All square roots are meant to be positive. The values X = + iq lead to 

 sin q^ X and cos q x as alternative real factors, whereas X = +^ q leads to 

 sinh q X and cosh q x. 



The following useful auxiliary formulas are easily obtained: 



2 2 2 1/ p 000 2 



q^ + q2 = 2q (/ 1 + ^ ; \ ' "^-2 ^^^ ' '^1*^2" "^ [6a,b,c] 



3 3 ^ ^ ^ 

 q^ ^2 ^1 ■" ^2 1; k h , _2 



+ 



q^ + q^ = 2q (l + 2? ) . [6d,e] 



12 -^1 -^1-^2 



Thias , in harmonic motion, M can be expressed in terms of the four 

 independent solutions of Equation [3a] as follows: 



M = e, sin q x + e cos q x + e sinh q^x + e, cosh q x [7] 



in which each coefficient e is the product of sin wt and an arbitrary 

 constant. Corresponding series for P, v, and 8v/9x or 6 are easily 

 obtained by substitution for M in equations previously written, 

 differentiating the series for v to obtain 6. 



At x= i, P = M = 0; whereas at x = , P = -F and M = -G, since F 

 and G represent actions on material lying toward positive x from the 

 terminal cross section and P and M represent actions toward negative x. 

 Thus, the boundary conditions for the present problem are (since 



P = -3M/3x): 



9M 

 atx=0: T7=F ; M=-G 



oX 



atx=«.: 1^=0 ; M=0 



3x 



Substituting the series for M and then setting first x = and then x = ^ 

 gives the four equations: 



* Compare with Equations [d8] of Reference 11, 



