28 M. C. Szily on the Deduction of the Second Proposition 



body lias remained unchanged ; so that equation (3) still re- 

 tains its validity ; consequently 



2m / (x / Bx +y / 8ij + z' Bz ) = 2T Bt (4) 



At the end of the time Bt the conveyance of energy com- 

 mences, corresponding to the former in such sort that the body, 

 after an equal period t, arrives at the same final state as by the 

 first path. Let the total quantity of energy now communi- 

 cated be accordingly Q + SQ. 



On the second path, where the change of state began St 

 later, it will also end St later — that is, at the instant t + St. 

 At this instant the coordinates, velocities, and acceleration- 

 components are not the same (x x . . . , x\ . . . , x'\ . . . ) as they 

 were at the end of the first transition, at the instant t, but dif- 

 fering from them by Bx x . . . , Bx\ . . . , and ha/ f . But now, on 

 the one hand, ,i\ . . . , and, on the other, x l + Bx Y . . . represent 

 the coordinates of the same particle in the same state, and only 

 so far differ from one another that the first correspond to the 

 instant t, and the latter to the instant t + Bt. Therefore equa- 

 tion (3) is applicable to these variations also ; that is, 



^m(x\Bx 1 + I/ \S I/l + z\Bz 1 ) = 2T 1 .Bt. . '. . (5) 



III. 



Equations (4) and (5) refer only to the commencement and 

 termination of the change of state, and cannot be applied to 

 the entire course of the transition. We must seek another re- 

 lation, one which remains valid at any and every stage of the 

 alteration. 



We will assume that on the first path the body, during the 

 time t, has arrived at a certain state (f^), which we will briefly 

 designate by the name " state Ax." In this state let the total 

 vis viva be T, the potential energy U, and the coordinates, ve- 

 locities, and acceleration-components of the particle m be x . . . . 

 x f ..., and x" . . . . 



On the second path the body comes in the course of the time 

 t (therefore in the instant T + Bt) into a certain state (f + Sf, 

 t) + Bij), which we will name "state A 2 ." The quantities be- 

 longing to A 2 differ only by their variations from those which 

 belong to A^ The particle m is now not in the position x . . 

 but removed Bx from it ; and in like manner the velocity- and 

 acceleration-components have suffered an infinitesimal variation 

 Bx r . . . , Bx". 



Let us now determine how much energy must be commu- 

 nicated to the body to conduct it, following the path denoted 

 by the displacements Bx, By, Bz, out of state Ai into state A 3 . 



