26 



Dr. Booth on the Rectification and 



From these considerations it easily follows that w, the por- 

 tion of the tangent between the point of contact and the foot 

 of the perpendicular, is either a maximum or minimum when 



d u 

 the radius of curvature is equal to p ; for then -r— = 0. 



u A 



It is also manifest that 



dp 



(32.) 



for 

 and 



hence 



2JT = Ta -ta = i:a-ck=p' -p-dpi 

 OT ^u + du, butOTx LtOT^^tT, tOT = d\; 



u = ~, since du dkisoi the second order. 



XVII. To apply this formula to the rectification of the 

 ellipse, let the centre be the pole, X the angle between the 

 perpendicular j9 from the centre on the tangent, and the major- 

 axe ; then, as the perpendicular is greater than the radius of 

 curvature towards the vertex of the curve which lies on the 

 major-axe, 



=> 



d\ — u; now;? = a v^l — e^sin^X; 



hence 



= a/rfX>/i 



e'' sirr \ ~ u. 



(33.) 



On the major and minor axes of the ellipse as diameters let 

 circles be described, and let a common diameter making the 



Fig. 4. 



angle <p with the 

 minor axis be drawn 

 cutting the circles in 

 m and n ; let fall the 

 ordinates ms and nt, 

 then W5=a:=asinf, 

 and nt = i/ = bcos(p. 

 Now these, it is easy 

 to show, are the co- 

 ordinates of the ex- a 

 tremity of the arc B Q of the ellipse measured from the vertex 

 of the minor axis. Differentiating x and r/, <p being the inde- 

 pendent variable, and substituting the resulting values in the 



common 



formula for rectification -^— = a / 1 + -3^, we find 

 dx V dx-^ 



s' = afd<p\^l -e2sin2(p (34.) 



If now the integrals in (33.) and (34.) be taken between the 

 same limits for X and <p, the values of the expressions under 

 the sign of integration will be equivalent, equal to K suppose; 

 hence s = K — Mj «' = K ; 



