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Periodic Field of Force. 19 



to the usually assumed type of viscous resistance, may be 

 written in the following form, 



U+£U.+ 2a/(0F(y) = 0, . . . . (1) 



where F(U) gives the distribution of the field, f (t) its 

 variability with respect to time, and 2a is a constant. If we 

 are dealing with oscillations about a position that would be 

 one of stable equilibrium if the field were constant, F(U) 

 may as an approximation be put equal to U. We then have 



(j + AU + 2a/(0U = (2) 



In the experiments described above, the periodicity of f(t) 

 is the same as that of the intermittence of the exciting 

 current. If an alternating current had been used, the fre- 

 quency of f(t) would have been double that of the alternations. 

 In any case we may write 



af(t) = cii sin nt -f a 2 sin 2nt -f a B sin Znt + &c. 



-h b + bx cos nt + b 2 cos 2nt + b B cos ?>nt + &c. . (3) 



Since U is shown to be periodic by experiment, we may write 



U^Ai sinp£ + A 2 sin 2j^ + A 3 sin3/?£ + &c. 



-f B -f B a cos pt + B 3 cos 2pt + B 3 cos 3pt + &c. (4) 



As a typical example of the even types of maintenance, we 

 may take the cases in which n—4p. We have 



af (t) = a-x sin Apt + a 2 sin 8pt + a 3 sin 1 2pt + &c. 



+ b Q + bi cos Apt + b 2 cos Spt + b 3 cos 12pt + &c. (5) 



In this case, and also in the case of the second, sixth, and in 

 fact in all the even types of maintenance, we find that the 

 quantities A 2 , A 4 , A 6 , &c, and B , B 2 , B 4 , &c, do not enter 

 into the equations containing A 2 and B : . We therefore 

 write them all equal to zero. The significance of this is that 

 with the even types of vibration maintained by a periodic 

 field of force, the even harmonics are all absent from the main- 

 tained motion. This result is fully verified by a reference to 

 the vibration-curves of the 2nd, 4th, and 6th types shown in 

 figs. 2, 4, and 6, PL II. It will be seen that the vibratory 

 motion of the armature-wheel has that type of symmetry so 

 familiar in alternating current curves, in which ail the even 

 harmonics are absent. In other words, the image of one- 

 half of the curve above the zero axis, as seen by reflexion in 

 a mirror placed parallel to this axis, is exactly similar to the 

 other half below it. 



C 2 



