488 BELL SYSTEM TECHNICAL JOURNAL 



cerned almost exclusively with aggregates of ' 'circuit elements" inter- 

 connected in various ways. It is also true that the most famihar form of 

 circuit equations is that which is similar to Kirchhoff's equations for the 

 steady current flow in networks of conducting rods, published^ in April 1845. 

 This form is applicable only to circuits. However, the application of 

 these ''Kirchhoff equations" to alternating currents, natural as it may seem 

 to us now, was not obvious one hundred years ago. The first equation for a 

 simple circuit consisting of a capacitor, an inductor, and a resistor in series 

 was published in 1853 by Lord Kelvin.^ Interestingly enough his approach 

 is based on the ideas applicable both to conventional circuits and to high- 

 frequency resonators. If q is the electric charge on one plate of the ca- 

 pacitor, the energy stored in the capacitor is q^/2C^ where the coefficient C 

 depends on the geometry of the capacitor. The magnetic energy of the 

 circuit is -| Lq^^ where q is the time rate of change of the charge, that is, the 

 current in the circuit, and L is a coefficient depending on the geometry of the 

 circuit. The rate of energy transformation into heat is Rq^, where i? is a 

 coefficient depending on the geometry of the conductors (and of course on 

 their resistivity). The law of conservation of energy demands that 



^W/2C + m'\ = -Rq-- (1) 



at 



When the differentiation is performed and q is cancelled, the usual form of 

 the equation is obtained. The coefficients of proportionality, that is, the 

 inductance L, the capacitance C, and the resistance R sum up and stress the 

 really important electrical characteristics of the circuit; the details of the 

 construction of the circuit are suppressed. 



It was Maxwell who formulated the general equations for electric net- 

 works by extending the application of a method developed by Lagrange for 

 mechanical systems. This Maxwell did in his last two lectures. In the 

 words of his student, J. H. Fleming:^ "Maxwell, by a process of extra- 

 ordinary ingenuity, extended this reasoning (the method of Lagrange) from 

 materio-niotive forces, masses, velocities and kinetic energies of gross matter 

 to the electromotive forces, quantities, currents, and electrokinetic energies 

 of electrical matter, and in so doing obtained a similar equation of great 

 generality for attacking electrical problems." 



Before discussing the Lagrange-Maxwell method more completely, let us 

 see if we can construct a network whose electrical properties would be the 

 same as those of a continuous medium. 



^ Annalen der Physik. 



2 Philosophical Magazine. 



' Philosophical Magazine, 1885. 



