84 BELL SYSTEM TECHNICAL JOURNAL 



In this case there is a one-to-one correspondence between the points of the 

 circumference of the unit circle and conical surfaces coaxial with the line 

 of sources. Such conical surfaces, called radiation cones, are loci of direc- 

 tions in which the radiation intensities are equal. If the separation between 

 the elements < X/2, the range of z is smaller than Itt and z describes only a 

 portion of the unit circle (Fig. 3A). Finally, \il > X/2, then the path of z 

 overlaps itself (Fig. 3B). Such a path, winding upon itself, will be called a 

 Riemann circle. In this instance, one and the same point on the circle may 

 correspond to several radiation cones; but if we regard different positions of 

 2 along its path as distinct points on the Riemann circle, then there will be 

 a one-to-one correspondence between the points on the circle and the 

 radiation cones. 



Since the radiation intensity is a periodic function of \p, the space factor 

 of a given array will repeat itself if the separation between the elements 

 is greater than one-half wavelength. 



Composition of Space Factors 



Since the product of two polynomials is a polynomial, we obtain the fol- 

 lowing corollary to Theorem I 



Theorem II : There exists a linear array ivith a space factor equal to the 

 product of the space factors of any two linear arrays. 



In other words, there is a linear array such that its radiation intensity 

 in any given direction is the product of the radiation intensities in this direc- 

 tion of any two given arrays. Thus we have 



a/*i = I flo + ais + QoJ -f- • • • -1- a„-i3"-^ ], 



\/^2 = I 6o + ^'is + hz- +•••-!- 6„,_is'"-i I, 

 _ _ (6) 



V^i V$2 = 1 (Go + ais + • • • + an-iS"-')(6o -f ^'is + • • • + ^',«-iS'"-^) I 



= I aoba + {aoh -\- aibo)z -\- (ao^2 + aih + Oobojz -f- • • • |. 



The coefficients of the expanded product represent the amplitudes and the 

 phases of the derived array. 



Naturally the process may be repeated and a linear array can be con- 

 structed with its space factor equal to the product of the space factors of 

 any number of linear arrays or to any power of the space factor of any array. 



For example, let us start with a pair of equal sources, represented by 



V^^\l + z\, (7) 



and construct a linear array with the space factor equal to the square of 

 (7). The field strength of the required array will be 



V^= |l + zl' = 11 + 2S + 2M. (8) 



