28 GEOMETRY OF VECTORS. [CHAP. II. 



plane, or to a couple whose axis is perpendicular to the plane, or 

 to zero. 



The single vector, and the couple, in the cases where the 

 system is equivalent to a single vector or a couple, are deter- 

 minate and unique. 



The conditions of equivalence of two systems of vectors localised 

 in lines lying in a plane are that when one system is equivalent 

 to a single vector the other is equivalent to a single vector of the 

 same magnitude and sense localised in the same line, when one 

 system is equivalent to a couple the other is equivalent to a 

 couple of the same magnitude and sense, when one system is 

 equivalent to zero the other is equivalent to zero. 



26. Analysis of vectors localised in lines in a plane. 



Take any origin 0, and any rectangular axes of x, y in the plane. 



Let X l} Y l be the resolved parts 

 parallel to the axes of one of 

 the vectors, and let P l (x l} y^} be 

 any point on the line in which it 

 x i is localised. Then the moment 



of the couple, when this vector is 

 replaced by an equal vector local- 

 ised in a line through and a 

 couple, is the moment of this 

 _ ,,. vector about 0, i.e. it is the sum 

 w . OBr of the moments of the resolved 



.Tig. JjO. 



parts X lt Y l about 0, and this 



sum is co l Y l y l X l . Hence the whole system is equivalent to a 

 vector localised in a line through 0, whose resolved parts parallel 

 to the axes are 5-3T and 2F, and a couple in the plane, whose 

 moment is S(#F yX). 



27. General analysis of vectors localised in lines. 



Take any origin 0, and any rectangular axes of x, y, z. Let 

 X, F, Z be the resolved parts parallel to the axes of one of the 

 vectors, and x, y, z the coordinates of a point in the line in which 

 it is localised. Introduce a pair of equal and opposite vectors 

 localised in a line through parallel to the line of this vector 

 and resolve them into components localised in the axes. The I 



