Abstract
In his early works, W. R. Hamilton set out to describe optical and mechanical systems by a single characteristic function. It is generally accepted that he based his theory on variational principles.
In this note the author shows that Hamilton derived the characteristic function V from the law of refraction and reflection in optics, and the equations of motion in mechanics, but not from variational principles. And she also shows that it was crucial for his description of optical-mechanical systems that differentiations of both characteristic functions have the same mathematical form； i.e. exact differential 1-form.
