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such can be the case only when vp, Qq, Rr are two and

two in the same plane; but all these lie severally in different

planes, viz. in the osculating planes at P, Q, R The radii

of absolute curvature therefore cannot meet and by their intersections form an evolute, unless all are in the same plane, or, in other words, unless the curve is plane. A surface however is generated by such radii of curvature, which is ruled, but of the class termed Skew.

There is also another remarkable property of evolutes of curves of double curvature, viz. when the polar surface is developed into a plane, they become straight lines all diverging from the same point. For consider two consecutive normal planes vp and <iq drawn at p and Q, and imagine the second one to turn about the polar line in vp, until the two planes coincide; then, as is manifest from the construction, Q falls on and coincides with p, and the line vp coincides with Qq, so that pq, the element of the evolute, coincides in direction with vp; and as a similar result would follow from a similar operation being performed on the other normal planes, it follows that the evolute of which pq is an element becomes a straight line emanating from p, the point into which the whole original curve becomes absorbed; and as a similar result is true of all the other evolutes, it follows that when the polar surface is developed, the evolutes become a pencil of straight lines diverging from the point into which the curve falls.

And as the length of an element of an evolute is not altered by the development, it follows that the element of an evolute is the shortest distance between the two extremities of the element. The evolute therefore is the geodesic between any two points on the polar surface.

390.] To enable the reader to obtain an adequate conception of the results of the last few Articles, the geometrical figure 129* is given.

Let Pi, p2, p3, Vi be successive points in a curve of

double curvature; and through the middle points Mi, M2, M3,

...of successive elements let the normal planes Li, Lj, L3, .. be

drawn, intersecting each other consecutively in the straight lines

* The figure is the same as that given by Monge in his "Application d'Analyse," and thence has been copied into most of the ordinary text hooks. Price, Vol. r. 4 c

Ai Bj, A2 B2, which are therefore the polar lines; and the

surface formed by the intersection of the normal planes is the developable polar surface; then, as all the elements of the curve are not in the same plane, the polar lines are not parallel, and therefore intersect consecutively, and thereby form an envelope, viz. the non-plane curve Qi Q2 Qs , which is the edge of regression of the polar surface.

Also let the normal planes Lj, L2 be cut by the osculating plane containing the elements Pi p2 and p2 P3, and which is therefore perpendicular to L! and 1^, and let the lines of intersection be M] ci, M2 Cj; then Ci is the centre of absolute curvature of the curve at Pj, and Pi Ci is the radius of absolute curvature.

Again, let the consecutive osculating plane be drawn containing the elements p2 p3 and p3 p±, and let its lines of intersection with L2 and L3 be M2 C2 and M3 c2; then c2 is the centre of absolute curvature of the curve at p2, and p2 c2 is the radius of absolute curvature. It is manifest now that the line M2c2 does not coincide with M2 Ci, because they are the lines of intersection of the same plane L2 by different planes; M2c2 therefore does not cut Ai Bi in the point ct; and therefore the consecutive radii and M2c2 do not meet. The successive centres of curvature therefore do not arise from the intersection of consecutive radii of curvature, and consequently these radii are not tangents to the locus of the centres; and therefore it

follows that the curve Ci c2 cannot be regarded as an evo

lute of the original curve. It is manifest however from the construction that such will be the case, if the original curve is plane.

The diagram gives us also a clear notion of the formation of evolutes. From Mx let any line MiBi be drawn in the normal plane, meeting at Dx the polar surface, to which it is tangential; and from Di let the line D! M2 be drawn to M2, the middle point of the next element; then this line lies in the consecutive normal plane, and is tangential to the polar surface, and has an element r>i D2 in contact with it; and let a similar process be continued on other consecutive normal planes; then there will

be described on the polar surface a curve Di D2 , such that

each successive clement on it being produced will pass through and be normal to the original curve, and such that the difference between two successive lines drawn from the new to the old curve is equal to an element of the new curve, as is manifest from the construction; therefore the curve i»i D2 is

an evolute to the original curve, which is called an involute relatively to it. Hereby also a developable surface will be

formed of which the curve Di D2 is the edge of regression,

the osculating planes to which are those containing the successive elements of the original curve.

