2 JAMES GRAY 



suitable external resistance. In a flagellum the changes in shape are 

 effected by bending each short length of the filament, and the external 

 resistance is provided by the viscosity of the medium. The waves of 

 changing curvature passing distally along the filament are an ex- 

 pression of the fact that each short element is not only bending but 

 also executing transverse movements in a plane normal to the axis of 

 vision and that the phase of bending reached by any element at any 

 instant is slightly ahead of that of its next posteriorly situated neighbor. 

 The form of the waves depends on the radius of curvature exhibited 

 by the elements when bent to their maximum extent and on the dif- 

 ference of curvature exhibited by adjacent elements at any particular 

 moment. In order to elicit a forward thrust from the medium, an 

 element must move with its surface inclined at an angle (0) to its 

 transverse path of motion; in other words, as it sweeps from side to 

 side, it must have a component of motion normal to, as well as one 

 tangential to its surface (Gray and Hancock, 1955). Under such 

 conditions a forward thrust ensues provided the coefficient of resist- 

 ance, C D , to movement in a direction normal to the surface is greater 

 than that, C L , to movement tangential to the surface; Hancock (1953) 

 has shown that for very thin and slowly moving filaments C N equals 

 2C L . Thus the propulsive force, dF, exerted by an element at any 

 instant is determined by three factors: (1) the tangential drag coeffi- 

 cient, C L ; (2) the element's velocity of transverse displacement, dy/dt; 

 and (3) the angle between its surface and its path of motion, dy/dx. 



dt dx 

 If we are prepared to assume that the form of the waves is the same 

 as that of a sine curve of appropriate length, A, and amplitude, b, 

 it is possible to express the speed of propulsion (V x ) of the filament in 

 terms of the form and frequency,/, of the waves passing distally along 

 it (Taylor, 1951, 1952; Hancock, 1953). This relationship is inde- 

 pendent of the absolute value of C L . 



Ijiflr I 1_ 



A" 



After allowing for a reduction in speed caused by the drag of an 



