REMARKS ON CERTAIN INTEGRABLE ONE-DIMENSIONAL MANY-BODY PROBLEMS

A heuristic argument, and a simple proof, imply that the well-known on e-dimensional many-body problems, characterized by the equations of mo tion x(j) = g2a-3SIGMA(k=1,k not-equal j)(n)f[(x(j)-x(k))/a] with f(y) = -p'(y) or f(y) = -{sinh(y)]-2}' = 2 cosh(y) x[sinh(y)]-3, remain in tegrable even if the parameter a is not constant, but varies linearly with time, a = a(t) = alpha + betat; and an invariance property of the se equations of motion with f(y) = 2y-3 is noted.