EAD1US OF CURVATURE. 341 



322. If x and y be each a function of a third variable t, 

 we have 



dy d*y dx d"x dy 



dy_~di d*y _~d? ~dt ~ ~d? ~di 



dx~ dx' dx* /dx\ 3 



~dt \dt) 



Using these values, we deduce 



i 



dt 



^ 



r d*y dx d'x dy ' 

 W 'dt~'Wdt 



For example, if t = s the arc of the curve measured from 

 some fixed point, then 



d*ydx d*xdy 

 ds* ds ds* ds 



since by Art. 307 + ] = 1 ...... .. (2). 



J \dsj \dsj 



1 d*y dx d z x dy 



r-d}d,-T[?ds ..................... w- 



By differentiating (2) we obtain 

 dx r^a; <fy 



Square (3) and (4), and add; thus 



From (3), by means of (4), we may also deduce 

 d*y d 2 x 



i = 5! = 



p dx dy 



ds ds 



323. If we put x = r cos 0, and y = r sin 6, we have from 



