In the event any claimant, during any current year, should appeal from the board of county commissioners, as provided for by law, from the amount allowed him by such board, the commissioners, in making their quarterly payments as above-provided for, shall estimate and allow such claimant the amount allowed him, and in the event the court should allow such claimant a larger sum than was allowed him by the board of county commissioners the amount so allowed by the court shall be considered and paid as above-provided for at the next quarterly settlement after such decision of the court.
NMS § 6-6-16