FIG. 2: The Lieber diagram for double collision processes whose solution is: x$_{2}$ = a ( $^{b}$− (1+ a + b ) x$_{1}$ ) /b ( a + x$_{1}$ ). The absolute value of x$_{2}$ is unity when cos( α$_{1}$ ) = ( a $^{2}$+ ab + $^{a}$−b ) / 2 ab. The cosine falls in the desired range when 3 ab + a $^{2}$+ $^{a}$−b ≥ 0 and b ≥ a. Thus the allowed region for B lies between the line b = a and the curve 3 ab + a 2 + a − b = 0. The latter is again a rectangular hyperbola with an asymptote at a = 1 / 3 in the first quadrant. The process C, or (21), is similar to B. If B is viewed under time reversal and with particles 1 and 3 exchanged, it is equivalent to C. Thus the values of ( a, b ) for which C is allowed are simply those of B with the roles of a and b reversed. Thus C is allowed if ( a, b ) lies in the region between the the line b = a and the curve 3 ab + b 2 + $^{b}$− a = 0. The latter is a rectangular hyperbola with an asymptote at b = 1 / 3. The allowed regions for the three binary processes are shown in the “Lieber diagram,” figure 2. 8