To distinguish this period thus defined from the former arbi- 

 trary one, we will designate it by i ; then 



r 



30 M. 0. Szily on the Deduction of the Second PwpQiition 



riod thus defined from the former ar 

 nate it by i ; then 



$e.<fc=i8Qj (7) 



and 



28Q=2Sr(T~T 1 + T )^; 



or, T denoting the mean value of T during the time i } and 

 putting for shortness 



T-Ti + To^C, 

 we have 



and hence 



^=28 kg (tit) (8) 



This equation agrees perfectly, in form, with that to which, 

 in a former memoir*, I reduced Hamilton's principle. I then 

 started from the hypothesis that 



Sw( l r / 1 S l r 1 + . . .)=Sm(#yta? + ...). 



If we drop this hypothesis, we see that the quantity C takes 

 the place of the T which occurs in the equation mentioned. In 

 the present deduction the other limiting assumption is also set 

 aside, viz. that the acting forces have a force-function. 



Equation (8) specifies how much energy must be conveyed 

 to the body, in order that, instead of the path ?7 =/(?)> it may 

 follow another deviating indefinitely little from this. The 

 same equation (8) we also apply to the case in which we seek 

 the amount of the energy which must be conveyed to the body 

 in order that it may pass out of a given state (j??) into another 

 state infinitesimally different (f + tf?, rj + drj). In correspond- 

 ence with this, we substitute for the symbol of variation the 

 differential-symbol, and write 



^=2cnog(«). ....... (9) 



So long as the body continues in the same state, r/Q = ; 

 and 



C=T-T + T=T. 



C being constant, i must also be constant. For each determi- 

 nate (f 17) state of the body, C and i have a certain determinate 

 value, which remains constant as long as the body is in the 



* Pogg. Ann. vol. cxlv. p. 300 ; Phil. Mag. [IV.] vol. xliii. p. 339. 