And as the first normal Pi Dj was drawn to a point Di, chosen arbitrarily, so might any other point have been taken on the polar line Bi Aj Qx; and thus there may be any number of evolutes on the polar surface, and the polar surface may be considered as the locus-surface of such curves.

It is manifest from the geometry that if the polar surface is

developed into a plane, Di o2 would become a straight line;

therefore Di D2 is a geodesic line on the polar surface.

391.] From the figure also may be deduced results, some of which are identical with those above investigated by algebraical methods.

Through the points Mu M2, , the middle points of successive elements, let there be drawn the binormals Mi Li, M2 L2,

which are parallel to the polar lines Aj Bj, A2 B2;

then the angle Ai Qi A2, being that between two successive polar lines, is equal to the angle of torsion. Also let the line M2 C2 cut the polar line Aj Bx in i; then we have the following values:

M1C1 = p = the radius of absolute curvature,
Mai = dp = the differential of ditto,
M[Qi = the radius of the osculatiug sphere,

*iQiA2 = the angle of torsion = deo = — •


Since then c2i may be considered as an arc of a circle, subtending an angle Aiq!a2 at Q1( we have

c2i = QiC2 x angle Aiqia2;

dp = QiC2 x da>,

dp do

••■ ^ = d. = KTs' (55)

and since Mi<-'iql is a right angle,

MjQ!2 = HUCiHCiQi2,

{the radius of the osculating sphere}2 = p2 + (R^) '> («*6)

'qici' K(tp

tauQiMiti = = (o7)

i Pi Ci p as

Also as M^cj, M2c2 are the lines of intersection of the normal plane at p2, by two consecutive osculating planes which are both perpendicular to it, the angle Cj Ma c2 is that between the osculating planes, and is therefore equal to the angle of torsion:

that is, ,

Ci M2 c2 = a«);

and as M2 Ci is perpendicular to Ai Bi, Cj I may be considered the arc of a circle subtending at M2 an angle rf&>; therefore

Cii = M2c! x angle CiM2c2,

= pdu>; (58)

and therefore if cic2 is joined, is an element of the curvelocus of the centres of curvature;

.-. CiC23 CiI3 + IC22,

= (fdtf + dp*. (59)

If the curve is plane da> = 0, and CiC2 = dp, in which case CjC2 is an element of the evolute of the original curve; but if

the original curve is non-plane, the curve CiC2 is not an


As Ci c2 is an element of the curve-locus of the centres of absolute curvature, it is equal to {dfa + dr? + rff2}* of the expressions given in equations (11) or (22) of the present Chapter; that is,

rfp2+PW = \dx+d.(^d.J) j' + + ...; (60)

whence, after several reductions,

392.] Now this last equation deserves attention; for on referring to equations (25) it appears, that the direction-cosines of the radius of absolute curvature are

U.% U.% U**, (62) as as as ds as as

and therefore on comparison of (61) with (29), it appears that the right-hand member of (61) is the square of the infinitesimal angle contained between two consecutive radii of absolute curvature. Therefore if da is the angle between these, viz. the angle between MiCj and M2c2 in fig. 129; then

and assuming K to be such that ds = K do, wc have

i = 7 + (65)

and calling — complex flexure, and HI the radius of complex

flexure, we have

(complex flexure)2 = (curvature)2 + (torsion)2. (66) Fig. 129 affords an easy geometrical proof of (64).

393.] On the osculating surface of a non-plane curve.

The osculating planes of a non-plane curve by their consecutive intersections form a developable surface, whose edge of regression is the curve itself; or the surface may be considered to be generated by the tangent lines in their successive positions. The tangent line therefore is the characteristic of the surface, which is called the osculating surface.

Let the equation to an osculating plane be

(f-ar) x + fo-y) Y + tf-z) i = 0; (67)

then the equation to the consecutive plane is

(f-ar) rfx + ft-y) dx + tf-z) dz = 0, (68)

"." xdx + Y dy + zdz = 0.

Eliminating therefore x, y, z from (67), (68), and the two equations to the curve, there will result an equation in terms of f, T), ( which will be that to the osculating surface.

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