Systems produced by the ordinary reflection of light ; it has there- 

 fore been thought advisable to give in the present Supplement, the 

 general demonstration of the formula, and some of its general conse- 

 quences. The demonstration is founded on the principles of the 

 calculus of variations, and on the known optical principle of least 

 action. The result deduced from these principles, is, that the co- 

 efficients of the variations of the final coordinates, in the variation of 

 the integral called action, are equal to the coefficients of the varia- 

 tions of the cosines of the angles which the element of the ray makes 

 with the axes of coordinates, in the variation of a certain homoge- 

 neous function of those cosines : this homogeneous function, which is 

 of the first dimension, being equal to the multiplier of the element of 

 the ray under the integral sign, and therefore to the velocity of that 

 element, estimated on the hypothesis of emission. It was proposed, 

 in my former Memoir, to call this result the principle of constant 

 action : partly to mark its connexion with the known law of least 

 action, and partly because it gives immediately the differential equa- 

 tion of that important class of surfaces, which, on the hypothesis of 

 undulation are called waves, and which, on the hypothesis of molecu- 

 lar emission may be named surfaces of constant action. But in the 

 present Supplement, it is proposed to designate the fundamental for- 

 mula by the less hypothetical name of the Equation of the Charade-' 

 ristic Function : because, whatever may be the nature of light, the 

 definite integral in this equation is, as we have before observed, a 

 function of the coordinates of its limits, on the analytic form of 



