Definition of linking number for disjoint submanifolds of the sphere- A problem from Milnor's book In problem number 13 of Milnor's 'Topology from the Differentiable Viewpoint', the linking number for two compact boundary-less manifolds $M,N \subset \mathbb{R}^{k+1}$ of dimensions $m,n$ such that $m+n =k$ is defined as the degree of the map $\lambda\colon M \times N \to S^{k}$ given by $\lambda(x,y)=\frac{x-y}{\|x-y\|}$.
The problem then asks to prove some properties of the linking number. The final question, where I need help, is 


  Define the linking number for disjoint manifolds of the sphere $S^{m+n+1}$.


I have no clue as to what the definition should be. I need this in order to proceed to the subsequent problems. Could somebody please give the definition, and a suitable reference ? Thanks !
Note that $M$ and $N$ do not cover all of $S^{n+m+1}$. If the point $P \in S^{n+m+1}$ is missed, then $M$ and $N$ live in $S^{n+m+1} \setminus \{P\}$, which is homeomorphic to $\mathbb{R}^{n+m+1}$. You can now use your old definition.

Note that, although this definition seems to depend on the choice of $P$, it does continuously so. Since the linking number is a (discrete) integer, it means it is independent of $P$.
you can find three different definitions
in bott and tu's book: differential forms in algebra topology page 229. The definitions by intersection number and differential forms are both in sphere.