Sorry if I’m wrong but I’m going to assume: “On the authority of” and also “ON the order of”. ‘On the authority of’ is usually someone like a parent, boss, teacher, landlord, (etc.) that gets to tell you what to do within the confines of their rules. ‘On the order of’ may mean an actual legal decree from a judge where the law tells you what to do within the confines of the legal system. Flouting authority, you may get spanked, fired, or evicted. Flouting an order, you may get arrested and taken to jail. But saying that they are kind of interchangeable depending