Using the Feynman-Dyson method, the cross section for the creation of an electron pair by a fast charged particle is calculated but in a manner more precise than that of Bhabha and others. The diiFerential cross section obtained here is valid as .long as the energies of participant particles are large ~ompared with the respective rest masses. The ambiguities in Bhabha's calculation :0 e also examined~ It is concluded that the theoretical value must be compared with the experimental results produced by high energy electrons (:2:; 10 Bev). § 1. lntrodnction The creation of an electron pair by a charged particle of spin 1/2 was investigated by Bhabha1', Nishina et al. and others!!), and their results have been applied for analysing highly energetic electromagnetic phenomena in cosmic rays underground. The results of such analyses seem to show that quantum electrodynamics is valid even at extremely high energy, say, i015 eV, though the comparison of the theories with the experiments is done indirectly3>. However, recent developments of the experimental techniques of photographic emul­ sion have made it possible to measure directly the cross section of the process in ques­ tion, called trident. Koshiba and Kaplon*4' have indicated that the experimental value of the cross section for tridents produced by a high energy electron is in disagreement with the theoretical _value given by Bhabha. It is an interesting and important problem whether there really exists a discrepancy between the experimental and the theoretical results, because it is believed that quantum electrodynamics gives a correct description of electromagnetic phenomena extensively. In order to clarify this point it will be necessary on the one hand that the experimental analyses of tridents be performed more accurately while on the other hand that the theoretical results be derived as strictly as possible following quantum electrodynamics. As is well known, Bhabha's results have been extensively used for analyzing cosmic ray phenomena. In his approach the incident particle is regarded as a classical one moving along a straight line with uniform velocity, the field of which is replaced by a classical field. This approximation is essentially based on the same assumption as used in