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By Jean Dieudonne

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8. The osculating circle at an ordinary point of a curve has contact of the sec ond order with the latter and all other circles which lie in the osculating plane and are tangent to the curve at the point have contact of the first order. ; 9. A necessary and sufficient condition that the osculating circle at a point have contact of the third order is that p = and I/T = at the point at such a point ; the circle 10. where 11. is said to superosculate the curve. Show p that any twisted curve and r are the When radii of first may be defined by equations of the form and second curvature at the point the equations of a curve are in the form /2 /I f fff f where has the significance of equation (12).

Moving In trihedral. we took 11 for fixed axes of refer ence the tangent, principal normal, and binormal to a curve at a of it, and expressed the coordinates of any other point of Q point the curve with respect to these axes as power series in the arc s is any point of the of the curve between the two points. Since M M such axes for each of curve, there is a set of Hence, its points. instead of considering only the points whose locus is the curve, we may look upon the moving point as the intersection of three mutually perpendicular lines which move along with the point, the whole figure rotating so that in each position the lines coin cide with the tangent, principal normal, and binormal at the point.

Substituting the values of #, /3, 7; X, /u-, i^ from (19) and (37) in the m, w, the resulting equations are reducible to expressions for , Hence, when the parameter u (42) is general, we have l=-(W- or in other form, 2 _ In consequence of (29) equations (42) (43) or by means of dzd 2 s be written: da dft dj ds ds ds (27), da Hence the tangent to the principal may 2 m dft dy do- da- to the spherical indicatrix of a curve is parallel normal to the curve and has the same sense. *C. , p. 31. CURVES IN SPACE 14 9.

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