August 7, 1908] 



SCIENCE 



181 



liquid, and acting on m. The term im^u, 

 then, is seen to express, in the first introduc- 

 tion of it, the kinetic energy associated with 

 the liquid as a necessary consequence of 

 moving m through it. The ratio of m^ to m 

 is calculable for various special assumptions. 

 Executing the differentiation with m, constant 

 gives directly 



X= {m + m,,)du/dt. (2) 



If we accept this as an " equation of motion," 

 just as it stands, and in the strict sense of 

 d'Alembert, it is obviously not such for m 

 alone, but for that mass plus liquid of con- 

 stant volume, it is true, but of varying iden- 

 tity. That feature of elusiveness in the mass 

 denoted by m^ has undoubtedly favored the 

 interpretation of the parenthesis as represent- 

 ing the " effective mass " of m under the con- 

 ditions, among which must be included that 

 X does not really comprise the total of ex- 

 ternal force acting on m, in conformity with 

 the suppositions underlying equation (1). The 

 completed equation of motion for m, in which 

 any resistance B — frictional or not — offered 

 by liquid must appear, is 



X — B = m du/dt. (3) 



yinee R = m^ du/dt, therefore, because du/dt 

 denotes the actual acceleration in both cases, 

 we have before us another instance of change 

 in reading, from mass-term to force-term, by 

 transposing in the equation. And, from the 

 point of view of equation (3), the power equa- 

 tion (1) can be adapted to the mass m ex- 

 clusively, by placing — Im^u' in the first 

 member, as the negative work of the force R. 

 As noted in connection with d'Alembert's 

 principle, each view is justified so long as the 

 proper context is retained, and we do not lose 

 sight of the mental device that harmonizes 

 them. A complete presentation includes both 

 views, and does not overlook, either, the possi- 

 bility of like alternative statement applying 

 to any equation of motion with corresponding 

 artificial basis. For example, if a mass m 

 is acted upon by forces X^ and X, that would 

 produce separately accelerations a^ and a,, it 



' See for example, Lamb, " Hydrodynamics," 

 p. 85, p. 130. 



is mathematically correct to write either form : 

 Xi -f- Xa =ma; Xi = ( m — mi)a; ( 4 ) 



if ffl = a^ -j- ffij is the actual acceleration for 

 the reference system used, and m^ = m a J a. 

 The " effective mass " of m, when the force 

 Xj is ignored (or unrevealed by first analysis 

 of the phenomena), would be greater or less 

 than m according to the sign of o.^ determined 

 by X^. The fiction indicated here would serve 

 no useful purpose in many classes of problems, 

 but it offers a certain convenience in treating 

 motions of bodies through media. The effect 

 due to inertia of the medium, or its equiva- 

 lent, finds adequate recognition by abolishing 

 the medium, and at the same time adding to 

 the inertia of the immersed body. The some- 

 what vaguely dispersed quality of the medium 

 finds definite location in the bulk of the body. 

 Wherever the circumstances are thus thor- 

 oughly understood, the matter of choice in 

 presentation is controlled completely by our 

 preference; it is enough that the equivalence 

 of two such modes of statement really covers 

 the points aimed at, the confessed fiction being 

 ranged with others like it in mathematical 

 physics. But it is clear that different types 

 of the external agencies called forces lend 

 themselves to calculation as pseudo-inertia of 

 the moving body itself with greater or less 

 facility, the change of front being easiest when 

 a resistance is involved whose magnitude is 

 proportional to the acceleration of the body, 

 as in the well-known hydrodynamical case 

 cited above. Another side of these differences 

 in the mathematical situation is the possibility 

 that they afford for making conditions of un- 

 ascertained physical nature reveal themselves 

 experimentally as arising from force rather 

 than from real inertia. Thus a resistance 

 proportional to displacement might be iden- 

 tified by adjustment to equilibrium, as in 

 stretching a spring, or charging a condenser; 

 " terminal velocity " is characteristic of .other 

 forms of resistance, which prove to be propor- 

 tional to various powers of speed. This second 

 group includes the obstructive electromotive 

 force of conductors to the passage of cur- 

 rent through them, beside the more visible 

 instances of such action. But a resietance pro- 



