55-] TRANSLATIONS. 2 / 



Thus in speaking of the motion of a railway train, we usually 

 regard the earth as fixed and can thus call the displacement oi 

 the train from one station to another an absolute displacement. 

 If, however, the motion of the earth with regard to the sun 

 be taken into account, the displacement of the train from 

 station to station is the relative displacement of the train with 

 respect to the earth ; and its absolute displacement would be 

 found by combining this relative displacement with the abso- 

 lute displacement of the earth (with respect to the sun regarded 

 as fixed). 



53. It follows that when the two displacements are transla- 

 tions the absolute displacement of the body will be found by 

 geometrically adding its relative displacement to the absolute 

 displacement of the body of reference. And conversely, the rela* 

 tive displacement of a body is found by geometrically subtracting 

 from its absolute displacement the absolute displacement of the 

 body of reference. 



54. Analytically, the composition and resolution of vectors is 

 merely a problem of trigonometry. Thus, the resultant of two 

 sectors is the diagonal of the parallelogram formed by the two 

 vectors as adjacent sides ; the resultant of three vectors is the 

 diagonal of the parallelepiped having the three vectors as con- 

 current edges. 



55. In the case of more than two or three vectors, however, 

 the solution by ordinary trigonometry would become rather 

 tedious, and it is best to proceed as follows : 



Assume an origin O and three rectangular axes Ox, Oy, Oz, 

 .and project each vector on the three axes ; let X> Y, Z be its 

 . projections. These projections X, Y, Z are three vectors whose 

 geometrical sum is equal to the vector. If n vectors were 

 originally given, we should now have them replaced by 3 n com- 

 ponents of which n lie in each axis. The components lying in 

 the same axis can be added algebraically ; let their respective 





