654 Prof. F. Slate on Force- Transformation, 



activity may be favoured by physical evidence ; but on 

 whichever one the preference thus falls, equation (5) or 

 some simple equivalent shows how to calculate it in terms of 

 the other, with due aid from a correcting partial derivative. 

 Presented in forms like 



= m^—fi'c 2 = (m 1 v 1 '-+ic/i!)(y 1 + ic) 



= z(miv+ ifiv)(i'i + ic) [Real terms], (6)< 



equation (1) sets in relief, first the idea of conservation 

 (equal gain and loss at a transfer), and secondly the needful 

 pairing of each alternative momentum with its own velocity- 

 factor, when the complex product is expanded. All these 

 aspects of the above more comprehensive situation embrace 

 essentials of that correlation between Newtonian and " non- 

 Newtonian" dynamics upon which this discussion turns.. 

 Beside the frequent appearance of (///, fi) just referred to, 

 keeping the last members of equations (1, 5) somewhat at 

 the front, mathematical prominence is assured to them 

 through the Lagrange function and its derivative *. But 

 this must not exclude the third member of equation (5). 

 Not only was its type put forward earlier as a central 

 necessity of general statement f, but it happens to offer 

 also for the present phase rather direct contact with the 

 systematic use of " proper time " and of " local time " 

 j eculiar to relativity. This distinction is largely superfluous 

 for our Newtonian plan, since without according it a place, 

 the main dynamical relations resting upon it have been 

 reproduced. That composite scheme of time-variables must 

 be truly secondary, if it be indeed carried into the funda- 

 mental equations through a constancy of inertia made 

 primary. Yet the newer doctrine takes so seriously what 

 centres upon an entire parity of time and coordinate, that 

 more adequate review of these points is in place, for which 

 the activity (vxT{) opens the way. A step or two in broader 

 terms can be added, before limiting ourselves by the Lorentz. 

 electron's assumptions. 



Define now 7 _ 



m = m x z = in' , v = v\/z; 



then ^, .— 



Q'Ezm'v^z = m^z 3 ' 2 . 



* The conception of kinetic potential has this consequence. Some 

 special coincidences have shown themselves alreadv : (III.), p. 104,-. 

 (II.), eq. (47). 



t In(I.),eq. (10): (II.), eq. (3). 



