2Cv2 M. C. Szily on the Dynamical Signification of the 

 or, inserting the mean values, 



i.SQssSm^&p+y^+^J +S(2.iT). . • (I.) 



This equation is, as remarked by Sir William Thomson in 

 c A Treatise on Natural Philosophy,' i. p. 233, valid for every 

 system wider all circumstances. 



Let us now investigate the signification of the here occur- 

 ring quantities. 



SQ is the integral of — ~ — between the limits and i ; and 



since — r — denotes the energy which the body receives from 



without during the time dt, its integral will denote the total 

 energy which it receives from without during the time of a 

 pulsation, viz. the energy imparted at the infinitely small 

 variation of state. 



The first sum on the right side of the equation, 



refers to t = — that is, to the first time-element of the state- 

 variation ; the second sum, 



Sm(/fe + . . . ), 



refers to t = i — that is, to the last time-element of the same. 

 Between these two instants lies just one period of the un- 

 changed body ; hence, as long as the state of the body does not 

 change, its phases at the instants and i are the same ; conse- 

 quently 



V { = V , T t =%, U i= U , 

 and 



U=t + i ; 

 therefore 



dti=dt , ST i= =ST , au^sUo. 



Further, the external works performed in the time-elements 

 dti and clt (the same infinitely small variation of state start- 

 ing from the same phase) will be likewise the same ; that is to 

 say, 



SWjdU = SW .dt 

 i i 7 



and hence 



sw t -=sw . 



Premising these, let us see the dynamical signification of 

 the sum 



£m(a/&e + . . . ). 



