of the Mechanical Theory of Heat from the First. 27 



the vis viva and potential energy of the body will be augmented, 

 and, in general, external work will be done. In the measure 

 in which we add more and more energy to the body, the state 

 becomes also more and more changed. Assume that the series 

 of successive changes is regulated in such manner that the con- 

 dition-variables satisfy at every instant the equation ?;=/(£), 

 or, in other terms, that the succession of the changes of state 

 (say, the path of the state) is given by the function/. We will 

 continue the importation of energy until the condition- variables, 

 corresponding to the relation 97 =/(!;), take the values % x and 

 wfc, Let the quantity of imported energy be then Q. With 

 the cessation of the importation of energy change of state 

 ceases also ; henceforward the variables f x and rj 1} and with 

 them the vis viva T x and potential energy Ui, retain their pre- 

 sent values. 



We have, consequently, conducted the body in a definite 

 manner from the initial state (£ *7o) m ^o the final one (fi^i), 

 and meanwhile conveyed to it the energy Q. Given the initial 

 state (fo^o)? fi na l state (fi^i), and the path of the changes of 

 state, the total energy Q to be conveyed to the body is also un- 

 ambiguously determined. Whether the importation of energy 

 takes place more quickly or more slowly, the sum of the im- 

 ported energy remains unchanged, if only the initial point, the 

 end-point, and the nature of the path remain unaltered ; in 

 other words, the initial and final states and the path of the 

 changes completely determine the total energy Q to be imported; 

 while the duration of the transition is not determined thereby. 

 With the same Q the duration of the transition, corresponding 

 to the mean velocity of the importation of energy, may take 

 any value whatever between andco . Let the duration be 

 denoted by t. The length of this duration, or the value of t, 

 depends entirely upon how we fix the mean velocity of the im- 

 portation of energy ; consequently the choice of what the 

 magnitude of t shall be, is entirely at our disposal. 



Let us now conduct the body from the same initial state 

 (fo^o) h 1 ^ the same final state (fi^i), but so that the present 

 series of changes (the second path of the state) shall be only 

 infinitesimally different from the just-described series (the first 

 path) — that is, so that the form of the function / in the equa- 

 tion 77=/(f) undergoes an indefinitely little alteration. The 

 importation of energy now commences not at the instant £ = 0, 

 but the infinitesimal time St later. Hence the coordinates of 

 the particles m at the commencement of the present change of 

 state will be no longer x Q , y , z , but different by &i' , Sy , 8z ; 

 and the same will be the case with the components of the velo- 

 city and acceleration. But during this time St the state of the 



