  The M\"{o}bius energy is one of the knot energies, and is named after its
M\"{o}bius invariant property. It is known to have several different
expressions. One is in terms of the cosine of conformal angle, and is called
the cosine formula. Another is the decomposition into M\"{o}bius invariant
parts, called the decomposed M\"{o}bius energies. Hence the cosine formula is
the sum of the decomposed energies. This raises a question. Can each of the
decomposed energies be estimated by the cosine formula~? Here we give an
affirmative answer: the upper and lower bounds, and modulus of continuity of
decomposed parts can be evaluated in terms of the cosine formula. In addition,
we provide estimates of the difference in decomposed energies between the two
curves in terms of M\"{o}bius invariant quantities.
