6 Prof. R. Clausius on the Relations hetween the 



the following, always consider that condition fulfilled, and ac- 

 cordingly put the difference which stands on the right-hand side 

 of the above equation =0, so that the equation will retain the 

 form (3«). 



The amplification introduced by Hamilton consists in the same 

 value not being attributed to the energy in both motions, a 

 change of energy being considered admissible, while yet the ergal 

 with the changed motion must still be the same function of the 

 space-coordinates as with the original motion. The equation 

 given by Hamilton for this case* is the following, in which E 

 signifies the energy : — 



mh{vH)=i$ft (4) 



The earliest form of my equation f is 



-(X8ff + Yfy + Z&8r)=g&;* + »w 8 81og*. . . (5) 



In this equation the force acting on the point is not limited by 

 any condition. Let us suppose, as before, that the force has an 

 ergal, and designate it by U, fixing at the same time the positive 

 and negative directions of the ergal so that the sum of the vis 

 viva and the ergal during the motion is constant, then the equa- 

 tion becomes 



-T-&e+-7-SyH — t-oV= — Sv^ + mv^Slogi; . (5 a) 

 ax dy * dz 2 



here, however, it is not assumed, as in Hamilton's equation, 

 that the function (designated by U) of the space-coordinates is 

 invariable, but this function may, on one motion passing into the 

 other, undergo a change independent of the alteration of the co- 

 ordinates. Let us imagine, for example, that the function con- 

 tains any quantities whatever independent of the coordinates and 

 therefore constant during the motion, these constants, in order 

 that Hamilton's equation may hold,- must have the same values 

 when the motion is changed as they had with the original mo- 

 tion. On the contrary, this is not necessary for the validity 

 of my equation, but with the transition from one motion to the 

 other the constants may change their values. This gives to the 

 sum 



dV s dV fi dV . 

 -jr- ox + -7- by + -y- bz, 

 dx dy * dz ' 



the mean value of which occurs in my equation (5 a), a peculiar 

 significance. We cannot replace it by the symbol 6TJ, if by this 



* Thomson and Tait, ' Treatise of Natural Philosophy/ p. 235. 

 t Sitzungsberichte der Niederrhein. Gesellsch.fur Natur- und HeilJcunde, 

 18/0, p. 174; Phil. Mag. S. 4. vol. xlii. p. 167. 



