PATHS OF MOVING PARTICLES SPRUNG 6l 



from which it is apparent that the point moves in an ellipse whose 

 semi-axes are a and b; for from (2) we get the equation of the ellipse 



x\ 2 (y 



If Ave put 



+ {V 



<» = ~y W 



then T denotes the entire time of revolution of the point in the 

 ellipse, for when t = T then both the coordinates and the com- 

 ponents of the velocity U x and U y attain the same values they had 

 at the time t = o. Since <o denotes the angular velocity of the 

 rotating surface, equation (4) shows that for the absolute motion 

 (elliptical) the time of revolution agrees with that of the rotating 

 surface. 



The absolute motion of the point can now be easily constructed. 

 Let us choose for example the time of revolution of the surface (and 

 of the material particle) as T — 24 seconds (compare fig. 1, page 

 62) and divide the circumference of a circle constructed on the di- 

 ameter 2a into 24 equal parts and from the points of division let fall 

 perpendiculars on the diameter, which in this case can be done by 

 joining the points in pairs by straight lines as the points of division 

 are symmetrically distributed with respect to 2a. The 12 diameters 

 which join the 24 points of division divide at the same time into 24 

 equal parts a circle constructed on the small axis 26; from the points 

 of division of this small circumference let fall normals to the di- 

 ameter 26, which is perpendicular to 2a, and prolong them on both 

 sides to the larger circle. If the time t is reckoned from the moment 

 at which the body is at the extremity of the radius b, then the 

 points of intersection of the two systems of normals which are marked 

 o, 1, 2, 3 . . . lie on the elliptical path characterized by 

 equations (2), (3) and (4) for T = 24, from which the correspond- 

 ing relative motion is derived in the following manner. 



A rotation of the surface about an angle to, 2co, 30* . . . cor- 

 responds to the absolute motion of the body up to the points 1, 2, 3, 

 ; evidently therefore we can find the positions at the mo- 

 ment t = o of those points I, II, III ... of the rotating sur- 

 face which will come in contact with the body after 1,2,3 

 seconds, by going toward them in a direction contrary to the mo- 

 tion of rotation of the surface along the concentric circles through 



