where

Here, the and the are the eigenvalues and eigenvectors obtained by solving Equation (786). Moreover, the and are arbitrary constants. Finally, we have made use of the fact that the two roots of are .

According to Equation (800), the most general perturbed motion of the
system consists of a *linear combination* of different modes. These
modes are generally termed *normal modes*, since they are mutually
orthogonal (because the are mutually orthogonal). Furthermore, it follows
from the independence of the that the normal
modes are also *independent* (*i.e.*, one mode cannot be expressed as a
linear combination of the others). The th normal mode has a specific
pattern of motion which is specified by the th eigenvector, .
Moreover, the th mode has a specific time variation which is determined by the associated eigenvalue, . Recall that
is *real*. Hence, there are only two possibilities. Either is *positive*, in which case we can write

(802) |

(803) |

The arbitrary constants and appearing in expression (801)
are determined from the *initial conditions*. Thus, if
and
then it is easily demonstrated from Equations (799)-(801)
that

(804) |

(805) |

(806) | |||

(807) |

Note, finally, that since there are arbitrary constants (two for each of the normal modes), we can be sure that Equation (800) represents the most general solution to Equation (782).