In case the sureties or any of them, of any public officer or employee shall permanently remove out of the state or become insolvent, or if from any cause an official bond shall be found insufficient, the Governor in the case of a state officer or employee, and the appropriate governing authority in the case of a local officer or employee, shall notify such officer or employee to appear at a day and place to be named within ten (10) days thereafter, to give a new bond with other sufficient sureties in a penalty equal to that of the former bond and with the like condition, or to show cause why the same should not be required. If such officer or employee shall fail or neglect so to do within a time to be designated, his office or position shall thereby become vacant and he shall cease to discharge any of the duties thereof. If a state officer or employee, the Governor shall cause the vacancy to be filled as in other cases; and, if a local office or employee, the vacancy shall be filled as in other cases of vacancy in local offices. From proceedings under this section there shall be no appeal; and if the officer or employee to be notified be without the state or abscond, he may be notified by publication as defendants in chancery are so notified.