614 KEPORT— 1887. 



opposed pole pieces, (iv.) The armaturo, or Ibat part of the machine carrying that 

 portion of the electric circuit which is subject to displacement under external 

 forces. 



The magnetic circuit is thus subject to magnetic forces due to the current in the 

 armature and the current round the magnet limb, ^^'e must therefore, in the 

 general case, take these as the two independent variables, which we may denote by 

 C and c. Now, it may be assumed with sufficient accuracy that in the magnet 

 limb the boundaries of the tube of magnetic induction are coincident with the 

 boundaries of the iron, and the cross section of the tube the same as the cross 

 section of the iron. Outside the limb a portion of the lines of force wUl complete 

 a magnetic circuit through external space, and will not enter the pole pieces. The 

 extent of this leakage or induction, from which no useful etlect is obtained, depends 

 upon the configuration of the machine and the degree of saturation of the iron, and 

 could be calculated therefrom ; but as it can be experimentally determined for any 

 machine with great ease, it is unnecessary to consider it further, and we may regard 

 the total induction in the magnet limb as greater than the induction in the pole 

 pieces in a constant ratio, wliich we will denote by ^,. It is usual to construct the 

 pole pieces of large section compared with the magnet limb, and hence the section 

 of the pole pieces may again be taken as the section of the tube of induction ; but 

 as the lines of force leave the pole pieces to cross the air space, we cannot ascribe 

 any boundary to the tube, but in every mHchine a portion only can pass through 

 the armature, and part must pass from one pole piece to the other by lines external 

 to the armature. Moreover, the relation between the two parts will not be a con- 

 stant one, unless the magnetic forces in the armature are constant, which can never 

 be the case. It is therefore necessary to consider the tube of induction, wliich 

 crosses the air space and enters the armature, as a variable portion of the whole 

 tube, the variation depending upon the magnetic forces in the armature. We may 

 denote the ratio of the induction through the pole pieces to the induction through 

 the armature by Un. Let Ai be the cross section of the magnet limb, li its length; 

 A„ the cross section of the pole pieces, /., the mean length of the tube therein ; A3 

 the cross section of the air space, comprising all the space through which the lines 

 entering the armature pass, /j its length ; A^ the cro.ss section of the iron of the 

 armature (if it contains iron), and /, the mean length of the ttibe of force therein. 

 Then the line integral of magnetic force taken round the circuit is : — 



I , V.f.J 1 , kJ / 1 /, 1 ^4 \r 



/xj 1 Ai fi, -A.,3 V3A3 fi^A,/ 



the n'a being the coefficients of magnetic induction for the several portions of the 

 circuit. For air the coefficient is unity, hence ^3 = !. I is the total induction in 

 the armature, which is assumed to be uniformly distributed over the tube. Now 

 the magnetic circuit is cut by the current in the magnet coils and the current in 

 the armature. Let n^ be the number of times it is cut bv the former, «., by the 

 latter. Then 



1 i/.r.T 1 I' J /L 1 L\ 

 M, ^^ AT ^J^J^X^ VA3 ^ ,, A> = '^(^'^^ + "^C). 



It must be noted that the direction in which the circuit is cut by c and is in both 

 cases taken to be positive. Referring to thi'ee rectangular axes and measuring the 

 induction along the axis of s, the current round the magnets along the axis of x, 

 and that in the armature along the axis of ?/, the above equation may be written 



This represents a surface the ordinate at any point of which is the induction 

 through the armature. Such a surface was first described by Dr. John Hopkinson 

 (' Lecture before the Inst, of C.E.,' April, 1883), and is called the ' characteristic 

 surface.' 



Having obtained a general expression for the induction in the armature, the 



