The Children of France

The children of France were the children of the King; the name is derived from the idea that the King embodied France. A child of France had no surname but that of "de France". Over time the title was extended to the children of not just the King but the Dauphin and even the King's eldest grand-son. Consequently, the Children of France were the children of the King - the present and the future Kings.
Technically, the Dauphin was also a child of France but due to his rank as the eldest child "of France" and the heir to the throne he would be styled simply as "Monsieur le Dauphin".

Then, in the 1630's, a new level was added to the concept: that of petit-fils/filles or grand-children of France. This was to apply to the children of a younger son of the King, originally it was created for Anne-Marie-Louise d'Orlèans (Duchesse de Montpensier) who was the only child of Louis XIII's brother.

All the daughters of France were styled with the extremely honorific "Madame" followed by their birth-name such as Madame Adélaïde, Madame Thérèse etc. However, it was important to add the name of the princess in question since the title of merely "Madame" belonged to the King's sister-in-law. The sons of France were not simply called "Monsieur" (since this was the title reserved for the younger son of the King) but instead their title such as the Comte d'Artois.
The title of Madame Royale was bestowed on the eldest daughter of the King who would forfeit the title when - or if - she married; then her younger sister would inherit it.

Both the children and grand-children of France were to be addressed as Your Royal Highness. They could all claim a chair in the presence of the King as well as the right to travel along with him and lodge near the King. They were NOT to shake hands with ambassadors and could only wear mourning for the death of a member of the royal family. Whenever the royal family officially entered a town they would be greeted with the sound of a cannon and a group of officials - at home they were the only ones besides the King and Queen who could dine alone whilst being watched by commoners.