bipolar and for the monopolar receiver, and in order to simplify the equations, let us write

Equations (15), (16) and (19) assume that the pull on the diaphragm due to i is in the phase with i; but with hysteresis and eddy currents present, the electromagnetic force will lag behind the current i by an angle 1; whence the force on the diaphragm due to the current i becomes, by eq. (19),

The e. m. f. induced in the coils by the motion of the diaphragm will be, in the absence of hysteresis,

and by differentiating equation (10) or (11), equation (22) gives to a first approximation

2Nax
R

However, it should be noted that there is also a hysteretic lag of flux with change of gap, and this will cause the induced e. m. f. to lag by a certain angle ẞ2 behind x, so that equation (23) should be changed

If L and R are the inductance and resistance of the receiver when damped, the impedance of the damped receiver will be

and if e is the instantaneous value of the impressed e. m. f. of the type Eet, we shall have

But owing to the influence of the e. m. f. of motion, the last equation becomes modified to

where Z' is the free impedance of the receiver.

This means that the impedance of the receiver has become increased, through the vibration of the diaphragm, by a motional impedance:

This motional impedance, being the reciprocal of the vector equation of a straight line with w as variable, is a circle for variable ω, and has a

A2

diameter depressed below the axis of reals by an angle ẞ1+ B2.

As to the relative values of B1 and B2 it seems reasonable that whether the change of flux of a circuit is caused by a small change of current, changing the m. m. f., or by a small change of gap-length, changing the reluctance, the angle of lag of flux behind the cause is the same; that is ẞ2 = ẞ1 B (Say). This is borne out by one of our experiments to be described below (see VI). With this equivalence substituted in equations above, we obtain,

=

Consequently, if we vary w from 0 to +, keeping the impressed e. m. f. and all other quantities constant, the motional impedance. Z' - Z has a circular graph through the origin, with its principal

A2

diameter of length depressed 23 below the axis of reals. Equation

r

(31) is the theoretical equation to the circular graphs of Figures 7 to 12. Replacing the vector z of equation (31) by its absolute value [z] and angle a, we have

are functions of w. The quantity A, involving 330 and , might be expected to vary with variation of w, but an examination of the experimental results shows that, with the excitations employed, not much error is introduced by considering A and also independent of w.

Equations (32), (33), and (34) are in convenient form for computation, and permit an easy determination of some of the important mechanical constants of the diaphragm.

For example, if we let wo be the angular velocity of impressed mechanical force for which the sustained vibration of the diaphragm is in resonance, we see from equation (6) above that

Now, if w, the angular velocity of the impressed electromotive force in

that a becomes zero; hence the value of w, which in the experimental circular graphs of Figures 7 to 10 lies at the remote end of the principal diameter is the w= wo for which the diaphragm in sustained vibration is resonant. This gives a simple and accurate method of determining wo for a telephone diaphragm.*

Again, let A be the logarithmic decrement per second of the diaphram, if vibrating under no external force, then by the theory of elasticity,

Differentiating (39) with respect to a, we obtain

That is, in the experimental circular graphs, the rate of change of w with change of a, at the remote end of the principal diameter, is the logarithmic decrement per second of the diaphragm. This quantity cannot, however, be obtained with the precision with which w can be obtained.

W2

Another method of obtaining A is by taking the values of w1 and which lie respectively 45° below and 45° above the principle diameter, these angles being measured at the origin, not at the center. For these points tan a is respectively + 1 and 1; whence from (39)

and by subtraction and division by 2 (w+w2),

4 For another method of finding wo from the humming tone of a telephone receiver, see a paper by A. E. Kennelly and W. L. Upson, Proc. Am. Phil. Soc., 1908, "The Humming Telephone."

Thus we have methods of determining both wo and A. The experiments, on the other hand, do not permit a direct determination of the quantities m, r, and s; but it would seem that by adding a known mass, as a small load, to the center of the diaphragm and repeating the series of measurements, these quantities should be capable of determination.

VI. COMPARISON OF EXPErimental ReSULTS WITH THEORY.

An examination of the experimental results with the aid of the theory above developed gives the following results, which may be called the characteristics of the several receivers (Table XI):

The method of obtaining these characteristics was as follows: The circular graphs of Figures 7 to 11 were plotted. The diameter of the motional impedance circle and the angle of depression of this diameter below the axis of R'-R could be measured off at once on the diagram. The value of w at the free end point of the diameter could also be read or obtained by interpolation; this w is the w, of the diaphragm. The logarithmic decrement per second A could have been obtained by either of the two methods derived in the discussion of the theory, equations (41) or (42); but a third method was employed; namely, by the use of the more general equation (39), in which several values of a and the corresponding values of w from the circular graphs were substituted, and the values of ▲ so obtained were averaged.