§ 13. Notice of neglect to file oath or undertaking. The officer or body making the appointment or certificate of election of a public officer shall, if the officer be required to give an official undertaking to be filed in an office other than that in which the written appointment or certificate of election is to be filed, forthwith give written notice of such appointment or election to the officer in whose office the undertaking is to be filed. If any officer shall neglect, within the time required by law, to take and file an official oath, or execute and file an official undertaking, the officer, with whom or in whose office such oath or undertaking is required to be filed, shall forthwith give notice of such neglect, if of an appointive officer, to the authority appointing such officer; if of an elective officer, to the officer, board or body authorized to fill a vacancy in such office, if any, or if none and a vacancy in the office may be filled by a special election, to the officer, board or body authorized to call or give notice of a special election to fill such vacancy; except that the notice of failure of a justice of the peace to file his official oath, shall be given to the town clerk of the town for which the justice was elected.