174 V. OSCILLATIONS AND CYCLIC MOTIONS. 



us therefore put instead of the above value of F r the value 

 whose real part agrees with the above, and having found a particular 

 solution of the differential equation, let us retain its real part only. 

 Thus we have instead of equations 63) n equations of which the r th is 



d 2 2i d*q z d*q n 



125) a r l i + #r2 2 ~^~ ' l~ a rn 



dq, dq, i dq n 



+Cr2qz H ----- h c rn q n 

 Guided by the result of 44, assuming 



these become 



( 



126) ; 



If we call the determinant of equation 65) -D(A) and the minor 

 of the element of the y th column and s th row D r (X), we have as 

 the solution of 126) 



127 ) 



Since D(Ji) = is the determinantal equation 65) for the free vibra- 

 tion, whose roots are A t , A 2 , . . . fan, we have 



128) 



where is the proper constant. 



Accordingly the denominator D(ip) is 



129) D(if) = C (V - V = <?(- ft + < (P - ".)) 



=1 =1 



The minors D rs (ip) are polynomials in ^ and the numerators are 

 therefore complex quantities, which however reduce to real ones if 

 the jc's are zero. We may write 



130) 



where S r and & r are real and & r vanishes with the sc's, and is small 

 if they are small. We thus have 



