in D.J Saunders 's book (The geometry of jet bundles) on page 25 he defines the vertical bundle by :
Given the bundle $(E,\pi,M)$ the subset $$\{\xi \in TE:\pi_*(\xi)=0 \in TM\}$$ is called the set of vectors vertical to $\pi$. It is denoted $V\pi$ and is a submanifold of $TE$.
The triple $(V\pi,\tau_E|_{V\pi},E)$ is a sub-bundle of $\tau_E$ which is called the vertical bundle to $\pi$.
Where $\tau_E:TE\rightarrow E$ denote the map which associates to each tangent vector the point of $E$ at which it is located.
Now on page 125 he introduces this bundle $(V\pi,\nu_\pi,M)$ Could any one explain me how this bundle is defined ?