For any increase in the coefficient, the spring in Fig. 2 lengthens, Q increases, and the frictional resistance on the surface of the pulley increases, both in consequence of the increase of Q, which increases the pressure on the pulley, and of the increase of the coefficient of friction.  Similarly for any increase of the coefficient of friction, the spring in Fig. 3 shortens, P diminishes, and the friction on the surface of the pulley diminishes so far as the diminution of P diminishes the normal pressure, but on the whole increases in consequence of the increase of the coefficient of friction.  The value of the friction on the surface of the pulley, however, is more constant for a given variation of the frictional coefficient in Fig. 3 than in Fig. 2, and the variation of the difference of tensions to be measured is less.  Fig. 3, therefore, is the better form.

A numerical calculation here may be useful.  Supposing the break set to a given difference of tension, Q-P, and that in consequence of any cause the coefficient of friction increases 20 per cent., the difference of tensions for an ordinary value of the coefficient of friction would increase from 1.5 P to 2 P in Fig. 2, and from 1.5 P to 1.67 P in Fig. 3.  That is, the vibration of the spring, and the possible error of measurement of the difference of tension would be much greater in Fig. 2 than in Fig. 3.  It has recently occurred to the author that a further change in the dynamometer would make the friction on the pulley still more independent of changes in the coefficient of friction, and consequently the measurement of the work absorbed still more accurate.  Suppose the cord taken twice over a pulley fixed on the shaft driven by the motor and round a fixed pulley, C.

For clearness, the pulleys, A B, are shown of different sizes, but they are more conveniently of the same size.  Further, let the spring balance be at the free end of the cord toward which the pulley runs.  Then it will be found that a variation of 20 per cent. in the friction produces a somewhat greater variation of P than in Fig. 3.  But P is now so much smaller than before that Q-P is much less affected by any error in the estimate of P. An alteration of 20 per cent. in the friction will only alter the quantity Q-P from 5.25 P to 5.55 P, or an alteration of less than 6 per cent.

[Illustration:  FIG. 4]

To put it in another way, the errors in the use of dynamometer are due to the vibration of the spring which measures P, and are caused by variations of the coefficient of friction of the dynamometer.  By making P very much smaller than in the usual form of the dynamometer, any errors in determining it have much less influence on the measurement of the work absorbed.  We may go further.  The cord may be taken over four pulleys; in that case a variation of 20 per cent. in the frictional coefficient only alters the total friction on the pulleys 11/4 percent.  P is now so insignificant compared with Q that an error in determining it is of comparatively little consequence.