Let us suppose that the transformation equations from a rectangular (x, y, z) system to the and systems are given by
Then there will exist a transformation directly from the system to the system defined by
The radius vector to point P is
in the system and
in the system.
From 4) we get
and from 5)
From 3) above we obtain
Substituting 7) into 6) and equating coefficients of on both sides we obtain
The vector A is given by
where C1, C2, C3 and are the contravariant components of A in the two systems.
Substituting 8) into 10) we get
Equating the coefficients of in 9) and 11) we get
End of proof.