^{page 175 note *} “La Fata Morgana,” Mem. della Societa degli spettroscopisti Italiani, xxxi., Catania, 1902; contains a full bibliography of the question, with extracts from the principal observations.

^{page 175 note †} “Mirages et réfractions anormales sur le lac Léman,” Bull. Soc. Vaud. Sc. nat., xxxii. 271, Lausanne, 1853–56.

^{page 176 note *} “Réfractions et Mirages: passage d'un type à l'autre,” Bull. Soc. Vaud. Sc. nat., xxxii. 271, Lausanne, 1898.

^{page 176 note †} Refraction phenomena are not always presented in nature so clearly or so simply as might be desired. There may be no doubt as to the general nature of the mirage, which may nevertheless be difficult to interpret. The phenomena originate at a far distance, in air more or less saturated with water vapour, and frequently masked by a veil of fog. The observations are beset with great difficulties. Had it not been so I should not have spent more than fifty years in arriving at the present explanation. The observation of refraction phenomena over the surface of a lake demands an intimate acquaintance with all its characteristics. An occasional traveller spending a few days on its shores cannot be familiar enough with the lake scenery to enable him to know what changes, if any, may have occurred. Prolonged residence in the neighbourhood of the lake and a keen interest in the ever-shifting illusions are essential to a complete study of the phenomena.

^{page 178 note *} The true horizon is the tangent cone to the surface of the lake, the vertex of the cone being at the eye of the observer, and the calculation being made on the assumption of no refraction; the circle of the true horizon is the curve of contact of the cone with the surface of the lake. The cone and circle of the apparent horizon are similarly defined in terms of the rays of light as they enter the eye after having been displaced by atmospheric refraction. These cones are so flat that they may be spoken of as planes.

^{page 178 note *} This is the line which separates the erect and inverted images in the usual mirage.

^{page 179 note *} Bull. Soc. Vaud. Sc. nat., xxxv. 25, Lausaune, 1899; see also Arch. d. Sc. phys. et nat., viii. 373, Geneva, 1899.

^{page 181 note *} If my hypothesis is sound, the instability which leads to the Fata Morgana may occur at the middle of the stretch round which the phenomenon is seen as well as at the extremities. In such a case we should see the Fata Morgana, at first single, splitting into two moving in opposite directions, the one to the right, the other to the left. This possible variation in the details of the illusion I have searched for in vain. Should it ever be observed, it will be an experimentum crucis, establishing the sufficiency of my hypothesis.