2(5 M. C. Szily on the Deduction of the Second Proposition 

 and, according to equation (2), 



SU=-Sm( ( / / ^+/% + c^) = 0. . . . (2 A) 



By forming tho two latter equations we readily obtain a re- 

 lation between the time-variations and the variations of the 

 coordinates; for if (1a) and (2a) be multiplied by dt and 

 added, we shall have 



'Zm(a/§al+. . . r— «"&e-~. . . .)dt = 0, 



or, in accordance with the signification of x' and a/ f t 



2m(dx$x / + >. . ,-dx'Bx-. . . .) = 0. 

 Now 



da/h = das(a/dx) — a/d&% ; 



consequently 



%m(dxh/ + ....+ a/Ux + ....) = c&ntQSfe + ....). 

 Further, 



da;$a/ + x'Mx = B(x'dx) = h(x' 2 dt) ; 



consequently 



d2,m(x'8x + ....) = S(2Tdt). 



As long as the state of the body remains constant, T also is 

 constant ; consequently, after indefinite integration, 



^m(x / Bx + ....) = 2T8t + constant. 



But now, for St = 0, 



Bx = ; 



consequently 



^m(x / Bx+^8i/ + z f Bz) = 2TBt (3) 



Equations (1), (2), and (3) remain valid so long, and only 

 so long, as the state of the body is unaltered. 



II. 



We will now pass to the consideration of the changes of 

 state. 



Let the initial state of the body (the state before the change) 

 be defined by the constants f and rj Q , Let the coordinates of 

 any particle m in the instant £ = be x , y , z ; the velocity- 

 components, x f , y' 0y z'o ; the total vis viva, T ; and the total 

 potential energy, U . If the state of the body were to remain 

 unchanged, then equations (1), (2), and (3) would continue 

 to hold good. 



But now, from the time £ = onward, we will convey to the 

 body successive quantities of energy. In consequence of this, 



