characteristic Quantities occurring in Central Motions. 5 



2. As for my second equation, I will here quote it only in the 

 form which it has for a single movable point, since that form is 

 sufficient for the following considerations, and the extension of 

 it to a system of any number of points would require analyses 

 of too great a length. 



This equation stands in connexion with those which express 

 the proposition of least action and Hamilton's amplification of 

 the same ; yet in an essential point it is different from them, as 

 will be immediately evident when I put here the two latter 

 equations, so that all three may be seen together. 



Let a material point move freely, under the influence of a 

 force, from a given initial point to a given final point. Then, 

 instead of this motion, imagine another motion of the material 

 point taking place between the same two limits in a path infini- 

 tesimally changed. If, now, the force acting on the point has a 

 force-function or (as I have proposed to name it) ergal, and if it 

 is further assumed that in both motions the ergal is represented 

 by the same function of the space-coordinates, and that the 

 energy (the sum of the ergal and vis viva) has the same value in 

 each, then the following equation (known as the expression of 

 the proposition of least action) holds : — 



8§v 9 dt = (3) 



If the time which the point requires for its motion be denoted 

 by i, then this equation can also be written thus : — 



8(c*i)=0 (3 «) 



Now this equation is valid even when the limits between 

 which the changed motion takes place are not the same as with 

 the original motion, provided they fulfil the condition that the 

 quantity 



doc ~ dy ^ dz ~ 



has the same value at the end of the motion as at the beginning. 

 This condition is fulfilled, for example, when both motions are 

 in closed paths and in each motion an entire revolution is con- 

 sidered, so that the point where the motion ends coincides with 

 the point where it begins. 



If, on the contrary, this condition relative to the limits were 

 not fulfilled, for (3 a) the following equation would have to be 

 substituted, 



m -(£*+ 1 * + % *) -g b+ 1% + * *), m 



in which the indices and , signify the initial and the final value 

 of the bracketed terms. For the sake of simplicity we will, in 



