564 NOTES. 



call the changes dq l ,dq 2 the coordinates of the displacement. We have 

 found that when the displacement is made so as to change only one of the 

 coordinates of the point q l or q 2 . the arcs are respectively ds^ = y'Edq l , 

 ds% = y r Grdq 2 , and that the angle included by them is given by 



F 



cos # = 



If now we have any displacement ds, whose coordinates are dq 1 , dq 2 , 

 and project it orthogonally upon the directions of ds 1 , ds 2 , we easily see 

 (Fig. 26) that the projections da L , d6 2 are 



*, = ds, + ^ cos & = 



da, = ds, + ds, cos 4> = ya dq 2 + 



We shall now, following Hertz, introduce the reduced component of the 

 displacement along either coordinate -line, defined as the orthogoneal pro- 

 jection divided by the rate of change of the coordinate with respect to 

 the distance traveled in its own direction. These reduced components 

 we shall denote by a bar, so that 



3) 



The fundamental property of these reduced components is found in the 

 equation giving the magnitude of the displacement 



4) ds 2 = dq^dq^ + dq 2 dq 2 , 



that is the square of an infinitesimal displacement is the sum of products 

 of each coordinate of the displacement multiplied by the respective 

 reduced component. 



In like manner the geometric product of two different displacements 

 ds, ds', whose coordinates are dq^, dq 2 , dq \ dq% is found to be 



' cos (ds, ds') = dxdx' -f dydy' + dzdz' 

 dx , dx -. \ fdx , , , dx 



( 



6) 



q 2 '-l-dq 2 dq l ') + Gdq 2 dq 2 ' 

 dq 2 dq 2 ' = dq^dq^ + dq 2 'dq 2 . 



