* It was about the beginning of 1784 that I investigated the foregoing demonstration, which, as the reader will see, is conducted after the method adopted by Sir Isaac Newton, in his demonstration of the 94th proportion of the first book of the Principia. I applied to my much esteemed colleague Mr Profosser Playfair, for his assistance in a case to which the foregoing demonstration may perhaps be thought not to extend, namely, when the motion of the light, and that of the medium, are perpendicular to the refracting surface. Before I had obtained a demonstration which pleased me, he favoured me with the following elegant analytical demonstration.

Let v be the velocity of a particle of light when it has arrived at the distance x within the refracting medium (x being counted from the point in which the particle began to be acted on, and being lest than the distance from that point at which the motion of the particle again becomes uniform.) Let f be the force adting on the particle at the distance x. Let a be the velocity of the incident light, and c the velocity of the medium in the opposite direction.

It is evident that the force f does not act on the particle during its passage through the whole space ẋ, but only daring its passage through the part . Therefore, , and , or . That is, 2v ṿ + 2c ṿ = 2/ẋ, and, taking the fluent, v ² + 2cu = 2∫fẋ + C ². But when 2∫fẋ =0, we have u ² + 2cu = a ² + 2ac, and therefore u ² + 2c u = a ² + 2ac+ 2∫fẋ. Let the fluent of 2fẋ (assumed, so that x shall be the distance at which the velocity of the light again becomes uniform) be supposed = g ². Then u ² + 2 cu = a ² + 2 ac + g ². Add c ² to both sides of the equation. Then v ² + 2 u c + c ² – a ² + 2a c + c ² + g ²; and therefore . But a + c is the relative velocity of the incident light, and v + c is the relative velocity of the refracted or accelerated light. Therefore the square of the latter exceeds the square of the former by the constant quantity g ². Now, g ² =: 2∫fẋ; and is therefore (by the celebrated 39th proportion of the first book of the Principia) the square of the velocity which a particle of light would acquire if impelled from a state of rest through the whole distance at which the medium arts on light.

Since the relative velocities, estimated in a direction parallel to the refracting surface, are not changed by the action of the refracting forces, it evidently follows from this demonstration that the difference between the squares of the relative velocities of the incident and refracted light, is equal to the square of the specific velocity of the medium, whatever may be the directions of the incident and refracted light, and therefore, that the final relative motion of the resrasted light is the same as if the medium had been at rest, and the light had approached it with the same relative motion. But although this demonstration would have been much more elegant, and more agreeable to the manner in which I have been accustomed to explain the refraction of light, I chose to retain the demonstration which I have given in the text, because I think that it gives me a better opportunity of exhibiting to the mind the whole motion of the light during its refraction or reflection. At the same time, I thought it my duty to communicate, with Mr Playfair's permission, his demonstration to the public.