218 SCIENCE PROGRESS 



u - real part of [2 {a s e^ . U (X s )} + 2(a "+ a"t+ . . . . a" a _^_ 1 t n -^ I )e ni ot . tj (m )] 

 -real part of 2^^[u;(m ) + aU;" 1 ^). t + Si^i) u;- 2 (m ) . t 3 + . . ..+ 



+ .. ^-^^-) U>o) ,;^] 



r U,(m s ) m t U : (m' s ) m - t U 3 (m" s ) m » t .-. 



- real part of 2 ■ ' P s e s ' +2 , ■ P' e s » +2 A , „\ P" s e 3 . 



r L s i A(m s ) s i s 2 A(m' s ) s » s 3 A(m s ) s 3 J 



a s , a ", a"i, etc., are arbitrary, giving the free vibrations. Of 

 course there may be free vibrations of the type given by X = m 

 when there are no forced vibrations of that type (m = m ) ; but 

 if there are forced vibrations of the type (m = m ) there are also 

 free vibrations, into which they may be absorbed, of the same type. 



There can be two double conjugate roots only of A, since 

 there are only four values of X. Similarly, there are not more 

 than two such roots of the type m . The signs ', ", '", etc., are 

 merely symbols when used with " a's " and " /3's " and not 

 symbols of operation. V^m,,), Ri(m ) are taken as possessing 

 /3' and /3" roots respectively. Hence 



v = the real part of [S{a s eV . V,(X S )} + S(a* +a' l ' | t 

 + ...a" a _^- 1 t a ^'- I )e m o t .V l (m ).] 



p„e m o t r 



- the real part of 2 ^__ [^(m,) + aV.-'K)! 



4- ^""gl/ +l) V/(m p )t^] 



- the real part of [2 ^ P s .V +1 ^ P ; e »V 



L s, A(m s ) s i s 2 A(m' s ) s j 



V 3 (m" s ) m . t -. 



+ 2 — ^- P" e 3 



r = the real part of [Ka^s 1 . R(A S )} + 2(a" + a",t 

 + a" a _^_ 1 t a -^"- I )e m o t . R,(m )] 



- the real part of 2 ^— } |_R »(m ) + aR x °- \m )t 



+ ... g(a -^J^ 0+l) R^(mo)t^"] 



t R i(m s ) m . t R^Cm'g ) m < t 

 5 -D rP e ' +2 A , , x P'e 2 

 s i A(m sl ) . s 2 A(m a ) s 2 



s g A(m s 3 ) s 3 



'} 



