This is an "ascending" horn. That is to say, when the third valve is pressed, the pitch of the horn is raised a whole step instead of being lowered by a step- and-a-half (minor third) as is the case on most three-valve instruments. This is accomplished by reversing the usual function of the third valve piston. The open horn air path includes the tubing of the third valve slide. When the valve is pressed the valve slide tubing is bypassed, shortening the length of the air path, thereby raising the pitch of the horn. For this reason a G-crook is used, since the rest of the horn corpus is the length of a standard F horn. (Horn math: the G crook plus the third valve slide equals the length of a standard F crook, therefore the open horn is pitched in F.)

Michel Garcin-Marrou explains the advantages of this system as follows:

    In short, one can conclude without fear of error that the positive and original contribution of the ascending system is, through the use of the third valve, to have at one's disposition a horn in G and thus reach the higher notes of the the horn's range with greater facility. Those who have played a little natural horn or a single F horn know well the difference in response between an F crook and a G crook, and how very much preferable it is to play a high g'' on a G horn than a high a'' on an F instrument. This is a rather important detail when one considers the time when this system was put into practice and the added playing security in the medium and high registers that it brought to horn players of that age.

The fingering chart for the ascending F horn is not significantly different from the standard (descending third valve) horn. In fact any fingerings using only the first or second valves alone or in combination remain valid although not necessarily preferred. Observe that the combination of first and third valve is never used on the ascending horn. (Why? Think about it.) Note also that some notes in the low register are not possible on the ascending F horn. For this reason horn makers offered both ascending and descending models. First and third ("high") horn players used the ascending horn, while second and fourth players used the descending horn, so as to have available the full lower range. The ascending system persisted in France throughout the first half of the twentieth century. Full double and compensating ascending piston valve double horns in F and Bb became the standard in France when they were finally overtaken by the larger bore rotary valve German models in the nineteen seventies.

A cut showing the ascending single F horn from the 1910 catalog of Herman Schoenaers who was Millereau's son-in-law and who took over the business in 1898.  

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Maxime-Alphonse (1880 - 1930) is best known for his series of six method books, Enseignement du Cor à Pistons, Deux Cents Études Nouvelles, Mélodiques et Progressive en Six Cahiers published by Alphonse Leduc & Cie. Paris, from 1920 to 1924. He dedicated these etudes to Edouard Vuillermoz (1869 - 1936) who at that time was solo horn in Monte-Carlo. Very little is known about the life of Maxime-Alphonse. In fact, according to information found on a web blog. "Maxime-Alphonse" was actually a pseudonym. His full name was Jean Marie Maximin François Alphonse with "birth name" Griet and he was s born December 31, 1880 in Roanne (Loire). He studied with François Brémond at the Conservatoire de Paris, placing third in the concours in 1900 and taking the Premier Prix (tied with Catel) in 1903. The morceau de concours for that year was Concertstuck by Georges-Jean Pfeiffer (1835 - 1908) and it was also the first year that the valved horn became the official instrument at the Conservatoire. In his professional career he was solo horn of l'Opéra Comique and also in the Concerts Pasdeloup which had been re-formed in 1918 after an absence of more than twenty years.

Maxime-Alphonse, Deux Cents Études Nouvelles, 4^e Cahier, p.4

L'élève observera, s'il joue le Cor avec le 3^e piston ascendent, les doigtés indiqué. Mais il devra également travailler des doigté ordinaires du Cor descendant, qui seront pour lui un bon travail de mécanisme.
Ibid., p.3

The student will observe, if he plays a horn with 3rd ascending third valve, the indicated fingerings But he should also work on the regular fingerings for the descending horn, which will be a good mechanical exercise for him.