Using these expressions and the above for ϵ$_{+}$ and ϵ − , we find that The process is completed when 1 has zero velocity so the condition for capture is: When this equation is combined with the definition of cos( θ ) , the simple result: sin(( n + 1) θ ) = sin( nθ ) , emerges. Aside from the trivial solution θ = 0 , which corresponds to no collisions at all, there are n solutions: The allowed region is now readily found by insisting that α be real, which is to say, − 1 ≤ cos( α ) ≤ 1 . If we solve Eq. (90) for cos( α ) and insist that it be real and between -1 and 1, we are left with: From this equation, it follows that θ cannot be π for that would yield the condition: 1 + a + b ≤ 0 . However, the remaining n possible choices for θ yield allowed regions which are bounded by the hyperbola: 1 + a + b = cot $^{2}$( θ/ 2) . The hyperbola is symmetric about a = b and has asymptotes at a = tan $^{2}$( θ/ 2) and b = tan $^{2}$( θ/ 2) . For example, if n=1, we are dealing with (32) which is the symmetric curve of the Lieber diagram. In that case, 27 1 + a + b ≤ ab cot $^{2}$( θ/ 2) . (99) θ = (2 m + 1) π/ (2 n + 1) , m = 0 , 1 , 2 , . . . , n (98) x sin(( n − 1) θ ) / sin( nθ ) = ( 1 0 ) M   1 0   (96) = x + ab ( x − 1) $^{2}$/ ( a + 1) / ( b + 1) . (97) ( 1 0 ) M n   1 0   = 0 ⇒ (95) = x n − 1 sin( θ ) (sin( nθ ) M − x sin(( n − 1) θ ) I ) (94) M n = ϵ$_{+}$P$_{+}$ + ϵ − P − (93) P − = M − ϵ$_{+}$I ϵ − ϵ$_{+}$ . (92) P$_{+}$ = M − ϵ − I ϵ$_{+}$ ϵ − (91)