motions in general dynamics 



189 



This gives 



d {u) = Q[ {u) h i''' 



e , lie ' 



In this the terms involving t outside trigonometrical signs are 

 iQ[{u)k ^1' •' • k-^ 



and putting therein i = 0, we have the determinantal equation for 

 the characteristic exponents 



= 0. 



*Ci - p. 



ta. 



P, i 



, iCo 



The form of this equation estabhshes the result in question. 

 If we put Q.^ (u) k = (tf (t), and, denoting the initial values of 

 the dependent variables by Xj^, x^, x^, put 



1,-1 (^ ^ ^ o\ _ /-^ ~ ~ o\ 



the general solution Q. (u) . {x^, x^, x^) becomes 



*(0 



or 



^2ie 

 <D3ie 



icit 



5 



) 





ic.,t 



k\J 



e , %te 



V'X , ^2 5 ^3 )■> 



^.oite"'"^ + ^.oc"^"^ 





12" 

 *22^ 





'32^^ 



>23« 

 ^33^ 



/^ J. ~ 0\ 



Wl 5 ^2 5^3 /' 



iCot 



iCit 



SO that Xi = z-^e " O^ + z^e " Q>-^^ + Zg^e ' ($13 + ^^O^a), 



a^2 = ^ie''*^21 + 22°e''''022 + Z^^e"'^ ($23 + *^^22), 

 ^3 = ^A"''<I>3i + 22°e"='<I>32 + H'e"'' (O33 + t^032), 



where the O's are periodic functions, and z-^, z^, z^ are arbitrary 

 constants of integration. 



