4 JAMES GRAY 



axis depends on the fact that the sperm is both yawing and rolling 

 as it moves forward (Gray, 1955). There are thus very substantial 

 differences between the observed movements and those which form 

 the basis of a calculated speed of progression. 



THREE-DIMENSIONAL WAVES 



From time to time — indeed, during this symposium — the concept 

 of three-dimensional waves has been invoked to account for the ro- 

 tation of a sperm about the longitudinal axis of its progression. To 

 this there can be no objection provided there is no ambiguity about 

 what is meant by a "three-dimensional" wave. It is convenient to 

 start an analysis of such movements from the assumption that all the 

 elements of a filament are bending simultaneously in two internal 

 planes normal to each other, e.g., dorsoventral and lateral. If two con- 

 ditions are satisfied — and only under such conditions — a filament 

 generating the waves could be wrapped round the surface of a cir- 

 cular cylinder during all phases of its activity; these conditions are 

 (1) that the maximum degree of bending should be the same in both 

 planes and (2) that the phase of bending reached by an element in 

 one plane is always one-quarter of a cycle behind that reached by it 

 in the other. When generating waves of this particular type the en- 

 velope of the filament would appear as a circle when viewed along 

 the axis of the circumscribed cylinder, and every element would move 

 round this circular orbit once during the passage of each complete 

 wave. If such terms as "three-dimensional" or "helical" are restricted 

 to waves of this particular form, misunderstanding is likely to oc- 

 cur. The elements constituting such waves cannot perform their cir- 

 cular motions against the resistance of the water without generating 

 couples tending to rotate the whole filament about the axis of the 

 wave in a direction opposite to that in which the elements are travel- 

 ing in their circular orbits (Fig. 1) (Taylor, 1952; Gray, 1953). The 

 net result of rolling the filament about the axis of the wave is to re- 

 duce the speed at which each element is traveling along its own or- 

 bit and to reduce the speed at which the whole wave is traveling 

 backwards relative to fixed axes and to the surrounding water (Fig. 

 2) (Gray, 1953). In the limiting case when the filament rolls through 

 360° with the same frequency as that at which the waves are travel- 



