DIFFERENTIAL AND INTEGRAL CALCULUS. Ill 



fdx\* fdy\* 



Since [ -j-I + -r = 1. we have 

 \ds) \dsj 



dx d' 2 x dy d*y _ ^ 

 ~~~* ~~ 



therefore 



ds ds* ds ds* VcfeV 



_ _ 

 dx ~ dy 



-^ "Ts \d~s -l ~ ( 



_--- 



' ds ^ ' ds 



But . (,- a )J + ( y -6)J = 0, 



or 



, . dy , 7 .dx 



, (xa}-^-(yb}-^~ ,, -- i v - ri.,v 

 x - a y - 1 v ; ^ Vl/ ; ds _ */(x - a f + y - b]*) 



dy dx 



therefore (T/ - b) ~ - (x - a) = p, 



, . / 7x / \ 



and since (y - b) - +(x-a) ^ = - 1, 



each of the quantities above = = ; or 



_ \ 



^ __ ~ ds ds* ds ds* ~~ V tUs* ) 

 ds ds 



These results are most readily found as follows : 



dx dy 



= = 



therefore -3-=- = sin -^ - 7- 



ds ds 



