It appears the answer to the math behind the french curve can be found in print. Ludwig Burnester was a geometer and best known for his work on the math behind mechanical linkages. Well, best known at the time because obviously in this forum he is best known for his work on the french curve. French curves were in use prior to Burmester but he formalized them by defining equations and a set of 28 tools to cover all useful curves. Generally they are sold as a set of 3 (sets of 8 and 28 are the other popular numbers) with one being "useful for hyperbolas, one for ellipses, and one for parabolas". However, this is clearly a simplification since two of these curves extend to infinity and the third is a regular closed curve and the tools are neither.
Burmesters writings were from late 19th and early 20th century and can be hard to find. Tracking down which one contains his curve work would be even harder. Fortunately the german engineer Otto Lueger references his work in his 7 volume Lexikon der gesamten Technik first published in 1894. This technical lexicon is now on it's 4th edition, last printed in 1972 in 17 volumes. Ten volumes of the 8 volume 1904 edition (? perhaps including supplements published in 1914 and 1920) have been digitized by zeno.org. As you would expect, the site is in german presenting a challenge to me in navigating. A search on Burmester (within the Lexikon and across the whole site) brings up work that all looks like it is related to his study of linkages.
The figure in creative commons of the 28 curves linked above is from this collection but sadly they neglect to say what page they got it from. The figure does include the word "Kuvenlineale" and finding this in the alphabetic index takes us to Zeichnen which includes the figure we are looking for. Here is the text referencing that figure:
"Im übrigen sind in Fig. 6 andre handelsübliche Kurvenlineale abgebildet. – Lineale der seither genannten Arten werden in der Regel aus Birnbaumholz hergestellt; man verwendet aber auch Eschenholz, Mahagoni mit Ebenholzeinfassung,Aluminium, Zink, Eisen, Hartgummi, Celluloid u.s.w. Die Dreiecke (Winkel) dienen zum Ziehen gerader Linien unter verschiedenen Winkeln, von welchen stets einer 90° beträgt. Man unterscheidetBöschungsdreiecke für einfüßige, anderthalbfüßige und zweifüßige Böschung; Kreisdreiecke (Bingscher Kreiswinkel), deren längere Katheten sich zu den Hypothenusen verhalten wie (0,25 · π)1/2 : 1, vgl. ; Oktogondreiecke, entsprechend den Winkeln zum Zeichnen eines Achtecks: Weichendreiecke mit Winkeln welche den Verhältnissen 1 : 8, 1 : 9, 1 : 10 u.s.w. entsprechen. Für den gewöhnlichen Bedarf verwendet man Dreiecke mit den Winkeln: zweimal 45° und einmal 90° oder 30°, 60° und 90°. Die Seitenlängen sind sehr verschieden; die größte Kathetenlänge geht bei den imHandel befindlichen größten Dreiecken bis zu 60 cm und beträgt bei den kleinsten Dreiecken etwa 7 cm."
Google translate: "In the remaining 6 other commercially available curve rulers are depicted in Fig. 6. - The rulers since those species are generally produced from pear wood; but are also used ash wood, mahogany with ebony edging, aluminum, zinc, iron, hard rubber, celluloid, etc. The triangles (angle) used for drawing straight lines at different angles, one of which is always a 90 °. It differs a hedge triangles for einfüßige, anderthalbfüßige and bipedal embankment; Circle triangles (Bingscher circle angle) with longer catheti behave to the hypotenuse as (0.25 · π) 1/2. 1, see ; Oktogondreiecke, corresponding to the angles of drawing of an octagon: time triangles with angles which the ratios of 1: 8, 1: 9, 1: 10, etc. comply with. For the ordinary needs to use triangles with angles: 45 ° twice and even 90 ° or 30 °, 60 ° and 90 °. The side lengths are very different; the largest Kathetenlänge is located at the imHandel largest triangles up to 60 cm and is the smallest triangles about 7 cm."
So none of that turns out to be helpful in finding out how Burmester drew his curves.
Meanwhile, Wikipedia says the french curves are derived from a Euler Spiral . While I could find no references linking Euler's spiral to French Curves (other than Wikipedia's say-so) it at least makes sense on two fronts. Euler's equations predate Burmester so they were there for him to use and Euler's Spiral produces a straight line as the limit approaches zero and a circle as the limit approaches infinity. In other words it creates a curve that will smoothly link a straight line to a circle which is exactly what draftsmen use a french curve to do (as well as linking curves to curves).