OBSERVATIONS OF CHANGE OF POSITION 29 



could be represented, as in this case, by the three sides of a 

 triangle. In other words, we may have one side of any tri- 

 angle representing in direction and magnitude the displace- 

 ment which takes place in a given interval of time, when a 

 body is simultaneously acted upon by two causes which would, 

 if they acted at different but equal intervals, produce dis- 

 placements represented respectively in magnitude and direc- 

 tion by the other two sides of the triangle. This generalisation 

 may be expressed in still another form. We may say that if 

 a body is subjected to conditions which would bring about a 

 certain speed in a certain direction, and also to further con- 

 ditions which would bring about, if the previous conditions 

 had not existed, another speed in another direction, then the 

 direction and magnitude of the resultant speed, which the 

 body really acquires under the joint conditions, may be calcu- 

 lated from the rectilinear distance to which the body would 

 have been moved if the component displacements had suc- 

 ceeded one another instead of being simultaneous. In case 

 of two causes, or two combinations of causes, which would 

 be capable in succession of giving to the body the speeds re- 

 presented by A c and c B happening to coincide in time, then 

 the line A B represents, in magnitude and direction, the resultant 

 speed, or the direction and rate of the resultant displacement. 

 We have therefore to remember, in connection with changes of 

 this kind, not only direction but speed, or rate of displacement. 



26. Further Consideration of Simultaneous Displacements, 

 In the last section we have learnt how to find the resultant 

 of two displacements. The method given is applicable to any 

 number of displacements. It is merely necessary to note that 

 our diagrams may be polygons instead of triangles, and that our 

 method only applies to displacements which take place in the 

 same plane. For displacements in different planes diagrams 

 on paper would not suffice, but wire models may be constructed 

 to exhibit both direction and magnitude. 



It is easy to see that if a body is acted upon by a cause 

 which is capable of giving a displacement equal and opposite 

 in direction to the resultant of displacements which would be 

 produced by the action of other causes, then the body is at rest 



