A deposition is a written declaration, under oath, made upon notice to the adverse party, for the purpose of enabling him to attend and cross-examine. In all actions and proceedings where the default of the defendant has been duly entered, and in all proceedings to obtain letters of administration, or for the probate of wills and the issuance of letters testamentary thereon, where, after due and legal notice, those entitled to contest the application have failed to appear, the entry of said defaults, and the failure of said persons to appear after notice, shall be deemed to be a waiver of the right to any further notice of any application or proceeding to take testimony by deposition in such action or proceeding.