The formal group law (cf. [Wikipedia][1], Ex. 1.6 of [nLab][2], [Hazewinkel][3]) derived from an analytic function or formal series $f(x) = x + a_2 x + a_3 x^2 + ...$ and its formal compositional inverse, perhaps derived from Lagrange inversion, $f^{(-1)}(x)$ is defined by

$$FGL(x,y) = f[f^{-1}(x)+f^{-1}(y)]$$.

I'm interested in earlier investigations of this formula and the closely related formula

$$F(x,t)=\exp[t \cdot g(x)d/dx]x = f[f^{-1}(x)+t],$$

where $g(x) = [df^{(-1)}/dx]^{-1}.$

Both occasionally pop up in MO (see [Q1][4], [Q2][5], [O3][6]).

So far the earliest presentations I know of are

1) by Abel for the $FGL(x,y)$ in 1826 (cf. "[From Abel's heritage: Transcendental objects in algebraic geometry and their algebraization][7]" by F. Catanese, p. 6, Thm. 2.1, and "[The Work of Niels Henrik Abel][8]" by Houzel, p. 24, Eqn. 5.)

2) by Abel for $F(x,t)$ in 1826 (cf. [Abel equation][9], 1826).

3) by Charles Graves for $F(x,t)$ in 1853 in  "A generalization of the symbolic statement of Taylor's theorem" in the Proceedings of the Royal Irish Academy, Vol. 5, (1850-1853), p. 285-287 (cf. p. 13 in [The Theory of Linear Operators][10] ... , Principia Press, 1936, by Harold T. Davis and MO-[Q4][11]).



Aware of any earlier presentations than Abel's or Graves?


  [1]: https://en.m.wikipedia.org/wiki/Formal_group_law
  [2]: https://ncatlab.org/nlab/show/formal+group
  [3]: http://www.numdam.org/item/AST_1979__63__73_0/
  [4]: https://mathoverflow.net/questions/291888/characterizing-positivity-of-formal-group-laws
  [5]: https://mathoverflow.net/questions/52241/formal-group-laws-and-l-series
  [6]: https://mathoverflow.net/questions/145555/why-is-there-a-connection-between-enumerative-geometry-and-nonlinear-waves/181534#181534
  [7]: https://arxiv.org/abs/math/0307068
  [8]: http://www.abelprisen.no/c53052/binfil/download.php?tid=53200
  [9]: https://en.m.wikipedia.org/wiki/Abel_equation
  [10]: https://catalog.hathitrust.org/Record/000616955
  [11]: https://mathoverflow.net/questions/214927/important-formulas-in-combinatorics/215203#215203