitly made by van der 

 Waals himself, who used, however, a somewhat 

 different method in deriving his equation. 



Chapter VI. — Ueber das Gleichgewicht der 

 lebendigen Kraft unter Gasmolekiilen. A 

 criticism is made of an assumption made by 

 Boltzmann in his treatment of the question 

 of thermal equilibrium of multi-atomic gas 

 molecules. Boltzmann's assumption is, that 

 in a collision between two molecules, whose 

 states of motion may be designated by A and 

 B, if A' and B' are the new states assumed 

 after impact, then, conversely, if the two mole- 

 cules have the states of motion A' and B' they 

 can after collision assume the states A and B. 

 The author takes exception to this general 

 assumption and proceeds to give his reasons 

 therefor. Then follows a simple proof that 

 in the case of mon-atomic gases the distribu- 

 tion of velocities given by Maxwell's law is 

 the only possible one. Finally a discussion of 



