By applying the transformation above from the stationary to the rotating frame three times (twice to d r d t and once to d d t [ d r d t ] ), the absolute acceleration of the particle can be written as: a = d 2 r d t 2 = d d t d r d t = d d t ( [ d r d t ] + ω ω × × r ) = [ d 2 r d t 2 ] + ω ω × × [ d r d t ] + d ω ω d t × × r + ω ω × × d r d t = [ d 2 r d t 2 ] + ω ω × × [ d r d t ] + d ω ω d t × × r + ω ω × × ( [ d r d t ] + ω ω × × r ) = [ d 2 r d t 2 ] + d ω ω d t × × r + 2 ω ω × × [ d r d t ] + ω ω × × ( ω ω × × r ) .