26 



GEOMETRY OF MOTION. 



[Si- 



When the resultant translation and one of its components are 

 given by their vectors, the process of finding the other com- 

 ponent is called geometric sub- 

 traction. It is effected, like 

 algebraic subtraction, by re- 

 versing the sense of the com- 

 ponent to be subtracted, and 

 then geometrically adding it 

 to the resultant (Fig. 15). 

 In other words, the geometric 

 difference of two vectors AB 

 and CD is found by geometri- 



Fig. 15. 



cally adding to AB a vector 

 equal but opposite to CD. 

 Thus, in Fig. 15, 02 is made equal and parallel to AB ; 21 is 

 equal and parallel to CD reversed, that is to DC\ 01 is the 

 required difference. 



51. The composition of translations by geometric addition of 

 their vectors (Art. 47) holds, not for successive translations only, 

 but, owing to the commutative law (Art. 48), for simultaneous 

 translations as well. This is easily seen by resolving the com- 

 ponents into infinitesimal parts. 



To obtain a clear idea of two simultaneous translations it is 

 best to imagine the body as having one of these translations 

 with respect to some other body, while the latter itself is sub- 

 jected to the other translation. A man walking across the deck 

 of a vessel in motion, an object let fall in a moving carriage, a 

 spider running along a branch swayed by the wind, are familiar 

 examples. 



52. This leads us to the idea of relative motion. 



Properly speaking, all motion is relative; that is, we can 

 conceive of the motion of a body only with regard to some other 

 body, called the body of reference. If the latter be regarded as 

 fixed, the motion of the former is called its absolute motion. 



