136 PRACTICAL MATHEMATICS 



When Ss is made infinitely small, 



and -y- = cos ....... (2) 



as 



Taking the first relation and differentiating both sides with 

 respect to s, 



d 2 y dx^ 2 _ d0 

 dx z ds = ds 



| cos 9 - sec 6 ^ 

 ete 2 ds 



ffiy 



dQ dx* 



dx* 



(1 + tan 2 

 Jx* 



i t*V/ JL 



and since _ = __ 



ds R 



R = 



dx* 

 For a very fiat curve ^ is very small compared with 1 



and SLi-a! 



a result which is used in the consideration of the deflection of 

 beams. 



84. The Co-ordinates of the Centre of Curvature. Let P be a 

 point on a curve, and let R be the radius of curvature at that 

 point (Fig. 42). The co-ordinates of P are x lt y v Let Oj be 

 the centre of curvature and x, y its co-ordinates. 



