38 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



2.212 The perpendicular from the origin upon the tangent to the curve 

 F(x, y) = o at the point x, y is: 



dF dF 



x + y 



dx 



dFv 



2.213 Concavity and Convexity. If in the neighborhood of a point P a curve 

 lies entirely on one side of the tangent, it is concave or convex upwards according 



as y" = -T^j is positive or negative. The positive direction of the axes are shown 

 in figure 2. 



2.220 Convention as to signs. The positive direction of the normal is related 

 to the positive direction of the tangent as the positive ^-axis is related to the 

 positive x-axis. The angle r is measured positively in the counter-clockwise 

 direction from the positive x-axis to the positive tangent. 



2.221 Radius of curvature = p; curvature = i/p. 



p ~ ds 



where s is the arc drawn from a fixed point of the curve in the direction of the 

 positive tangent. 



2.222 Formulas for the radius of curvature of curves given in the three forms 

 of 2.200. 



f (dy\ 2 \ i 



..I 1 W/ _(i/*)i 



( b ) o = 



dx d 2 y dy d 2 x ( /(Px\ 2 fd 2 y\* fd 2 s 



dt dF~Tt di? ( \dPj " h W I " \& ( 



If s is taken as the parameter /: 



(b i . I g 



|/a_Fy /afyii 



I \ ;w / r \ ^^, / I 



(c) 



_ d*F dF_dF_ 

 dx>\dy ~ 2 dxdy dx dy 



