Is there any consensus on what is meant by Lagrange's equations of the first kind? Joos and Freeman define them as follows:
Coordinates are given in terms of a rectangular Cartesian coordinate system. Given a system of $N$ point-masses $\left\{ m_{\iota}\mathfrak{r_{\iota}}\right\} _{N},$ a set of $N$ corresponding forces $\left\{ \mathfrak{F}_{\iota}\right\} _{N}$ both indexed component-wise by $\left\{ i=1,2,\dots,3N\right\},$ a system of $K$ holonomous constrains $\left\{ f_{j}\left[\left\{ x_{i}\right\} _{3N}\right]=0\right\} _{K}$ and a set of $K$ provisionally undetermined real number constants $\left\{ \lambda_{j}\right\} _{K};$ Lagrange's equations of the first kind for this system are the $3N$ equations $$\left\{ m_{i}\ddot{x}_{i}=F_{i}+\sum_{j=1}^{K}\lambda_{j}\frac{\partial f_{j}}{\partial x_{i}}\right\} _{3N}.$$
Wikipedia (https://en.wikipedia.org/wiki/Lagrangian_mechanics) gives
$$\frac{\partial L}{\partial\mathfrak{r}_{\iota}}-\frac{d}{dt}\frac{\partial L}{\partial\dot{\mathfrak{r}}_{\iota}}+\sum_{j=1}^{K}\lambda_{j}\frac{\partial f_{j}}{\partial\mathfrak{r}_{\iota}}=0$$
as Lagrange's equations of the first kind. Wells gives yet another definition of what he calls the Lagrangian equations, but does not distinguish between first and second kind.
