RADIATION CHARACTERISTICS OF AN ANTENNA. 227 



PART III. 



The Mutual Term in Power Determination. 



16. The Trigonometric Relations. — In Section 14, equation 

 (56), it has been shown that the power radiated through an element 

 of surface consists of three terms in the form 



rfp= f {EeH^-\- E^H^+ 2 cos a Eg H^)dS. 



The first two of these terms we have already discussed. Putting 

 in the values of E^ and H^ from equations (19) and (55) the remaining 

 power term, which we have for convenience called mutual power, be- 

 comes in the time average 



,_ P cos a dS . Az \ in • /d t\ I 



dp = s • — ^r-- — r sm — < cos\p sm B — sm {B cos i/'j r 



TTC r-Q sm d sm\f/ To ( ) 



< cos B cos (A cos 6) — sin B cos d sin {A cos d) — cos G r . (82) 



In forming this equation we have multiplied the expression for Eg 

 of eq. (19) by the expression for H^, eq. (55). The product so obtained 

 contains terms involving sin r cos t plus terms involving cosV. The 

 time average of the sin r cos r terms is zero ; while the time average 

 of cos^ T is I; these facts have been used in forming (82). 



To be able to integrate equation (82) we must replace a, z, \p and 

 dS by their values in terms of 6, 4> and Tq. By Fig. 3, 



z = rocos 6, (83) 



dS = ro- sin B dd d4>. (84) 



In the spherical triangle of Figure 10, a is represented, as defined, as 

 the angle between d and \p, while opposite to a the side is ■k/2. The 

 important trigonometric relation in a spherical triangle is as follows: 



I. The cosine of any side is equal to the product of the cosines of 

 the two other sides plus the continued product of the sines of these 

 sides and the cosine of the included angle. 



