306 PROCEEDINGS OF THE AMERICAN ACADEMY. 



This is the equation to which the empirical equation (1) corresponds, 

 provided the height of antenna H is proportional to the capacity of 

 the antenna Csi assumed localized. The condition that C3 should be 

 localized, and particularly the conditions that the incident waves 

 should be undamped, persistent and single-valued, were not fulfilled in 

 the illustrative experiment cited above, and therefore it does not seem 

 important to enter into a detailed comparison of equation (28) with 

 equation (l). This has been partially done in the previous account of 

 the experiment. 3 



The manner in which the resonance relation is affected by the resist- 

 ance of Circuit III, corresponding to the antenna circuit, is shown in 

 the curves marked 7^3 -^ = .01 and rjs^ = .1 of Figure 5. The resistance 

 R4 of Circuit IV, corresponding to the detector circuit, is without effect 

 in determining the form of these curves, which represent the general 

 resonance relation. On the other hand, R4 does have an effect in 

 determining at which point of the curve of resonance relation the re- 

 sonance is best, and R4 is also significant in determining the sharpness 

 of resonance. Some computations on this subject are given below. 



VL On the Optimum Resonance Relation. 



Let us next examine the conditions for best resonance. We are 

 still concerned merely with the steady-state vibration of the coupled 

 circuits of Figure 1, under the action of the impressed sinusoidal 

 electromotive force. The conditions for best resonance provided 

 M^o)^ > R3R4 are given in the equations (21) and (22), which after 

 substitution from (15) become respectively 



(29) 

 and 



(30) 





(^--iL=*^|(^^^"^-^=^'> 



Dividing both sides of equation (29) by L3W and employing the nota- 

 tion of equation (26) we have 



(81) 



\w J opt VsVi 



' G. W. Pierce, Physical Review, 20, 220, 1905. 



