This is a math question, even if it may seem an economics one. I'll try to explain all the economics in this question.
I've got the following production function, where $Y$ is the product, $L$ is the labour used in production, while $K$ is the capital. The $M$ subscript just stands for "manifacturing sector".
$Y_M=f_M (L_M, K_M)$ (1)
which is assumed to be a first degree homogeneous function. With this assumption, we may write
$Y_M=L_M m (k_M)$ (2)
Where $k_M=\dfrac{K_M}{L_M}$ and $m$ is the average productivity of labour.
Which transformation has been performed in order to obtain (2) and what's the influence of the function's first-degree homogeneity in it?