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We first propose a pedestrian review of the Coordinate Bethe Ansatz for the$\delta$-Bose gas on an infinite axis. The scalar product of Bethe states,briefly reviewed, provides the first keystone to compute a compact expressionfor the Matrix Elements of the Particle Number Operator as conjectured by V.Terras, the main result of the following reflection.

Delta-Bose Gas: the Matrix Elements of the Particle Number Operator as a Determinant