Physical Aspects of Protoplasmic Streaming 229 



resultant is reduced. Consequently, the resultant wave undergoes 

 pronounced variations in amplitude. In the former case the result- 

 ant is "constructive", whereas, in the latter case it is "destructive". 



When two simple tones, the frequencies of which differ slightly, 

 are sounded together, a throbbing is heard. The two notes are said 

 to produce "beats." This is a fact well-known to acoustics, and is 

 represented in Figure 14 where the thin lines show two component 

 waves, the ratio of the frequencies of which is 8:9. The resultant 

 of these two component vibrations is shown by a thick line. The 

 amplitude increases, when crest meets crest and trough meets 

 trough, whereas, it decreases when crest meets trough. The number 

 of beats per unit time is equal to the difference in the frequencies 

 of the components during the same unit time. Therefore, the longer 

 the period of one beat, the less the difference between the frequencies 

 of the two components. 



Although the component harmonic curves of different wave 

 lengths can no longer be compounded into a single harmonic curve, 

 as shown on the graph in Figure 14, the resultant is, nevertheless, 

 periodic, if the wave length of the components are commensurable. 

 If, on the other hand, the wave lengths are incommensurable, the 

 period of the resultant curve is infinite, i. e., the resultant is non- 

 periodic. 



It is extremely interesting to compare the above-mentioned result- 

 ant wave with the experimental curve of the motive force generated 

 in protoplasm. The alternation of waxing and waning periods bears 

 a strong resemblance to the phenomenon of beats. In fact, increase 

 and decrease in the amplitude of the dynamoplasmogram (Figs. 7 

 and 8) are, in all probability, due to the interference of intraplasmic 

 forces, the rhythmic frequencies of which are slightly different from 

 each other. At the waxing periods, two main component rhythms 

 of the motive force reenforce each other, whereas, at the waning 

 periods these two main groups oppose each other. In Figure 7, there 

 are about eight waves in each "beat" of the dynamoplasmogram. 

 Since one of the waves must gain a wave length on the other for 

 each beat produced, one can say that the period of one of the two 

 main components must differ about Vs from that of the other. As 

 one "beat" continues here for about 11 minutes, it is possible to 

 evaluate the approximate difference in the period between the two 

 components for about 10 seconds. In the case of Figure 8, the period 

 ratio between the two main components will be ca. 3 : 4. 



But the explanations above-mentioned are not quite enough. 



