The city clerk shall keep in his office a book to be known as the `Fireman's and Policeman's Pension List.' This book shall give a full and complete history and record of the action of the board in granting a pension to any person or persons. Such record shall give the names of the firemen and policemen to whom or to whose dependents such pensions have been granted, the date when such pension was first allowed and the reason therefor. Whenever a pension shall be allowed to any person or persons by the board as herein provided, it shall be the duty of the city clerk to transmit the names of all such persons to the city treasurer who shall register such names in his office as persons entitled to such pensions.