37-39] RELATIVE MOTION. 39 



39. Relative coordinates and relative motions. Let 



#D 2/i > z \ be the coordinates of a point A at time t referred to 

 axes with origin at 0, # 2 , y 2 , z^ the coordinates of a second point 

 B at the same time referred to the same axes, and f , 77, f the co- 

 ordinates of B at the same time referred to parallel axes through 

 A. Then f, 77, f are called the coordinates of B relative to A. 



We have # 2 = #1 + ?> 1 



2/2 = 2/1 + *?, > ............. .......... (1). 



*-* + ] 



Let accented letters denote at time t' the quantities that cor- 

 respond to unaccented letters at time t, thus let #/, y/, s/ be the 

 coordinates of A', the position of A at time t'. Then as before 



By subtraction we deduce 



^-^ 2 =-^)+(r-a] 



y'-y*-(yi'-yi) + (*f-i), ............ (2). 



*,'-*=(*i'-*i) + r-D.J 



The terms on the left are the components parallel to the axes 

 of the displacement of B. 



The terms in the first brackets on the right are the components 

 parallel to the axes of the displacement of A. 



The terms in the second brackets on the right are the com- 

 ponents of the displacement of B relative to parallel axes with 

 origin at A. 



Thus we have the result: The displacement of a point B 

 relati-ve to axes at is compounded of the displacement of a 

 point A relative to the same axes and the displacement of B 

 relative to parallel axes through A. 



By dividing equations (2) by t' t and passing to the limit 

 when t' t is indefinitely diminished, or, what is the same thing, 

 by differentiating equations (1) with respect to t, we find 



# 2 = #i + | y 2 = 2/i + 7 ?> *a = fc + i 

 and by differentiating again we find 



#2 = #i + ?> & = & + *?, 2 2 = *! + !;. 



