EQUATION TO NOEMAL. 307 



This result may be written, 



,7 



} -0')=r* (2). 



If we put - = u, and = u, then 

 r r 



1 dr du 



'Hence, dividing both sides of (1) by r, we obtaiu 



'' 



or u' = u cos (ff - 0) + sin (& - 6}. 



282. To find the polar equation to the normal at any point 

 of a curve. 



Let SP=r, PSx = 0, 



SN=r, NSx = 6', 



N being any point in the normal ; then 



SP sin 8NP 



SN sin 



sin I - 



therefore - = sin (&' 6} tan < + cos (6' 



This may be written 



, d 



and may be transformed into 



u' = u cos (ff -&}- w'^sin (0' - 0). 



