January 19, 1900.] 



SCIENCE. 



is to him that we owe the equations of mo- 

 tion of a particle of a ' perfect fluid.' This 

 is an ideal fluid, moving without friction, 

 or subject, in technical terminology, to no 

 tangential stress. But while no such fluids 

 exist, it is easily seen that under certain 

 circumstances this assumed condition ap- 

 proaches very closely to the actual condi- 

 tion ; and, in accordance with the method 

 of mathematico-physical science in untang- 

 ling the intricate processes of nature, prog- 

 ress has proceeded by successive steps from 

 the theory of ideal fluids toward a theory 

 of real fluids. 



The history of the developments of hydro- 

 mechanics during this century has been very 

 carefully and completely detailed in the re- 

 ports to the British Association for the Ad- 

 vancement of Science of Sir George Gabriel 

 Stokes,* in 1846, and of Professor W. M. 

 Hicks,t in 1881 and 1882. And the his- 

 tory of the subject has been brought down 

 to the present time by the address of Pro- 

 fessor Hicks before Section A of the Brit- 

 ish Association for the Advancement of 

 Science in 1895, and by the reportj of Pro- 

 fessor E. W. Brown to Section A of the 

 American Association for the Advancement 

 of Science in 1898. It may suffice here, 

 therefoi'e, to glance rapidly at the salient 

 points which mark the advances from the 

 state of the science as it was left by La- 

 grange a hundred j^ears ago. 



The general problem of the kinetics of a 



* ' Report on recent researches in hydrodynam- 

 ics, ' Eeport of British Association for the Advance- 

 ment of Science for 1846. 



t ' Report on recent progress in hydrodynamics,' 

 Reports of British Association for the Advancement 

 of Science for 1881 and 1882. 



X ' On recent progress towards tlie solution of prob- 

 lems in hydrodynamics,' Proceedings of American 

 Association for the Advancement of Science for 1898. 

 See also Science, November 11, 1898. 



Reference should be made also to Professor A. E. 

 H. Love's paper ' On recent English researches in 

 vortex-motion, ' in the 3Iatliematisclie Annalen, Band 

 XXX., 1887. 



particle of a ' perfect fluid' is easily stated. 

 It is this : * given for any time and for any 

 position of the particle its internal pressure, 

 its density, and its three component veloc- 

 ities, along with the forces to which it is 

 subject from external causes; to find the 

 pressure, density, and velocity components 

 corresponding to any other time and to any 

 other position. There are thus, in general, 

 five unknown quantities requiring as many 

 equations for their determination. The 

 usual six equations of motion, or the equa- 

 tions of d'Alembert, contribute only three 

 to this required number, namelj"^, the three 

 equations of translation of the particle, 

 since the three which specify rotation 

 vanish by reason of the absence of tan- 

 gential stress. A fourth equation comes 

 from the principle of the conservation of 

 mass, which is expressed by equating the 

 time rate of change of the mass of the 

 particle to zero. This gives what is techni- 

 cally called the equation of continuity. A 

 fifth equation is usually found in the law of 

 compressibility of the fiuid considered. f 



ISTow, the equations of rotation, as just 

 stated, refuse to answer the question 

 whether the particles proceed in their 



* The statement here given is that of the ' historical 

 method,' -vvhioh seeks to follow a particle of fluid 

 from some initial position to any subsequent position 

 and to specify its changes of pressure, density and 

 speed. What is known as the ' statistical method, ' 

 on the other hand, directs attention to some fixed vol- 

 ume in the fluid and specifies what takes place in that 

 volume as time goes on. 



t The five equations in question are 



1 3p 



dl p 3j;' 



dt p dy' 



dio 



■0, 



P=f{p)\ 



= z- 



dt ' p dz' 

 in which p is the pressure and p is the density at the 

 oentroid {x, y, z) of the particles ; V is its volume ; 

 «, V, w are its component velocities ; and X, Y, Z 

 are the force components per unit mass arising from 

 external causes. 



