94 DIFFERENTIAL AND INTEGRAL CALCULUS. 



LP 



and similarly sin</> = limit of sin SQP= limit of -~-p 



QP 



r . f sinA0 arc QP d6 



= limit of r - . -j- T-^ = r -^ . 

 As chord <)P <& 



J<9 



and tan = r -7- . 



dr 



(Of course since ic = r cos 0, ?/ = r sin 0, these results could 

 be deduced from the former in x and y, and it will be a 

 useful exercise so to obtain them). 

 Hence we have the equations 



dx z 



or if t be any other variable in terms of which we can 

 'express #, y, ?, 0, 



dt \dt 



It is often most convenient to express x and y the 

 coordinates of any point of a curve in terms of i/r ? the 

 angle which the tangent makes with the axis of x (or any 

 other fixed straight line). Thus, if the curve be the 

 parabola 



dy 2a 



y =* ax , fx = ^ = tan ^ 



or ?/ = 2acot>/r J x = aco^^jr. 



In this case it is better to put \TT ty for >/r, so that ty 

 will be the angle which the tangent makes with the axis 

 of ?/, and we shall now have x, ?/, ty start together ; and 



dx 2a sin-^r 



therefore ^ = 2a tan* sec * = -- 3 ^ , 



dy 2a ds 2a 



j, = T-T j therefore -, - - = -7 . 

 dty cos* 7 dty cos* 7 



