TRANSACTIONS OF SECTION A. 7390 
second slide-wire of equal resistance to the slide-wire on which the galvanometer 
contact works, the zero for the galvanometer slide-wire being taken at a point 
so far along the wire that the piece between this point and the end shall be equal 
to the length of the other slide-wire between the end and the other contact of the 
galvanometer circuit. Nickel steel wire is used for the slides. 
6. The Division of an Alternating Current in Parallel Circuits with 
Mutual Induction. By FREDERICK BEDELL. 
A divided circuit with mutual induction between the two branches is the same 
as a transformer with the primary and secondary circuits connected in parallel. 
The problem may be treated in the same manner as that of the transformer. The 
electromotive force equations for the two circuits are similar, the internal electro- 
motive forces in each being equal to the same impressed electromotive force. The 
electromotive force of mutual induction will be positive or negative according to 
the sense or direction in which the coils are connected. If the coils are connected 
so that the ampere turns of the two coils assist each other, the electromotive force 
of self and mutual induction will be of the same sign, and the coefficient of mutual 
induction will be positive. If the coils are connected so that the two oppose each 
other, the electromotive force of mutual induction will be opposed in sign to that 
of self-induction. The coefficient of mutual induction may accordingly be plus M 
or minus M. Writing the electromotive force as a function of the time, the 
electromotive force equations for the two circuits are: 
e=f(t)=R,z, + L, De, + MDz, ; 
e=f(t)=R,2, + L,Di, + MDz, ; 
where e and 7 represent current and electromotive force, R and L represent resis‘- 
ance and self-induction, and D stands for the operator & . The solution of these 
equations gives us the values for the currents in the two circuits, and their phase 
relations. Where the coils are opposed and nearly similar, the angle of phase 
ene between the currents depends largely upon the amount of magnetic 
eakage. 
The graphical treatment of the problem shows this relation more clearly. The 
electromotive force to overcome the resistance of each circuit is represented by a 
vector in the direction cf the current. The electromotive forces of self and mutual 
induction are at right angles to the currents in their respective circuits. This 
gives us three vectors for the electromotive forces in either circuit, and the sum of 
these three vectors in either circuit is equal to the electromotive force impressed 
upon the two circuits. The direction of the vector representing the electromotive 
force of mutual induction depends upon the sense in which the coils are 
connected. 
The equivalent resistance and self-induction of the two coils together, whether 
they are additive or opposed, may be found by resolving the electromotive force 
into two components, one in the direction of the main current, and the other at 
right angles to it. The resultant of these components may be obtained graphically 
and from them the values of the equivalent resistance R1, and the equivalent self- 
induction L'. The equivalent resistance and self-induction of their branches may 
be obtained in the same manner. 
Particular cases may be discussed by assuming definite values for the constants 
of the circuits or definite relations between them. 
