104 DIFFERENTIAL AND INTEGRAL CALCULUS. 



Coordinates of the. centre of curvature are 



ds . 2a sin !// 2a sin i// 2 . sir 



d\l/ 1 - sini// (1 sin i//) 2 (1 sin $)* ' 



ds 2a cos< (2 sini// 1) 2a cosi// 



2a sin' 2 A/r T72 16a 2 /I -f sin>|r\ 3 



Or A = 7 i . , . JL = ; ; , 



(1-smi/r; 2 ' 9 Vl-sin^/' 



from which the equation of the locus of the centre of cur 

 vature is readily found, 



~:\ 2 sinojr 



V/O 



a 1 sin 



'*(=?)'' 



. 



The curve represented by this equation is also drawn in 

 fig. 27. 



DIFFERENTIAL CALCULUS. (5). 



1. The tangent to any curve at a point (05, y) makes 



sJ/lt xV/v 



with the axis of x an angle whose tangent is -j- , cosine -y- , 



and sine -j- , 5 being the arc of the curve measured from 



a fixed point to the point (#, y] ; also the tangent makes 

 with the radius vector at a point (r, 6) an angle whose 



. de . dr de 



tangent is r -7-,, cosine -7- , and sine r -y- . 



2. Prove that the length of the tangent at any point 

 of the curve x*+y% = a* intercepted between the axes of 

 coordinates is always a. 



3. The curve 9a?/ 2 = (x + a) (x - 2a) a is such, that if at 

 any point P the normal meet the axis of y in A", and PM 



