A MATHEMATICAL THEORY OF LINEAR ARRAYS 



89 



Each binomial represents the directive pattern of a pair of elements sepa- 

 rated by distance /. Hence 



Theorem III: The space factor of a linear array of n apparent elements 

 is the product of the space factors of (n — 1) virtual couplets with their null 

 points at the zeros of v^^: ti , tt , • • • /„_i . 



Accordingly the radiation intensity of an array is equal to the square 

 of the product of the distances from the null points of the array to that 

 point z on the unit circle which corresponds to the chosen direction (Fig. 

 7B). To each null point lying in the range of z, there corresponds one 

 and only one cone of silence provided each null point is counted as many 

 times as z happens to pass it in describing the complete range. 



Z= I 



Fig. 8 — The null points of a uniform linear array and the point s = 1 representing the 

 direction of the greatest radiation divide the unit circle into equal parts. The 

 hollow circles represent the null points and the solid circles the points of maximum 

 radiation. 



By summing the geometric progression (9) the radiation intensity of 

 a uniform array can be represented as follows 



V* = 



Z — 1 

 Z - 1 



(20) 



Hence the null points of such an array are the «-th roots of unity, ex- 

 cluding z = \. Since s is a unit complex number/ any power of it is also 

 a unit complex number. Moreover, each multiplication by z = e rep- 

 resents a displacement through an arc of ^}/ radians. Hence the n-ih roots 



^ A unit complex number is a complex number whose absolute value is equal to unity. 



