GEOMETRICAL REPRESENTATION OF THE EQUATION OF MOTION. 29 



683. The motion represented by equation (16) is susceptible 

 of a simple geometrical representation. The position of a point 

 on its trajectory is at any moment the projection of a movable 

 body which would traverse, with a constant angular velocity y, a 

 logarithmic spiral (Fig. 132) in which the constant angle of the 

 tangent with the radius vector will be denned by the equation 



cot </> = -. 

 7 



The equation of this curve is, in fact, 



p = Ae~y* cot * = A<?- e '. 



If we count the time from the moment at which the radius vector 

 OM is vertical, the projection of the movable body on the horizontal 

 is at the time / x at a distance from the pole defined by the equation 



x = A<?~ 6 ' sin yt. 



The deflections E and E' correspond to the points of the curve where 

 the tangent is vertical. These points are given by the intersection 

 of the curve by a right line BB' passing through the pole O, and 

 making the angle <f> with the vertical. 



684. In the most general case in which the directing couple, 

 owing to disturbing actions, is no longer proportional to the de- 

 flection, we may develop the function f(x) by Taylor's theorem. 

 We shall have then, observing that/(^ ) = 0, 



If we assume that the resistance is proportional to the velocity, the 

 equation of motion is of the form 



(22) -j- + 2 - + w 2 ^ + ;z' 2 ^ 2 + ... = 0. 



at at 



When the deflections are so small that the terms of an order 

 higher than x 2 may be neglected, we may take as integral of 

 equation (22) 



^- 2e M i+cos 2 y (/-/ ) . 



