of only particular parts of curves; and they intersect in a straight line, which, in accordance with the nomenclature of M. de Saint Venant, and taken from Monge and Lancret, we will call the polar line: the polar line manifestly passes through the centre of absolute curvature, and is perpendicular to the osculating plane. Consider again a third consecutive normal plane; this will intersect the second one in a straight line, which is a consecutive polar line; and so on; thus a developable surface will be formed by the intersection of such normal planes ; and as it is also generated by the polar lines according to the properties of developable surfaces investigated in the last Chapter, it is called the Polar Surface.

Also a curve is formed by the intersection of the polar lines; for the lines of intersection of the first two and of the last two of three consecutive normal planes lie in the second plane, and, not being parallel, necessarily intersect, and must by their intersection form a curve of double curvature, which is the edge of regression of the polar surface. To this edge of regression therefore the polar lines are tangents; and as two consecutive polar lines are perpendicular to two consecutive osculating planes of the original curve, it follows that the angle of curvature of this edge of regression is equal to the corresponding angle of torsion of the primitive curve. Also it is manifest from the mode of generation that the normal planes of the primitive curve are osculating planes of the edge of regression; and therefore the angle of torsion of the edge of regression is equal to the angle of curvature of the primitive curve. We cannot however hence conclude, that the curvatures and torsions of the two curves are reciprocally equal, because the lengths of the corresponding elements of the two curves are not always equal.

385.] Investigation of mathematical expressions connected with polar surfaces and polar lines.

Let, n, be the current coordinates of the normal plane; (x, y, z) the point on the curve at which it is drawn; then the equation to the normal plane is

(-x) dx + (n-y) dy + (5—z) dz = 0;

and the equation to the consecutive normal plane is

(−x) d2x + (n-y) d2y + (5—z) d2≈ = ds2;

(46)

(47)

by means of which, and of the two equations to the curve, x, y, z

may be eliminated, and the resulting equation in terms of §, 7, 5 is that of the polar surface.

386.] Again, to find the equations to the polar line.

Let έ, n, be the current coordinates of the line; then, as it passes through the centre of absolute curvature, and is perpendicular to the osculating plane, or is parallel to the binormal, its equations are, by means of (22),

387.] Also to find the coordinates of the point of intersection of two consecutive polar lines: that is, the coordinates to a point on the edge of regression of the polar surface: we must again differentiate (47), whereby we have

(§—x) d3x + (n− y) d3y + (5— z) d3z = 3 ds d2s;

and by cross-multiplication between (46), (47), and (49), (§—x) (x d3x+yd3y+zd3z) = 3 ds d2s x-ds2dx,

(49)

(50)

(51)

and similar values are true for the symmetrical expressions. Therefore by means of (41),

By means of which and the two equations to the curve, x, y, z may be eliminated, and the two resulting equations in terms of έ, n, will be those to the edge of regression of the polar surface.

388.] Let us further consider these properties in relation to a sphere which has contact with the primitive curve.

As a circle is definite when it passes through three consecutive points on a curve, and cannot in general pass through more, so a sphere is definite when it passes through four points, provided that these four points have not such a position as to give a singular form to the sphere, as for instance to make it a

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

P dx Rdp px ds ds ds ds3'

d. +

nay = p

ds

Ρ dz

8- = ρ d. +

dx

ds

Rdp pz

ds ds ds ds3

(53)

are the direction-cosines of the radius

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

R

ture, and of a line equal in length to

dp

ds

measured from the

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 + (dp)2 } *.

(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, dš, 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 (x, 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, dn y-n dš 2

αξ
do

Also

=

=

,

=

r

do

r

do

— rdr = (x−§) d§ + (y −n) dn + (z−5) d¿,

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 èp produced, the arc qr should lie in a produced, and so on: and