Physical Aspects of Protoplasmic Streaming 239 



Using the same model, the case will now be considered in which 

 the two cranks are turned slowly with the same angular velocity 

 (change in pressure due to positive and negative acceleration of 

 water is neglected) , and the air-pressure in compartment B is con- 

 stantly adjusted so as to stop the movement of water in the connecting 

 tube. Since there is no exchange of water between the two reser- 

 voirs under such circumstances, their levels show motion which is 

 exactly the same as that of the pistons; this is simple harmonic 

 motion. The combination, either addition or subtraction, of two 

 simple harmonic motions having the same frequency gives rise to 

 another simple harmonic motion of the same frequency. This being 

 so, the balance-pressure which holds the water in the tube at a stand- 

 still and is equal to the pressure that is equivalent to the difference 

 between the two levels, can be expressed by a simple harmonic 

 function of time. This holds true, no matter what the phases and 

 amplitudes are, provided the angular velocities of the two cranks 

 remain the same. 



The next case to be considered is that in which the two cranks 

 have slightly different angular velocities. The motion of the pistons 

 is of a simple harmonic nature, but each motion has different frequen- 

 cies. At some point in the time scheme, the pulsation of the pistons will 

 be in the same place, but the next moment the phase of the pulsation 

 in one reservoir will have advanced beyond that of the other. After 

 a certain period of time, periodic movement of the pistons in both 

 a and b will attain phases which directly oppose each other. 



If, when the two pistons pulsate with slightly different fre- 

 quencies, one plots the change in balance-pressure of the model 

 against time, there will result a curve which reveals a pattern with 

 alternating periods of increase and decrease in amplitude exactly 

 as in the case of beats (cf. Fig. 14) . When the pulsation of the two 

 pistons are in the opposite phases, the amplitude of the curve will 

 reach its maximum (waxing period) . When the two pulsating 

 motions are in the same phase, the amplitude of the curve is at its 

 minimum (waning period) . 



It is likely, though not necessarily so, that in one plasmodium, 

 when its dynamoplasmogram reveals a wave pattern resembling beats 

 (cf . Figs. 7 and 8) , the rhythmic frequency of the protoplasm in one 

 compartment is not the same as the frequency of that in the other, 

 just as in the case of the two reservoirs, when each has a different 

 pulsating frequency. 



