402 



SCIENCE. 



[N. S. Vol. IX. No. 220. 



X. Components Simple Harmonics at Right 

 Angles to Each Other, with Wave-Length Ratio, 

 1:2. Transverse Space Wave. 31. Harmonic 

 Gurves.—W\i\\ the preceding cam axles, let 

 the rear ends of the leaves be lifted upon 

 the horizontal back plate and adjusted for 

 the same component amplitude (Fig. 4). 



Space waves of this and the following 

 kind may be conveniently termed Lissajous 

 waves, since their sunshine shadow on a 

 screen normal to the axis of motion is al- 

 ways the appropriate Lissajous figure. 

 Starting the waves with the initial eccen- 

 trics towards each other, the harmonic 

 curve has a meandering space form, char- 

 acterized, however, by its sunshine shadow, 

 which is the specific bow-shaped 1:2 Lissa- 

 jous, concave toward the cams. Dephas- 

 ing the rear axle +90° produces the sym- 

 metrical 8-shaped figure ; +90° farther the 

 bow-shape again, this time, however, convex 

 toward the axles of the cams; +90° far- 

 ther returns the 8-shape described in a di- 

 rection opposite to the preceding. The 

 intermediate cases are assymmetrical 8's, 

 but not well given unless the balls are small 

 enough. 



The harmonic curves themselves present 

 no marked complexity. Seen from above 

 they contain two wave-lengths ; seen from 

 the front but one wave, each in the appro- 

 priate phase at the origin. This gives a 

 very clear anal3'sis of the occurrences. The 

 wave envelope in the bow-shaped cases is a 

 gutter. 



32. Waves. — The waves corresponding to 

 the above space harmonics are instructive. 

 If the figure of the compound wave is to be 

 preserved, i. e., if its shadow Lissajous is to 

 remain fixed, both component waves must 

 advance with rigorously the same velocity. 

 This implies double rotation (double fre- 

 quency) for the rear waves of shorter wave- 

 length. The direction of rotation in the 

 shadow is particularly well marked. For 

 initially opposite or for like phases at the 



origin the figure is alike 8-shaped, but 

 when horizontal pointers on the front axle 

 correspond to down on the rear or up on 

 the rear the rotation is clockwise or coun- 

 ter-clockwise respectively in its upper half; 

 etc. 



33. Case X. tvith Component Velocities Un- 

 equal. — If the velocities of the component 

 waves are unequal, but of the same sign 

 (pulley 2:3, for instance), the compound 

 wave continually changes form, as is best 

 shown by the sunshine shadow. This is 

 identical with the Lissajous curve for two 

 tuning forks of the same amplitude, but 

 with period ratios slightly different from 

 1:2. If the speeds of the two axles are 

 equal (pulleys 1 : 1) a single rotation of the 

 crank produces all the Lissajous between 

 two occurrences of the same figure. 



If the component periods are equal, but 

 of opposite sign, stationary wave conditions 

 appear for this case. Particles at the ends 

 of the compound wave oscillate in any 

 fixed Lissajous ; the intermediate particle 

 has the inverse figure. In general the perma- 

 nent vibration figures vary proportionally 

 to the distance apart of the particles. The 

 sunshine figure is reproduced for 1/2 rota- 

 tion at the crank. One may note the con- 

 trast that, whereas the particles themselves 

 vibrate in the elliptical Lissajous series, the 

 sunshine shadow produces the 2 : 1 series. 



If the component periods are unequal 

 and opposite in sign the figures drift as 

 above. The transitional case is given when 

 but one axle rotates. 



XL Component S. H. Motions Coplanar 

 with Wave-Length Ratio, 2:S. 34. Plane 

 Harmonics and Waves. — The front cam axle 

 is replaced by one containing 3 wave- 

 lengths, with adjustments as above (Fig. 

 3). The curves of this series are more 

 complex than the preceding, and if the de- 

 phasing be eifected in steps of 90° each, 16 

 marked forms of curves may be exhibited. 

 Among these the symmetrical types are 



