THEORY OF VIBRATIONS 55 



It has been thought worth while to recapitulate these ele- 

 mentary matters because they have interesting illustrations in 

 the present subject. Thus if x, y be rectangular coordinates, 

 and we write 



z = x + iy, ........................... (8) 



the equation z = Ce int , ........................... (9) 



where C may of course be complex, expresses that the vector C 

 is turned in a time t through an angle nt in the positive direc- 

 tion. It therefore represents uniform motion in a circle, with 

 angular velocity n, in the positive direction. The radius of this 

 circle is given by the "absolute value" of G, which is often 

 denoted by | G ; thus if C = A + iB, where A and B are real, we 

 have C \ = \I(A Z + -B 2 ). In the same way the equation 



z =G' e -int ........................ (!0) 



represents uniform motion in a circle, with angular velocity n, 

 in the negative (or clockwise) direction. 



We come now to- the application to linear differential 

 equations with constant coefficients. From our point of view 

 the simplest case is the equation 



of 4. In order that every step of the work may admit of 

 interpretation, we associate with this the independent equation 



0, ................. -....(12) 



as in the theory of the spherical pendulum. The two may be 

 combined in the one equation 



-' +*-. ..................... < 13 > 



which may indeed be regarded as representing directly, without 

 the intermediary of (11) and (12), the law of acceleration in the 

 spherical pendulum and similar problems. To solve (13) we 

 assume z = Ce**, and we find that the equation is satisfied 

 provided X 2 -f n 2 = 0, or X = in. Since different solutions can 

 be added, we obtain the form 



int , .................. (14) 



