54 



DYNAMICAL THEORY OF SOUND 



alters the length in a certain ratio r, and turns it through a 

 certain angle a. These quantities are defined by 



rcosa = a, r sin a. = b, (1) 



the quadrant in which a lies being determined by the sign 

 attributed to cos a or sin a by (1). We have then 



a + ib = r (cos a -I- i sin a) (3) 



Hence a symbol of the form cos a + i sin a denotes the opera- 

 tion of turning a vector 

 through an angle a without 

 alteration of length; in par- 

 ticular the symbol i denotes 

 the operation of turning 

 through a right angle in the 

 positive (counter-clockwise) 

 direction. 



The symbol 

 w = cos + i sin (4) 

 may be represented by a unit 

 vector OP drawn from in 

 the direction 6. If we regard Fi g . 23. 



this as a function of 0, and if 



w + Bw be represented by OP', the angle POP' will be equal 

 to BO. The vector PP' which represents Bw will therefore have 

 a length BO, and since it is turned through a right angle 

 relatively to OP, its symbol will be iBO.w. Hence 



.(5) 



It is easily shewn that the only solution of this equation 

 which fulfils the necessary condition that z = 1 for 6= 0, is 



w = e, (6) 



where e ie is to be taken as denned by the ordinary exponential 



series. Thus 



e* = cos0 + *sin0 (7) 



We may add that the "addition-theorem" of the exponential 

 function can now be derived immediately from the geometrical 

 representation. 



