292 VII. DYNAMICS OF ROTATING BODIES. 



we have for the general solution , 



| = 4 1 ' e"" + Af e~ if " + Af e"" + A ( ? e' 1 ", 

 n = 4 1 ' "" + Af e-"" + Af e 1 " + Af e~ ivt , 



where we have by the first of equations 136) , 



4 1 ' ._ 2) ' 3) > 



1 /LA 1 " - - > _ ~ 



>* 1~ ' 



Introducing the values of the J. 2 's in terms of the A,? a, and writing 



145) 4 1) +4 2) =, 4 1) -4 2) =-ift4 3) +4 4) =', 4 3) -A^=-in', 



we have, replacing exponentials by trigonometric terms, 



| = cos lit + (Ism [it + ' cos v^ 4- /3 f sini/tf, 



146) e _ u. 2 c 



?? = -- (j3 cos ^t asi 



with the four arbitrary constants a, ft ', /3', or putting 



147) a = ^ cos s l9 /3 = ^ sin 1; a' = A% cos 2; /3' = A 2 sin 



| = A l cos (^^ fj) -f -^2 cos ( v ^ ~ ^2)7 



148) ft 2 - 



sin t - f -- sin v - 



Accordingly the motion may be described as the resultant of two 

 elliptic harmonic motions of frequencies ^? ^7 the directions of the 



axes of the ellipses being coincident, and given by the directions in 

 which the system can make a principal oscillation when the gyrostat 

 is not spinning. The absolute sizes of the two ellipses are arbitrary, 

 but the ratios of the axes, and the phases, are determined by the 

 nature of the system and the rapidity of the spinning. 



Calculating the coefficients in 148) from the values of /it, > v, > 



149) 



bv 



