ANA />'//' QBOMXTB7 (Ctt. vi. 



thru a transformation back t<> tin- original axrs (\\liicli must 

 give again the original equation) would raise the degree, 

 \\liirh lias juM lit-cn shown to In- impossible ; lu-nee all these 

 transformations leave the degree of an equation unchanged. 



II. POLAR COORDINATES 



76. Transformations between polar and rectangular sys- 

 tems. ( 1 i / imitin from a rectangular to a j 



system, and vice versa, the origin an, I 

 x-axis coinciding respectively u'ith tin- 

 pole and the initial line. L< t u\' 

 and OY be a given set of rectangular 

 axes, and let OX and be the initial 

 line and pole f<>r the systt-m of polar 

 coordinates. Also let P, any point in the plane, have the 

 coordinates x and y when referred to the rectangular a 

 and p and 9 in the polar system (Fig. 61), then 



OM=OPcosXOP; 



i.e., * = P co8e 



similarly. f/ = pgin9. 



These are the required formulas of transformation when, but 

 only when, the rectangular and polar axes are related as 

 above described. 



Conversely, from formulas [27], or directly from Fig. <!. 



p = Va? 4- ;A cos0=^ - , and sin 0= 9 



which are the required formulas of transformation I'mm 

 polar to rectangular axes, under the above condition-. 



