Say the object is composed of N pieces with masses 
. Call the
displacment vectors between these pieces and the axis
distances 
between the pieces and the axis, then
When the axis is displaced by a vector 
, then we want to compute
where 
is the
displacment vectors between these pieces and the new axis.
Let's relate the to 
And since 
(dropping the subscript for convenience)
Now plugging this into 1.57 we have
The last term contains 
. Dividing this by M, this
would be the center of mass in the plane perpendicular to the axis. It is
reckoned about the center of mass, so by definition, this must be zero.
That is, if you calculate the center of mass of an object when the
origin of the coordinate system is the center of mass, you get zero. So
we only have the first two terms in the above equation. So we get
eqn. 1.55.
