statistical mechanics 27 



20. Let us now examine how the other integrals oï motion behave at the 

 collisions. The integrals of areas in the problem of two bodies are the following, 

 expressed in absolute coordinates: 



[Ui ^l^x — ^'i ^i) + '"2 (.'/a "'2 — '^2 ^2) = ^1 



where c^, c, , are the constants of areas. 



Are these integrals valid at a collision between and ? 

 In examining this question we denote by 



til t t I 



the coordinates after the collision. 

 Consider now the expression 



c/ = '>^h il/i' ^1' — ^1' ^1') + ^^2 iVi' ^^2' — ^2' ^2') 

 and compute its value immediately after the collision. 

 Then we have 



^1' = «/i =^2 = ^2'' 

 ^1' = "I = ^2 = ■^2'' 

 wliereas m,', ^t^', etc. are given through the formulae (40). 

 Consequently we get 



, , i 'f^l + "*2 ^^2 '"1 ^1 + »"2 *^2 



+ ^■^^ ^ 1 1- I _j_ v^i 2 1 2' 



OTj + ^"^^ ^ ' 1^ ' H- 2 2 2' 



and 



^2 {P2' < — < O = _|! (^2 '"^2 — ^2 î^s) + !^ 4, '""i ~ ' 



so that 



w«i (2/1' — ^1') + ^"2 iVi ^2 — ^2 '«^2') = 

 = [y^ — .2^ w J + W2 [y, tv,^ — 2,^ v.^ ) = . 



Hence the theorem: The integrals of the areas are unaltered at the collisions. 



21. Consider next the energy integral. 

 We get from (40) 



m, u,'^ = ^ — Tg- [m/ -t- + ^2 , 



'1 ""1 



(m^ + Wg) ' ^ 



