12G i/.r//<- 01 OM1 ///r [Cii. VL 



3. What doe* the equation y*-2r i -2y + 6r-3 = become when 

 the origin is removed to (|, 1), directions of axo HIM -hanged? 



4. Find th.- equation f>r the straight line y = 3x -f 1 wlu-u the origin 

 is removed to the point (1, 4), directions of axes unchanged. 



5. Construct appropriate figures for exercises 1 and 4. 



72. Transformation from one system of rectangular axes 

 to another system, also rectangular, and having the same 

 origin : change of direction of axes. 



Let OX and OF be a given pair of rectangular axes, and 

 let OX 1 and OY 1 be a second pair, with Z XOX 1 = 6, the 



angle through which the first pair 

 of axes must be turned to come 

 x into coincidence with the second. 



B J^ 



r \f Also let P, any point in tin- 



x_ plane, have the coordinates x 

 and y when it is referred to the 

 first pair of axes, and x 1 and y 1 



referred to the second pair. The problem now is to 

 express x and y in terms of x f , y 1 , and 6. Draw the or- 

 dinates HP, M'P, and QM\ and draw M'R parallel to tin- 

 s-axis; then 



OM = OQ + QM = OM f cos - M'P sin 6, 



* = *' cos e-i,< sin 6,1 



and similarly, y = x' sin 6 + y' cos 0, J 



which are the required formulas of transformation from one 

 pair of rectangular axes to another, having the same or 

 hut making an angle 6 with the first pair. 



1 Ili.-so formulas are more easily obtained, in fact, they can 

 be read directly from the figure, -if one recalls Art. 17, and considers 

 that the projection of OP equals the projection of 0\F -f the projection 

 of M'P, upon OX and OYin turn. 



NOTE 2. The reader will observe that a combination of the ti 

 formation of Art. 71 \\ilh that of Art. 7:2 will transform from one ].air 

 of rectangular axes to any other pair of rectangular axes. 



