......

such can be the case only when Pp, 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 Pp and qq drawn at P and Q, and imagine the second one to turn about the polar line in Pp, until the two planes coincide; then, as is manifest from the construction, q falls on and coincides with P, and the line pp coincides with qq, so that pq, the element of the evolute, coincides in direction with Pp; 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 P1, P2, P3, P4 be successive points in a curve of double curvature; and through the middle points M1, M2, M3, of successive elements let the normal planes L1, L2, 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 books. PRICE, VOL. I.

4 C

......)

A1 B1, 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 Q1 Q2 Q3 which is the edge of regression of the polar surface. Also let the normal planes L1, L2 plane containing the elements P1 P2 therefore perpendicular to L1 and L2, section be M1 C1, M2 C1; then c1 is the centre of absolute curvature of the curve at P1, and P1 C1 is the radius of absolute curvature.

be cut by the osculating and P2 P3, and which is and let the lines of inter

Again, let the consecutive osculating plane be drawn containing the elements P2 P3 and P3 P4, 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 C is the radius of absolute curvature. It is manifest now that the line M2 C2 does not coincide with M2 C1, because they are the lines of intersection of the same plane L2 by different planes; Mg Ca therefore does not cut A1 B1 in the point c; and therefore the consecutive radii M1 C1 and M2 C2 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 C1 C2 .... cannot be regarded as an evolute of the original curve. It is manifest however from the construction that such will be the case, if the original curve is plane.

M1

The diagram gives us also a clear notion of the formation of evolutes. From M1 let any line M1 D1 be drawn in the normal plane, meeting at D, the polar surface, to which it is tangential; and from D1 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 D1 D 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 D1 D2 such that each successive element on it being produced will pass through and be normal to the original curve, and such that the differ

ence 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 D1 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 D1 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 P1 D1 was drawn to a point D1, chosen arbitrarily, so might any other point have been taken on the polar line B1 A1 Q1; 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, D1 D2 ...... would become a straight line; therefore D D ...... 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 M1, M2, ......, the middle points of successive elements, let there be drawn the binormals M1 L1, M2 L2, ...... which are parallel to the polar lines A1 B1, A2 B2 .. then the angle A1 Q1 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 A1 B1 in 1; then we have the following values: M1 C1 = p = the radius of absolute curvature,

M2I dp

=

=

the differential of ditto,

M1Q1 = the radius of the osculating sphere,

A1 Q1 42 = the angle of torsion = dw =

ds

R

Since then c2I may be considered as an arc of a circle, subtending an angle A1Q1 A2 at Q1, we have

and since MC1Q is a right angle,

M1Q12 M1C12 + C112,

=

{the radius of the osculating sphere}2 = p2 + (Rd2)2; (56)

ds

(57)

Also as M2 C1, 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 c1 M2 C2 is that between the osculating planes, and is therefore equal to the angle of torsion : that is,

C1 My C2 = dw;

and as MC is perpendicular to A1 B1, CI may be considered the arc of a circle subtending at M2 an angle do; therefore

C1 = M2C1 × angle C1 M2 C2,

=

pdw;

(58)

and therefore if c12 is joined, c1 c2 is an element of the curvelocus of the centres of curvature;

If the curve is plane do = O, and c12 = dp, in which case C1 C2 is an element of the evolute of the original curve; but if the original curve is non-plane, the curve C1 C2, evolute.

is not an

As C C is an element of the curve-locus of the centres of absolute curvature, it is equal to {d§2 + dŋ2 + d¿2} of the expressions given in equations (11) or (22) of the present Chapter; that is,

dx

dp2 + p2 dw2 = { dr+d. (~* d. d/z)}'+
(22
ds ds

whence, after several reductions,

ds ds

ds ds

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

Ld. ds

dx Ρ
dy ρ dz
d.
d. ;
ds ds ds' ds ds

(62)

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 do is the angle between these, viz. the angle between M1C1 and M2 C2 in fig. 129; then

dy

dz 2

{ d. ( { d. d)) } + { d. ( &. d. d)}, (63)

ds ds

ds ds

(64)

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

(complex flexure) = (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

(−x)x+(n−y)x+(-2) z = 0;

(67)

then the equation to the consecutive plane is

(§−x) dx + (n−y) dx + (5−z) dz = 0,

(68)

x dx + y dy + z dz = 0. .

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