108 Mr. W. J. M. Rankine on the General Law of 



in the formula (1), which is transformed from the actual to the 

 potential form in consequence of the change of state dV, 

 The sum of those three quantities is as follows : 



rf.Q=(l + L)rfQ+Q.^.<nr. ... (2) 



If from this expression be subtracted the potential energy 

 developed, that is to say, given out in overcoming resistance, 



the result will be the algebraical sum of the energies, actual and 

 potential, acquired on the whole by the substance in passing 

 from the total actual energy Q and state V to the total actual 

 energy Q + c?Q and state \ ^dY ; viz. 



rf^=rf.Q-^.<;V=(l+L)</Q+(Q^-l)^.<iV. . (B) 



Now this quantity must be the same, whether the change dQ 

 or the change dY be made first, or both simultaneously ; other- 

 wise by vaiying the order of making those changes, the sum of 

 energy in the universe, actual and potential, might be changed, 

 which is impossible. Therefore the above expression must be 

 the complete difi'erential of a function of the energy Q and state 

 V ; that is to say, 



rfV-(/Q\^^Q ^)dY~'^d(^^' dY' 

 consequently 



L=/.(Q)H-Q.g, 



/'(Q) being a function of Q only, to be determined by experi- 

 ment, of which /(Q) is the primitive. 



Thus we obtain, for the total energy, actual and potential, 

 acquired by the substance, in consequence of the changes of 

 total actual energy from Q to Q + dQ,y and of state, from V to 

 V-t-rfV, the formula 



^=rf.Q-^.U=(l+/.(Q) + Qg)<;Q+(Q^-l)^.rfV 

 =<^.{q+/(Q)h-(Q,VO/K-'^}' 



in which the symbol d,V is used to denote the total potential 

 energy really developed ; while / -^g- . dY is a partial integral 

 computed for each value of Q, as if that value were constant. 



(3) 



