TRANSACTIONS OF SECTION A. 615 



electromotive force in the electric circuit, when displaced, can be deduced by 

 Faraday's law. 



Consider now the application to alternate current machines. Such machines 

 are usually multipolar. In machines of the disc type the number of poles is even, 

 and the armature is divided mto sections corresponding to the number of poles, and 

 revolves uniformly between them. The tube through any one pair of opposed 

 poles and back through another need only be considered, and the total effect of the 

 machine obtained therefrom by summation. Suppose the iron of both magnets and 

 armature so arranged that no currents are induced therein. There is then only one 

 electric circuit to deal with. The whole current in one section of the armature 

 cuts the magnetic tube passing through the section, as many times as there are 

 convolutions. Let m be the number of convolutions. The current x round the 

 magnets is usually derived from independent sources, and maintained constant. 

 For each such constant value the characteristic surface becomes a curve giving the 

 relation between the induction through the armature and the current in it. The 

 areas Aj, A„,and A^, and the lengths l^, I,, l.^, and l^ are constant, but the area A3 

 is a periodic function of the time, and can be expressed by a series of cosines, the 

 coefficients of the series being determined by Fourier's theorem from the dimensions 

 of the machine. If the equation of the characteristic be dilierentiated with regard 

 to the time, we shall obtain an equation of the form 



A 2/" + B y = periodic function of t, 



when B is constant and A is a periodic function of t, but usually assumed to be 

 constant, and called the ' self induction ' of the machine. 



In general no current continuous in direction can be obtained by continuous 

 rotation of any part of the electric circuit, unless arrangement is made for reversing 

 the current at a certain stage of each revolution. To diminish the oscillation of 

 the current the electric circuit is divided into a number of sections, arranged 

 symmetrically on the armature, the current in one or two of which only is 

 reversed at "a time. If the number of sections be even and equal to 2ni, one 

 half will be in series, and one half the total current will pass through each half, 

 except at the instant of commutation. At such time two sections are short cir- 

 cuited, and form complete circuits in which the current will be determined by the 

 induction through them at the time ; and the number of sections in series will be 

 m — 1. If the number of sections be odd and equal to 2m + 1, one section only will 

 be commuted at a time, and at that instant there will be m sections in series. At 

 other times there will be ?» + 1 sections in series on one side and 7» on the other, 

 and consequently there will be a superposed current flowing through the armature 

 only, due to the inequality in the number of sections in series in the two halves. 

 In one revolution of the armature the tube of induction through it will be cut four 

 times by each section, and if the plane of commutation is symmetrical with regard 

 to the tube of induction the current in one half the sections will cut it in the 

 opposite direction to that in the other half In this case, Jij = ''• But any dis- 

 placement of the plane of commutatiou from the symmetrical position will cause 

 the current in a greater number of sections to cut the tube in one direction than in 

 the other. Let A. be the angular advance of the plane of commutation, and ??i the 



number of sections in the armature ; then «., = — ' The value of X may be fixed 



for any given machine, or varied at pleasure, or may be determined to avoid spark- 

 ing at the time of commutation of a section. The general discussion of the value 

 of X to effect this has not yet been attempted. For the present X must be regarded 

 as independent. The general equation of the characteristic surface becomes for a 

 continuous current machine 



^^^W^ ^^'/^(B- ^3i;-4/.(|>4.«...-4x.,, 



X being reckoned positive when the displacement is in the direction o rotation. 

 If no current passes through the armature, i/ = o and v^ may be taken as constant 



