[continued from previous message]   
      
      F_son_on_ground - F_father_vs_son = m_son a                       (23)   
   i.e.   
      F_father_vs_son = F_son_on_ground - m_son a                       (24)   
   so that by (17)   
      F_father_vs_son   
         = F_son_on_ground - (m_son/(m_father+m_son))   
                             (F_son_on_ground - F_father_on_ground)     (25)   
      
   As a check, (21) and (25) should give the same result: (21) says   
      F_father_vs_son   
         = (1 - (m_father/(m_father+m_son)) F_father_on_ground   
              + (m_father/(m_father+m_son)) F_son_on_ground             (26)   
         =   (m_son   /(m_father+m_son)) F_father_on_ground   
           + (m_father/(m_father+m_son)) F_son_on_ground                (27)   
   while (25) says that   
      F_father_vs_son   
         =   (1 - m_son/(m_father+m_son)) F_son_on_ground   
           + (    m_son/(m_father+m_son)) F_father_on_ground   
         =   (   m_son/(m_father+m_son)) F_father_on_ground   
           + (m_father/(m_father+m_son)) F_son_on_ground                (29)   
   and we see that (27) and (29) are in fact identical.   
      
      
      
   If the father wants to loose the pull-of-war for a while, this same   
   analysis still applies with each '>' changed to a '<' and vice versa:   
   To loose, the father decreases /F_father_on_ground/ so that   
      F_father_on_ground < F_son_on_ground   
   and hence by (3) /F_net_on_father_and_son > 0/ and so both people   
   accelerate to the right (i.e., /a > 0/).   
      
      
      
   So, inclusion, now that we look at this system more carefully, we see   
   that the rope was actually a bit of a distraction.  *The actual determinant   
   of who wins is the balance between how hard the father pushes on the   
   ground (the force /F_father_on_ground/) versus how hard the son pushes   
   on the ground (the force /F_son_on_ground/).*   
      
   If /F_Father_on_ground = F_son_on_ground/ then the pull-of-war is a   
   tie, and (7) says that the direct forces /F_father_vs_son/ between father   
   and son are necessarily equal in magnitude to the forces pushing on the   
   ground.   
      
   If /F_father_on_ground > F_son_on_ground/ then the father wins and   
   (17) gives the father's and son's acceleration to the left (/a < 0/).   
   In this case the direct forces /F_father_vs_son/ betwen father and   
   son have magnitude in between those of the father's and son's forces   
   on the ground, as given by (14), and in more detail by (27) and (29).   
      
   Finally, if /F_father_on_ground < F_son_on_ground/ then the son wins   
   and (17) gives the father's and son's acceleration to the right (a > 0).   
   In this case the inequalities in (14) are reversed, and we have   
      F_father_on_ground < F_father_vs_son < F_son_on_ground            (30)   
   In this case the direct forces /F_father_vs_son/ betwen father and   
   son still have magnitude in between those of the father's and son's   
   forces on the ground, and are given (27) and (29).   
      
   --   
   -- "Jonathan Thornburg [remove -color to reply]"    
      (he/him; on the west coast of Canada)   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)