II. THOMAS DOUBLE-COLLISION KINEMATICS 5 Without loss of generality we may consider the collisions to occur in a plane, the xy-plane. If we express the velocities of the three particles as complex numbers and write them as the components of a three dimensional, complex vector, as: ( v$_{1}$$_{x}$ + iv$_{1}$$_{y}$, v$_{2}$$_{x}$ + iv$_{2}$$_{y}$, v$_{3}$$_{x}$ + iv$_{3}$$_{y}$ ) $^{T}$, $^{then the velocities before and after a collision can be related by a 3}$× 3 complex matrix [13]. If we denote the matrices associated with collisions (1), (2) and (3) by S$_{1}$, S$_{2}$, and S$_{3}$, then While solving these four equations is straightforward, we will use a technique which will make the calculations in Section III simpler. The equations of conservation of energy are: We will now determine the mass ratios a and b for which these processes are allowed. First we look at process A, that is, (31). The initial velocities of particles 2 and 3 are zero and u will be the initial velocity of particle 1. After the first collision, (3), the velocities of 1 and 2 are u ′ and v . After the second collision, (1), the velocities of 2 and 3 are u ′ and w . The equations of conservation of momentum for the two collisions are: It is convenient to work with the ratios of the masses of particles 1 and 3 to the mass of particle 2: a = m$_{1}$/m$_{2}$ and b = m$_{3}$/m$_{2}$ . The mass ratios a and b which lead to scattering may be found simply by solving the equations of conservation of energy and momentum at each collision. It is also convenient to refer to the binary collisions using the number of the particle not involved in the collision, the ”spectator particle.” We will use parentheses to distinguish between particle numbers and collisions numbers. So, for example, (3) refers to a collision between particles 1 and 2. A capture process will be described by a string of numbers enclosed by parentheses corresponding to the collisions in the order they occur from left to right. Thus the three double collision processes we have considered, A, B, and C, can be referred to as (31), (32), and (21) respectively. A sketch of their trajectories is given in figure 1. a u = a u ′ + v (1) v = u ′ + b w . (2) 1 2 au 2 = 1 2 au ′ 2 + 1 2 v 2 (3) 1 2 v 2 = 1 2 u ′ 2 + 1 2 bw $^{2}$. (4)