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SCIENCE PROGRESS 



X s being a root as on page 212, and the last term containing four 

 terms which are the free vibrations. 



A«t x 



p =the real part of 2(a s e s )P (X s ). 



the real part of {2 -^Pe s ' +2 — - ^P'e W^-^P'e 9 * } 



A(m s ) 



1 



s 2 A(m' s ) s 2 



s 3 A(m" s ) s 3 



q =the real part of S^eV). Q/Xg)}. 



the real part of { 2 SjS^P ,***<+ 2 |^P' Si e ra V + 2 %^P'' S3 e m V}. 

 * s,A(m Si ) s > s„A(m' S2 ) s ' s 3 A(m' S3 ) s » '■ 



Instead of the free vibrations being proportional to W^X), 

 Pi(^-)i Qi(^)» they could have been taken proportional to W 2 (A<), 

 PA), QM or W s (\), P s (\), Q 3 (V). 



Note that 



A(m) = 



Wm 

 g + w 



M 



w 



W 

 Z P + m C0S ^o 



g p 



- F™ + M p 

 g p 



7 wu w . 



g 



m 



- F-4-L 

 g q 



and that W,(m), W 2 (m), W 3 (m); P^m), P 2 (m), P 3 (m) ; and 

 Qi(m), Q2(m), Q 3 (m) are the cofactors of the constituents re- 

 spectively of the first, second and third columns. To prevent 

 confusion I might mention that there is no connection between 

 the coefficient " P " of the disturbing forces and the cofactors 

 " P " of the determinant. 



The conclusions re — large forced oscillations, etc., when 

 A(m) = 0; re — the failure of A(m) to be zero if the gusts are 

 of the permanent periodic type ; re — the vanishing or limiting 

 of the forced oscillations, i.e. a forced oscillation can be large, 

 only if there be present, in the co-ordinate directly acted on, a 

 free vibration of the same period and real exponential as those 

 of the disturbing force ; and also re — the vanishing of the forced 

 oscillations by means of two other conditions are the same 

 as on the pages referring to the symmetrical oscillations. 



Complete Solution 



In the general case A(m) = o may have a roots equal to 

 m (= — n + ip ) and W^S), say, may have fi roots also equal 



