ANALYIK './"V/.//M . \ I. 



the axes to which it is refer n-d. /;.//.. tin- line A,/,, \\hen 

 referred to the axes OX and OK, has the equation 



y = tan 6 x + 6, 



but when referred to the axes O'X' and O 1 Y\ the former 

 of which is parallel to the given line, its equation is y = c. 



For these, and other reasons, in the study of curves and 

 surfaces by the methods of analytic geometry, it will ol't.-n 

 be found advantageous to transform the equations from one 

 set of axes to another. 



It will be found that the coordinates of a point with 

 reference to any given axes, are always connected 1>\ simple 

 formulas with the coordinates of the same point \\hn it is 

 referred to any other axes. These relations or formulas 

 for the various changes of axes are derived in the next 1 \v 

 articles. 



I. CARTESIAN COOKDIN ATKS ONLY 



71. Change of origin, new axes parallel respectively to the 

 original axes. Let OX and OF be the original axes, O'X' 

 and O'F' the new axes, and let the coordinates of the n-\v 



origin when referred to tin- 

 original axes be h and &, i.e., 

 0' = (A, &), where h = OA and 

 k = AO'. Also let P, any point 

 of the plane, have the coordi- 

 nates Z and y when it is ref< 

 to the axes OX and n Y. and a/ 



and y' when it is referred to the axes O'X' and O 1 F' 

 Draw MWP parallel to the y-axis ; then 



OM= OA -f AM = OA + O'M', 



T 



[281 

 and similarly, !/=/' / 



