July 31, 1908] 



SCIENCE 



155 



member of the equation then measures the 

 remainder of effective force only, and exhibits 

 the necessary magnitude of the equilibrant 

 that would change the conditions of the prob- 

 lem from those of acceleration to those of 

 equilibrium, or zero acceleration. The " re- 

 versed effective force," if superposed upon 

 the forces actually operative, says d'Alembert, 

 would prevent the actual accelerations, and 

 bring about equilibrium that did not in fact 

 occur. This conception of equivalence be- 

 tween the differing modes of statement in the 

 two members of such equations is prominent 

 with d'Alembert and Lagrange, and entirely 

 in accord vnth out every-day use of equationa 

 of motion to evaluate any one of the three 

 quantities force, or mass, or acceleration, when 

 the corresponding values of the two others are 

 known.^ The advance made by d'Alembert, 

 therefore, is in the direction of devising a 

 static measure for unbalanced forces by gen- 

 eralizing the procedure when we determine 

 weight active by hanging a body from a spring 

 balance. It is parallel to the zero method of 

 the laboratory, that seeks the measure of any 

 unknown quantity in terms of independent 

 conditions adjusted to compensation of its 

 effects. This point of view sets in a proper 

 light the limited sense in which d'Alembert's 

 principle brought dynamics within the scope 

 of statical equations, and disposes effectually 

 of the obscurity or confusion involved in 

 " forces of inertia," or the recently substituted 

 term " kinetic reaction." The extension of 

 d'Alembert's principle to modern generalized 

 dynamics does not modify essentially this con- 

 ception of the method; we are still dealing 

 with relations between force and inertia — the 

 doing of work, and the quality of storing 

 energy in a particular way. Clear thought 

 in a new field is not furthered by meeting a 

 paradox at its threshold; for nobody accepts 

 literally the dictum that finite acceleration is, 

 ' D'Alembert's " force of inertia " is merely a 

 loose expression for (m) ; it does not denote 

 ( — mih) . Lagrange uses the phrase " force re- 

 sulting from inertia" as describing {m'cc) , with 

 unchanged sign. See d'Alembert, " Traitg de dy- 

 namique," ed. 1758, p. x; Lagrange, "Mgcanique 

 analytique," ed. 1853, Vol. 1, p. 282. 



as a general statement, consistent with zero 

 values of force, and force-moment, applied to 

 a given system that has inertia. 



Equation (1) may be recast mathematically 

 in several ways; and some of its equivalents, 

 being adapted more closely to certain aspects 

 of physical thought, are obviously helpful as 

 well as legitimate. But for clearness the 

 name " equation of motion " shall be confined 

 here to the above primary mode of formu- 

 lating the idea. This was adopted by the old 

 masters as segregating causes from results, 

 terms of each class appearing by themselves in 

 one member of the equation. We may de- 

 scribe these as " force terms " and " m.ass- 

 terms " respectively. So soon as homogenous- 

 ness in this sense is disturbed, the equation is 

 altered in prima facie physical meaning. 

 Even removing terms from one member to the 

 other ; so that a force- term is now interpretable 

 as a mass-term, or vice versa; may be re- 

 garded as passing to a new problem, concerned 

 with different masses, or modified forces, or a 

 new classification of the effects of force. 

 Some typical instances are the following, 

 purposely taken on familiar and elementary 

 ground : 



1. Denoting by (P) and (B) the aggregates 

 of positive and negative external force, re- 

 spectively, thought of as acting on a single 

 mass (m), for simplicity, we have theT;ype 



P = R-\-mx. 



(2) 



Here the negative forces have been transferred 

 to the second member, and the equation now 

 expresses directly the fact that the forces (P) 

 overcome the resistances (iJ), and produce 

 acceleration as well. {R) may represent dis- 

 sipative or conservative agencies. If the 

 latter, equation (2) is preliminary to express- 

 ing storage of energy in both forms. 



2. Subtracting {R) from both members of 

 equation (2) gives 



P — B=(B — B) +mbc. 



(3) 



This puts to the front the idea that the total 

 force (P — R) sets up static stress (=ti2) to 

 an extent determined by the resistances, the 

 remainder becoming effective as a volume dis- 



