## A T Petit's Programme for his Astronomy thesis

We give below a translation of the Programme which Petit submitted to the Faculty of Science in Paris on 18 December 1811:

The Theory of Astronomical Refraction has been taken as the subject.

Refraction in general.

- Assuming that the different motions of light are the result of the action of attractive or repulsive forces, the principle of least action must be verified in all the hypotheses of refraction; we conclude this from relations by means of which we can determine the law of the velocities of light, when we know the directions of the light rays, and vice versa.
- Applications to cases of ordinary refraction, and extraordinary refraction in crystallized substances.
- Special examination of the case of ordinary refraction. This is the subject. Demonstration of this law, by the principle of the conservation of "forces vives." Discussion of the different circumstances presented by the motion of light in environments whose refractive power varies in some way. Determination of the angle corresponding to the total refraction. Explanation of a mirage.
- Means of determining the refractive power of diaphanous bodies and opaque bodies.

*Astronomical refractions.* - Application of the preceding results to the successive attractions which the different layers of the atmosphere exert on the luminous molecules which pass through them. Differential equation of the motion of light.
- Integration of this equation. To do it generally, one must know the law according to which the density of the atmospheric layers varies with their height. The two limits of this law are a constant density, and a decreasing density in geometrical progression, for heights in arithmetic progression. Examination of the first case. The resulting refraction is much too low.
- The second hypothesis presupposes a uniform temperature throughout the whole of the atmosphere. Integration of the differential equation with this assumption. It gives too strong a refraction.
- Integration of the differential equation, assuming that the density of the air decreases in arithmetic progression for layers of equal thickness. This supposition gives too small a refraction, but closer than that which results from a constant density. The true constitution of the atmosphere is therefore intermediate between this last supposition and that of a uniform temperature.
- Integration of the differential equation with a hypothesis composed of the two preceding ones. The resulting formulas for the refractions and the decrease of the heat of the air agree with the observed phenomena.
- Formula which gives the refraction for all heights above 12°. Discussion of the elements which enter into this formula.
- Examination of the influence of air humidity on refraction.
- Terrestrial refractions. Determination of the formulas that express them.