Mechanical Equation to the Motion of Material Points. 329 



tions and repulsions which the moving points exert on one an- 

 other, and which, according to any law, depend on the distance, 

 it is known that the ergal can be expressed very simply. If the 

 force which two points whose masses are m and m x exert on one 

 another at the distance r is represented by mm$ (r), in which a 

 positive value of the function corresponds to an attraction — and 

 if, further, 



then the ergal is determined by the equation 



U = Xmm^ (r) , 



in which the summation embraces all the combinations of the 

 given points in pairs. Accordingly, for this case the preceding- 

 equation is transformed into 



VZmm^ir) = 5) -q- W + Smv 2 8 log i. . . (40) 



We will now assume, specially, that only two material points 

 are given, the masses of which are m and m lt and which move 

 about each other under the influence of their mutual attraction. 

 In this case, if we denote by letters to which an index is attached 

 all the quantities which relate to the second point, we can write 

 the preceding equation without signs of summation, thus : — 



mmfifyir) = — Sv* + ~ Bv'l + mv^S log i + m^lS log i v 



Since, however, in such a motion of two points about each other, 

 both points have the same period, i^ssf, and hence the last two 

 terms can be combined. By bringing at the same time the first 

 two terms on the right-hand side under a common sign of varia- 

 tion, we can write : — 



mmficf) (r) = \h(mv* -f m^) + (mv 2 + m^S log i. . (41) 



To this equation we can give a still more simplified form. 

 For this purpose the relative velocity u of the two points may 

 be introduced ; it is determined by the following equation : — 



2 _ (dx _ dx\ 2 (dy _ dytf + ( d l__ &V f 42^ 

 u -\dt dt) + \dt dt) + \dt dt)" { **' 



But now it may be readily seen by resolution of the bracketed 

 terms that the following equation holds good : — 



(dx dxA* , . r /dxV , /dxtfl ( dx dxA 2 



On the condition we have supposed, that the two points move 



