14 LORD ROTHSCHILD 



or helical, that is to say three-dimensional. There is no doubt that 

 the bull spermatozoa swimming in the unusually viscous medium 

 passed two-dimensional waves along their tails, so that we can say that 

 bull spermatozoa do sometimes swim in this way. But, for photo- 

 graphic purposes, a cover slip was put on the drop of sperm suspen- 

 sion. This means that the spermatozoa were in an abnormal environ- 

 ment, in the sense that helical waves might have been interfered with 

 by there being too little distance between the microscope slide and the 

 cover slip. According to van Duijn (personal communication), bull 

 spermatozoa always propagate helical waves along their tails when the 

 distance between the microscope slide and the cover slip is more than 

 40 microns. Rikmenspoel had, I think, the same view in 1957. I am 

 certain this statement is not true when bull spermatozoa are ex- 

 amined in media with a viscosity of 2000-3000 centipoises. I have no 

 apparatus with which to decide whether the waves are planar or heli- 

 cal when they are in their normal medium, at 37°C. I may have con- 

 tributed to the differences of opinion about helical and planar waves 

 by misinterpreting photographs of the apparently spiral tracks of 

 sea urchin sperm heads under dark-ground illumination (Rothschild 

 and Swann, 1949). Having drawn attention to this misinterpretation 

 and having said that the waves which travel along sea urchin sperm 

 tails are two-dimensional, Gray (1955) went on to show that, while 

 propagating such planar waves, sea urchin spermatozoa rotate at a 

 frequency of about two revolutions per second, whereas the tail 

 wave frequency is twenty times as great. There is, of course, a dif- 

 ference between a planar wave, which periodically alters its plane 

 relative to fixed axes, and a helical wave. The equations character- 

 izing the two types of motion are, in cylindrical coordinates r, B, x 



r = b sin k(x + VJ) (1) 



and 



(2) 



k(x + VJ) 



where b is amplitude, k = 2tt/a, A is wavelength, V w = r\, the 

 velocity of waves relative to the sperm head, v is frequency, w = 2,-n-x, 



