86 LECTURES TO SCIENCE TEACHERS. 



in a plane is known if the motion of any two points (i.e. of a 

 line) in it be known. The motion of any body having 

 conplane motion is known if the motion of a plane section 

 of it, parallel to the plane of motion, be known. Such a 

 plane section of it is, of course, simply a plane figure moving 

 in its own plane. The motion of any body having conplane 

 motion (as in nine cases out of ten in machinery), can therefore 

 be determined by the determination of the motion of two 

 points. In speaking now, therefore, of the motion of a line 

 for shortness' sake it must be remembered that we are really 

 covering all cases of conplane motion of solid bodies. 



In Fig. 4 PQ and T? 1 Q 1 are two positions of the same plane 

 figure, or plane section of a body having conplane motion. 

 If now we have two positions (in the same plane) of any 

 plane figure, we know that the figure can always be moved 

 from the one to the other by turning about some point in the 

 plane. The position of the point 0, about which the figure 

 can be turned from the position PQ to the position P L Q 1 

 can be found at once by the intersection of the normal 

 bisectors to PP 1 and QQ V The motion of PQ in the 

 plane is, of course, its motion relatively to the plane, and 

 therefore relatively to any figure (as A B) in the plane. 

 Such a point as we have found here is called a temporary 

 centre, because the turning or motion takes place about it for 

 some finite interval of time. It will be remembered that not 

 only the two points and PQ of the figure, but every other 

 point of it, must have a movement about this same point 

 at the same time. Now suppose we have some further posi- 

 tion of the same figure, as for example at the position marked 

 P 2 Q 2 , we can find in the same way the centre about which the 

 figure must be turned to move from P 1 Q 1 to P 2 Q 2 . We may 

 indicate this point as r Similarly taking other positions of 

 this figure P^Q S and so on, we can find other points, 2 S , &c. 

 By joining the points OOfl^D^. we obtain a polygon, and if 

 the figure in its motion come back to its original position the 

 polygon also comes back on itself, and passes again through 

 the point 0. Such a polygon, whether it be closed in this way 

 or not, is called a central polygon ; its corners are the tem- 

 porary centres of the motion of the figure. 



I have pointed out that all the points in the figure PQ move 

 round during the motion from PQ to P^. They move 

 round necessarily through some particular angle, the angle 



