Mabch 17, 1899.] 



SCIENCE. 



405 



difference both in the period and the velocity 

 of the component circular waves of like 

 wave-lengths. 



42. Unequal Component Wave- Lengths. — 

 With the front and rear cam axles still 

 respectivelj' left and right, if more turns be 

 put on one than on the other, the harmonic 

 curves will become helical. In other words, 

 the compound of two plane simple har- 

 monic curves of the same wave-length ratio 

 and phase difference at the origin will now 

 be inscribed on a regular helix. If the 

 axles be rotated with the same angular 

 velocity in opposite directions the compo- 

 nent harmonics have the same period, but 

 diflFer in velocity. The vibration lines in 

 the compound wave remain fixed for each 

 particle, but their directions differ in alti- 

 tude proportionally to their distance apart. 

 The rotary polarization so obtained is due 

 to a difference in the velocities of the circular 

 components. The helix may be rotated as 

 a whole by dephasing the initial particles. 



43. If the axles are turned with unequal 

 velocities the helical compound wave must 

 rotate as a whole about the common axis of 

 the component circles, in consequence of 

 the continuous and like dephasing at all 

 cam pairs. Rotary polarization is again 

 due both to difference of velocity and of 

 period, as in § 41. If, however, the period 

 of rotation at the cam axles is proportional 

 to the wave-lengths of the helices the 

 velocities of the components will be the 

 same and the continuous rotation occurring 

 due merely to difference in the periods of 

 the components. Hence, if the oscillation 

 in the first particle of the compound wave 

 is always supplied parallel to a given line 

 the rotary polarization obtained will be due 

 simply to the difference in the periods of the 

 components. 



44. Right-Handed Circular Compioiient Har- 

 monics. — The same amount of rotation as in 

 the last cases will be obtained when the 

 wave-length of one of two equal right and 



left cam axles is increased and that of the 

 other decreased by half the stated increase 

 of the single axle in § 42. It will even be 

 obtained when both cam axles are right- 

 handed screws or both left-handed screws, 

 alike in all respects but differing in 

 phase by 180°, subject as before to counter 

 rotation (cross-belted). But, whereas the 

 rotary polarization in the preceding case, 

 § 42, is due solely to normal advance of 

 the circular waves, it is now due to the 

 independent counter rotations impressed by 

 outside agency. The two right-hand helices 

 specified, being opposite in phase, constitute 

 a series of stresses in equilibrium and pro- 

 duce no displacement. 



If the cam axles of equal wave-lengths 

 rotate with the same velocity the compound 

 wave is a helix, but with each of its par- 

 ticles in the same phase. The neutral posi- 

 tion is thus a line of balls in the common 

 axis encircled by the lever ends, and this 

 may be used as a test on the adjustment. 

 Each particle persists in its line of vibration, 

 and their locus is a helix which expands 

 and contracts in diameter rhythmically. 



45. If the two axles rotate unequally 

 swiftly the component circular waves ad- 

 vance unequally swiftly and the line of vi- 

 bration of each particle or the contractile 

 helix as a whole rotates around the common 

 axis. 



46. Finally, in two right-handed cam 

 axles of equal amplitude, but different 

 wave-length, the resultant harmonic curve 

 will be the compound of corresponding 

 plane harmonics, but inscribed on the cor- 

 responding helix. For rotations of the same 

 angular velocity (equal periods) the helical 

 wave will not rotate as a whole. For un- 

 equal periods it will so rotate. 



Some of these cases are more important 

 than others. Their application is a question 



«f°P*i<^^- Carl Bards. 



Beown University, 

 Peovidkncb, K. I. 



