Examination of witnesses mandates examination-in-chief, cross-examination, and re-examination, permitting new matter only with court permission. Section 143 sets the order of examinations: witnesses are examined-in-chief, then cross-examined, then re-examined. Examination-in-chief and cross-examination must relate to relevant facts, though cross-examination need not be confined to matters from the examination-in-chief. Re-examination addresses explanations of matters raised in cross-examination; if new matter is permitted by the Court during re-examination, the adverse party may further cross-examine on that matter.