plane; this is manifest geometrically, inasmuch as an infinite number of spheres may be made to pass through three points, but of these that which passes through the fourth point has a definite radius; and also algebraically, because the equation to a sphere involves four arbitrary constants, and these may be expressed in terms of the coordinates of four given points. Consider then a sphere to be so placed as to pass through a given point on a curve, and to touch the tangent to the curve at the point; the sphere passes through two consecutive points on the curve, and its centre may be at any point which is equidistant from these two; it may therefore be any where in the normal plane. Suppose also that the sphere passes through a third consecutive point in addition to the former two, then its centre must also be in the second consecutive normal plane, and must therefore be in the intersection of these two normal planes; that is, it must be in the polar line, but it may be at any point in that line; that line therefore may be considered as the locus of the centres of spheres which pass through the same three consecutive points on a curve. Suppose again that the sphere also passes through four consecutive points, then its centre must be in the point at which the polar line is intersected by its consecutive polar line, and is therefore at a definite point, and the radius of the sphere is of definite length. The point then at which the centre of such a sphere must be placed is a definite point on the edge of regression of the polar surface; such edge therefore may be defined to be, the locus of the centres of spheres which pass through four consecutive points on a curve of double curvature.

Therefore έ, 7, of equations (52) are the coordinates to the centre of the osculating sphere, and έ-x, n-y, -z are the projections on the coordinate axes of its radius. Now after some long but not difficult reductions, equations (52) assume the forms

of curvature, and

px ds3'

......

are the direction-cosines of the polar

line, it follows that the projections on the coordinate axes of the radius of the osculating sphere is equal to the sum of the projections on the same axes of the radius of absolute curva

Rdp

ture, and of a line equal in length to measured from the

ds

centre of curvature along the polar line; and therefore as the polar line is perpendicular to the radius of absolute curvature,

the radius of the osculating sphere = {p2+(d)}* (54)

ds

And if the radius of absolute curvature of a curve is constant for all points of the curve, so that dp = 0, then the centres of absolute curvature and of the osculating sphere are coincident; and if at a point on a curve p is a maximum or a minimum, the same result follows.

And by differentiating (52) or (53) we may find d§, dŋ, d¿, and thereby the length of an element of the edge of regression of the polar surface; and thence the curvature and torsion of the edge of regression in terms of the coordinates of the corresponding point of the primitive curve.

389.] The above investigations lead us immediately to an inquiry respecting those properties of curves of double curvature which are analogous to evolutes of plane curves.

Let a normal line be drawn at any point of a curve of double curvature; it will be in the normal plane at the point, and will therefore touch the polar surface.

Now conceive a second and consecutive normal plane to be drawn; it will meet the first normal line on the polar surface, and at the point of meeting let a normal line be drawn to the curve: and conceive again a third consecutive normal plane to be drawn, and to meet the second normal line: and another normal line to be drawn to the curve, and by a method similar to the former one: and so on; then will a curve be described on the polar surface, the elements of which are elements of these successive normal lines, and which curve is such that if a perfectly flexible and inextensible string is fixed at any point of it, and of such a length as when stretched will reach to the curve; then, if it be wrapped round the polar surface and along, and tangential to, the curve thereon described, the extremity of

the string will describe the original curve of double curvature. On this account the curve described on the polar surface is called the Evolute, and the original curve is called the Involute with respect to it.

Thus, let (r, y, z) be the point on the original curve at which the normal line is drawn, and let the point on the polar surface at which the normal meets it be (, n, ), and let do be a lengthelement of the evolute; and let r be the distance between the two points; then, as r is to be a tangent to the evolute,

and taking the negative sign r-σ = a constant; so much of the string therefore is taken off from its length by the wrapping, as to leave the remainder equal to the distance of the point on the old curve from the point on the evolute where the wrapping ends.

Hence if from two points on an evolute tangents to the evolute are drawn to the involute, the difference of their lengths is equal to the length of the arc of the evolute between the points of contact.

As the basis of the construction of the evolute thus far has been an arbitrarily chosen normal line at a given point of the original curve, so may any other normal line be taken; and thus there may be any number of evolutes, all of which will be on the polar surface, and which may therefore be considered as the locus surface of such evolutes.

The locus of the centres of absolute curvature is not an evolute, although it is a curve described on the polar surface; and for this reason; suppose P, Q, R to be consecutive points on the curve, and p, q, r ... to be the centres of curvature corresponding to the points; then, if the line pqr...... were an evolute of PQ R the arc pq should lie in the line pp produced, the arc qr should lie in aq produced, and so on: and

......

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 og 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.

be cut by the osculating and P2 P3, and which is

Also let the normal planes L1, L2 plane containing the elements P1 P2 therefore perpendicular to L1 and L2, and let the lines of intersection be м1 C1, M2 C1; then c1 is the centre of absolute curvature of the curve at P1, and P, C1 is the radius of absolute curvature.

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 C2 is the radius of absolute curvature. It is manifest now that the line Mg 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 A, B, in the point c1; 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.

The diagram gives us also a clear notion of the formation of evolutes. From м, let any line M1 D1 be drawn in the normal plane, meeting at D1 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 Dg 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