Let {$}k{$} be an uncountable algebraically closed field of characteristic zero and let {$}\backslashmathscr X{$} be a nonsingular cubic hypersurface in {$}\backslashmathbb P{\^{}}5{$} over {$}k{$}. We prove that, for a very general hyperplane section {$}\backslashmathscr Y{$} of the cubic {$}\backslashmathscr X{$}, there exists a countable set {$}\backslashXi {$} of closed points on the Prymian of {$}\backslashmathscr Y{$}, such that, if {$}\backslashSigma {$} and {$}\backslashSigma '{$} are two linear combinations of lines of the same degree on {$}\backslashmathscr Y{$}, then {$}\backslashSigma {$} is rationally equivalent to {$}\backslashSigma '{$} on {$}\backslashmathscr X{$} if and only if the cycle class of {$}\backslashSigma -\backslashSigma '{$}, as a point on the Prymian, is an element of {$}\backslashXi {$}.