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9 contact of a circle with a curve curvature the

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9. Contact of a circle with a curve. Curvature. * The general equation of a circle, viz. ( x - a ) 2 + ( y - b ) 2 = r 2 , (1) contains three arbitrary constants. Let us attempt to determine them so that the circle has contact of as high an order as possible with the curve y = f ( x ) at the point ( ξ, η ), where η = f ( ξ ). We write η 1 , η 2 for f 0 ( ξ ), f 00 ( ξ ). Differentiating the equation of the circle twice we obtain ( x - a ) + ( y - b ) y 1 = 0 , (2) 1 + y 2 1 + ( y - b ) y 2 = 0 . (3) If the circle touches the curve then the equations (1) and (2) are satisfied when x = ξ , y = η , y 1 = η 1 . This gives ( ξ - a ) 1 = - ( η - b ) = r/ p 1 + η 2 1 . If the contact is of the second order then the equation (3) must also be satisfied when y 2 = η 2 . Thus b = η + { (1 + η 2 1 ) 2 } ; and hence we find a = ξ - η 1 (1 + η 2 1 ) η 2 , b = η + 1 + η 2 1 η 2 , r = (1 + η 2 1 ) 3 / 2 η 2 . The circle which has contact of the second order with the curve at the point ( ξ, η ) is called the circle of curvature , and its radius the radius of curvature . The measure of curvature (or simply the curvature ) is the reciprocal of the radius: thus the measure of curvature is f 00 ( ξ ) / { 1 + [ f 0 ( ξ )] 2 } 3 / 2 , or d 2 η 2 1 + 2 3 / 2 . * A much fuller discussion of the theory of curvature will be found in Mr Fowler’s tract referred to on p. 324 .

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[VII : 151] ADDITIONAL THEOREMS IN THE CALCULUS 334 10. Verify that the curvature of a circle is constant and equal to the reciprocal of the radius; and show that the circle is the only curve whose curvature is constant. 11. Find the centre and radius of curvature at any point of the conics y 2 = 4 ax , ( x/a ) 2 + ( y/b ) 2 = 1. 12. In an ellipse the radius of curvature at P is CD 3 /ab , where CD is the semi-diameter conjugate to CP . 13. Show that in general a conic can be drawn to have contact of the fourth order with the curve y = f ( x ) at a given point P . [Take the general equation of a conic, viz. ax 2 + 2 hxy + by 2 + 2 gx + 2 fy + c = 0 , and differentiate four times with respect to x . Using suffixes to denote differen- tiation we obtain ax + hy + g + ( hx + by + f ) y 1 = 0 , a + 2 hy 1 + by 2 1 + ( hx + by + f ) y 2 = 0 , 3( h + by 1 ) y 2 + ( hx + by + f ) y 3 = 0 , 4( h + by 1 ) y 3 + 3 by 2 2 + ( hx + by + f ) y 4 = 0 . If the conic has contact of the fourth order, then these five equations must be satisfied by writing ξ , η , η 1 , η 2 , η 3 , η 4 , for x , y , y 1 , y 2 , y 3 , y 4 . We have thus just enough equations to determine the ratios a : b : c : f : g : h .] 14. An infinity of conics can be drawn having contact of the third order with the curve at P . Show that their centres all lie on a straight line. [Take the tangent and normal as axes. Then the equation of the conic is of the form 2 y = ax 2 + 2 hxy + by 2 , and when x is small one value of y may be expressed ( Ch. V , Misc. Ex. 22) in the form y = 1 2 ax 2 + ( 1 2 ah + x ) x 3 , where x 0 with x . But this expression must be the same as y = 1 2 f 00 (0) x 2 + { 1 6 f 000 (0) + 0 x } x 3 , where 0 x 0 with x , and so a = f 00 (0), h = f 000 (0) / 3 f 00 (0), in virtue of the result of Ex. lv . 15. But the centre lies on the line ax + hy = 0.]
[VII : 152] ADDITIONAL THEOREMS IN THE CALCULUS 335 15. Determine a parabola which has contact of the third order with the ellipse ( x/a ) 2 + ( y/b ) 2 = 1 at the extremity of the major axis.

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