156 



SCIENCE 



[N. S. Vol. XXVIII. No. 709 



tribution of force producing local accelera- 

 tion. The connection of equation (3) with 

 the lost forces of d'Alembert is visible at 

 once. 



3. Separate the forces to which magnitude 

 may be assigned arbitrarily from those whose 

 magnitudes are fixed by conditions of the 

 system like displacement, velocity, accelera- 

 tion. Call the former group {A) and the 

 latter (iS). Then the form of equation 



A = S-\-mx (4) 



makes the second member a function of ele- 

 ments specified for the system, while the first 

 member is independent of such elements. 

 Such a segregation is convenient for mathe- 

 matical handling of the differential equation, 

 but {A) and {8) are both external forces, in 

 the original sense of that term. We need, 

 perhaps, to remind ourselves of this fact, when 

 we find {A) alone described as external, in 

 opposition to " forces exerted upon the system 

 by itself," or inner forces.' 



4. The eSects of a force-aggregate (X) be- 

 ing in general to bring about changes of mag- 

 nitude in some momenta, and of direction in 

 others, that separation of results may be indi- 

 cated by the notation in both members of the 

 equation of motion, giving 



X=M+D = mxM + fn%D. (5) 



According to that supposition, then, 



X — D = X — mxD = mxM=M. (6) 



One reading of equation (6) carries out the 

 separation referred to; it measures explicitly 

 the force devoted to producing change of mag- 

 nitude in momentum. Another legitimate 

 interpretation connects the change in force 

 from (X) to (X — D) with a definite change 

 of reference system. But alongside of these 

 we find surviving still a third, to the effect 

 that (M) is the real force-total in this case 

 (retaining the reference system and mass \m- 

 changed), resulting from the combination of 

 (X) with centrifugal force. A similar un- 

 clearness allows the " centrifugal couple " of 

 Euler's equations to masquerade as an external 

 force-moment. These forms of confusion are 



' See, for instance, Abraham and Fiippl, " Elek- 

 trizitat," Vol. 1, p. 195. 



reasonably looked upon as survivals from the 

 days when the process of vector addition to 

 momentum by force was grasped less com- 

 pletely. The changes in direction seemed al- 

 most a side issue, to be deducted before pro- 

 ceeding to the serious measurement of force. 

 We still find the thought followed without 

 flinching to the case where (71/) happens to be 

 zero, and leaves " equilibrium " between (X) 

 and (D).' 



The significance of such current forms, 

 which may justify citing them in the present 

 connection, lies in the mingling of force-terms 

 and mass-terms common to them all. This 

 encourages an undiscriminating attitude trans- 

 ferred from the field of mathematics, toward 

 the terms included in equated expressions, 

 which may easily obliterate certain phases of 

 physical thought. To inquire whether a par- 

 ticular distinction of this sort is profitable is 

 one way of exercising discrimination. It is 

 proposed to raise this question presently,' as 

 regards mass-term and force-term, especially 

 where those conceptions are employed with the 

 wider meaning of recent usage. We may ad- 

 vance toward that end by considering first the 

 form into which d'Alembert's principle is 

 thrown, in preparation for the equation of 

 virtual moments. 



2(AX)b— 2(AmS) = 0. 



(7) 



How is this to be understood from the phys- 

 ical point of view ? If their original meaning 

 is attributed to the summations, and equation 

 (7) is nothing but a transposition of equation 

 (1), the second sum can not represent forces 

 actually applied to (m), since by supposition 

 these are accounted for completely in the first 

 sum. Neither can this be an equilibrium 

 equation for the mass (wi), so long as the 

 second sum does not vanish. D'Alembert, 

 however, detected in 



— 2(Ama;) 







a new sense, by associating it with the other 

 force-terms as their equilibrant. Or, foUow- 

 • Goodman, " Mechanics," p. 204 ; cf. Klein imd 

 Sonmierfeld, " Theorie des Kreisels," p. 141, etc. 

 These instances do not fatand alone. 



