introduction: flagellar propulsion / 



erates waves, the form of the latter cannot be such as allow the fila- 

 ment to be wrapped over the surface of a circular cylinder.* 



On the other hand, if the orbit of rotation of each element about 

 the axis of the wave is elliptical instead of circular, forward pro- 

 pulsion can occur even if the frequency of roll is the same as that of 

 the waves. An elliptical orbit could be produced in at least two ways: 



(1) by a difference in the maximum degree of bending of an ele- 

 ment in the dorsoventral and lateral planes while maintaining a 

 phase difference of one-quarter of a cycle between the two cycles, or 



(2) by an alteration in the phase difference of the cycles but keeping 

 the maximum degree of bending the same in the two planes. With 

 a phase difference of less than a quarter cycle, the orbit of each ele- 

 ment about the axis of the wave is elliptical. In the extreme case 

 when the two bending movements are either in phase or 180° out 

 of phase with each other, the path of movement of each element is 

 a straight line inclined at an angle of 45° to its dorsoventral axis; 

 the wave then becomes "two-dimensional." Risk of ambiguity would 

 be reduced if less emphasis could be placed on the "dimensions" of 

 the waves and more on the movements of constituent elements 

 relative to the longitudinal axis of the system. 



The planes of transverse movement of the elements of the tail of 

 a spermatozoon could be determined if it were possible to observe 

 the form of its envelope when viewed along the longitudinal axis 

 under conditions which effectively prevent rolling of the whole fila- 

 ment. If rolling occurs, an elliptical orbit is transformed into a com- 

 plicated curve comparable with those figured by Perrin (1906) in 

 Cristaspira when in active undulatory movement. Much valuable 

 information could perhaps be obtained by photographing a sper- 

 matozoon in the normal way using very short electronic exposures 

 and lenses of very small depth of focus, thus detecting and measur- 

 ing any changes in the plane pf an element's transverse movement 

 during each complete cycle. 



It is generally assumed that the movement of the elements of a 

 flagellum relative to each other is entirely due to bending deforma- 

 tion and that one element cannot be displaced relative to its neigh- 



* If the base of a flagellum is fixed to a cell in an epithelium (as in the choano- 

 cytes of a sponge), individual elements are effectively prevented from spinning 

 about their own longitudinal axes; they can then move in circular orbits and yet 

 provide an efficient means of creating a current of water. 



