270 MOTION UNDER A VARIABLE FORCE 



if we introduce new constants C, D to replace the constants 

 Acospe and Asmpe. The other two equations have similar 

 solutions, so that we can take the complete solution to be 



x = C cos pt + D sin pt, (103) 



y = c f cos pt + D 1 sin pt, (104) 



z = C" cos pt + D" sin pt. (105) 

 We can always solve the equations 



f<7 + rC"+s<7"=0, (106) 



\D+rD'+sD"=Q, (107) 



and so obtain values of r and s for which these relations are true. 

 Let us multiply equations (104) and (105) by these values of r and 

 s, and add corresponding sides to the corresponding sides of equa- 

 tion (103). We obtain 

 (x + ry + sz) = ( C + r C f + sC") cospt + (D + rD' + sD") sinpt 



= 0, (108) 



since equations (106) and (107) are satisfied. The meaning of 

 equation (108) is that for all values of t we have the relation 

 x 4- vy + sz = 0, and, therefore, that throughout its motion the 

 particle remains in the plane of which this is the equation. 



The axes of coordinates have been supposed to be chosen arbi- 

 trarily. We can always choose the axes so that the plane in which 

 the whole motion takes place is that of xy. The motion is then 

 given by two equations of the form 



x = C cos pt + D sin pt, 

 y = c r cos pt + D f sin pt. 

 Solving for sinpt and cospt we obtain 



C'xCy 



D'x Dy 

 - - 



so that on squaring and adding, we obtain 



( C'x - CyY + (D'x -Dy)* = ( C'D - CD')*. 



