192 PIERCE, 



that is useful from the useless radiation as heat losses and from the 

 radiation in useless directions. 



It is the purpose of this paper to give a treatment of this problem. 

 Such a treatment is, so far as I know, up to the present entirely lacking, 

 but the method here employed is that developed by Abraham ^ in a 

 very remarkable paper entitled Funkentelegraphie und Elckirodynmnik. 

 In that paper, Abraham obtained theoretically the characteristics of 

 a straight oscillator vibrating with its natural fundamental and 

 harmonic frequencies. The present work is an extension of Abra- 

 ham's method to the much more difficult problem of an antenna with 

 a flat-top and with added inductance at the base. 



2. Inadequacy of the Conception of an Antenna as a 

 Doublet. — Apart from the brilliant investigation by Abraham, all 

 other attempts at the treatment of the radiation from an antenna 

 assume that the antenna is a Hertzian Doublet. This is onlv a verv 

 crude approximation to the facts, for the derivation of the electromagnetic 

 field about a doublet assumes that the length of the doublet is negligible in 

 comparison with a quantity that is itself negligible in comparison with the 

 wavelength. 



Hence, the doublet theory will apply in all of its essentials to an 

 antenna, only provided the length of the antenna is not greater than 

 one ten thousandth of the wavelength emitted. Of course, it may be 

 that at great distances from the oscillator, the theory that it is a 

 doublet may not introduce any large errors into certain problems 

 such as the propagation over the surface of the earth; but the present 

 treatment shows that the doublet theory does introduce large errors 

 into computations of such quantities as the electric and magnetic field 

 intensities and the radiation resistance of an antenna. It seems 

 probable that other problems also should be revised in such a way as 

 to replace the conception of the antenna as a doublet by the view of 

 it as an oscillator that has a length comparable with one quarter of 

 the wavelength. 



3. Method of the Present Investigation. — In the present in- 

 vestigation, a doublet of infinitesimal length is assumed at each point of 

 the antenna. This is the device used by xVbraham. These elementary 

 doublets are free from the objection regarding their lengths, as they 

 are of infinitesimal lengths, while the wavelength is that due to the 



1 M. Abraham: Physikalische Zeitschrift, 2, 329-334 (1901). 



