In harmonic vibration. Equation [2a] becomes: 



2 3P 



yu) V = - — 



An equation containing M alone is easily deduced, thus: 



3M . 2 3^M 



P = - 9^ ; . . po) V = — 



3x 



3v . 3y 3^v 8^M 

 Y = -^ - 2oP; . . T^ = — 2 + 2a p 



and, substituting this last expression for Sy/ 9x in Equation [2d] and 



then substituting for v yields: 



k 2 



M • 1 3 M ^ 3 M 



ax 3x 



or 



i!^+2au.'^-^ M = [3a] 



^ 3x2 ^^ 



For convenience, introduce q and ^, defined thus: 



q 



Xx 2 P 



Then, substituting M=e sin ojt and also 2ouu = 2q C into Equation [3a], 



and dividing through by M, yields the following equation as the necessary 

 and sufficient condition for X: 



X*^ + 2q^^^ - q'^ = 

 Thxis , by solving X = -q 5 + l/q^ + q and taking first the minus and 

 then the plus sign before the radical and again extracting the square root, 

 it is found that either X = + iq-, , or ^ = + q.p » where 



^=q|/yr77^,; q^ = q\^ 



5^-5 [5a,b] 



Compare with equations in Appendix [D2] of Reference 11. 



