PQ, QR,.

are equal elements, and each to ds. Produce PQ to T, making QTPQ=ds. From T draw TN at right-angles to the surface, and meeting the surface in N; join QN, TR, RN. Then the plane PQTN is evidently the normal plane to the surface at P, and PQ, QN are consecutive elements of the geodesic of which PQ is the first element; consequently this geodesic touches the curve PQR at P, and has contact of the second order at least with the curve made by the normal section at P. Now the line NR indicates the deviation of the curve PQR from the geodesic PQN which touches it at P; and for this reason the deviation is called geodesic curvature. And as in the fig. thus described TQR is the angle of absolute curvature of PQR, TQN is the angle of absolute curvature of the geodesic PQN, so is the angle RQN the angle of geodesic curvature of PQR. Let p and p' be the radii of absolute curvature of PQN and PQR respectively; then by the definitions. of these we have ds = pLTQN; ds = p'LTQR. Analogously let us suppose ds= p" NQR; then p" is called the radius of geodesic curvature*. Now these three infinitesimal angles form at q the vertical angle of the tetrahedron, of which the three adjacent sides QT, QR, QN are approximately equal: hence the angles are proportional to the opposite sides; so that

Px TN = px TR = p′′ x NR.

(84)

Let the angle which the osculating plane of PQR makes o with the normal plane PQN; that is, let RTN = 4; then

the former of which equations is Meunier's theorem; see Art. 414, Vol. I; the latter gives the value of the radius of geodesic curvature in terms of the radius of absolute curvature.

Since for a geodesic 4 = 0, the radius of geodesic curvature of a geodesic is infinite.

342.] The following are also theorems of considerable importance.

(1) Let o, fig. 49, be a point (x, y, z) on a given surface F(x, y, z) = 0; and from o let a series of geodesics OP, OQ, OR... be drawn infinitesimally near to each other; and from them let

* Liouville in the appendix to Monge, Application d'Analyse, &c., pp. 56%, 574; and Bonnet, Journal de l'Ecole Polytechnique, Cah. XXXII.

Op =

...

=

equal lengths OP OQ = OR, ... 8; be cut off; and through the extremities P, Q, R let the curve PQR... of which the current coordinates are §, n,, be drawn; this curve, which is the locus of all points the geodesic distances of which from o are equal to s, is called a geodesic circle, whose radius is s, and whose centre is o. Similarly from the same centre o a series of concentric circles may be described. Now since for any given circle the radius is constant, and that radius = the distance from (xo, Yo, Zo) to (§, n, 5), = s, the variation of s between these two points vanishes; consequently the coordinates of these points must satisfy the integrated part of (65). As one of the limits, viz. (x。, Yo, zo), is a fixed point, there are no variations of its coordinates; but for the other limit, viz. (§, ŋ, Š), we have

in which dέ, dŋ, d are the projections on the coordinate axes of

an arc-element of the geodesic circle, and

dx dy dz
ds' ds
ds' ds

are the

direction-cosines of that element of a geodesic radius which meets the circle at (§, n, Š); consequently these two elements are perpendicular to each other, and each of the radii is cut at rightangles by the geodesic circle; and thus the geodesic circle intersects orthogonally a series of geodesic radii originating at a given point.

This theorem may also be demonstrated by the geometrical process of infinitesimals. Let OP and oq be two consecutive geodesics of equal length, and PQ be an element of the curve which joins the extremities of the series of similar geodesics of equal length. Then PQ is perpendicular to both op and oq. For if PQO is not a right-angle, let us suppose it to be greater than a right-angle. From q let qr be drawn at right-angles to PQ and intersecting OP in T; then considering the triangle PQT, as PQ is an infinitesimal and PQT is a right angle, PT is greater than QT; so that adding or to each, OT+TP is greater than OT+TQ; but OT+TP = OP=OQ; therefore oq is greater than OT+TQ; that is oq is not the shortest line from o to Q, as it is assumed to be. Consequently oqp is not an obtuse angle; in the same way it may be shewn that it is not an acute angle: and therefore it is a right-angle.

(2) Hence also conversely, if every geodesic through o to

...

points on the curve PQR. cuts that curve at right angles, the lengths of the geodesics are equal.

(3) By a similar process it may be shewn that if from all points of a line drawn on a given surface geodesics of equal length are drawn perpendicular to that line, the locus of their extremities is a curve which is perpendicular to the geodesics. Lines on a surface thus related are called geodesic parallels; such are PQR, P'Q'R',... in fig. 48; the number of them is of course infinite.

(4) If from a point o on a surface two geodesic radii yectores OP, OQ are drawn to two points P and Q on a line on the surface, and infinitesimally near to each other, then, if PQ = ds, and OPQ = 4,

OP-OQ = ds cos p.

(87)

As these theorems are fundamental in the demonstration of geometrical theorems concerning straight lines and planes, they are applicable to the proof of theorems concerning lines on surfaces in reference to geodesics, when the lines are of infinitesimal length and may be considered straight, and when surface-elements are considered which are approximately planes. The truth of the following theorems is evident from these principles.

(5) If two points are taken on a surface, and a curve is drawn on it such that the sum of the lengths of the two geodesics drawn from any point of it to the two given points is constant; the tangent at any point of the locus is equally inclined to the two geodesics. Such a curve is called a geodesic ellipse, the given points being the geodesic foci, and the two geodesics drawn from any point on the curve to the foci being the geodesic focal radii.

(6) If two points are taken on a surface, and a curve is drawn on it such that the difference of the geodesic lengths from any point on it to these two points is constant, the tangent at any point of the locus bisects the angle between these two geodesics. The locus thus defined is called a geodesic hyperbola.

(7) A series of geodesic confocal ellipses is intersected orthogonally by a series of geodesic confocal hyperbolas.

343.] The following are examples in which the preceding theory of geodesics is applied.

Ex. 1. Determine the geodesics on the sphere, x2+ y2 + z2 = a2. In this case from (66) we have

whence by integration, if h1 is an arbitrary constant,

where he and h, are also arbitrary constants :

.. h1x+h2y+h, z = 0,

(88)

which is the equation to a plane passing through the origin, which is the centre of the sphere, and consequently intersects the sphere in a great circle. Hence a great circle is a geodesic on a sphere; and the geodesic joining two given points on a sphere is the arc of a great circle. As the great circle passing through two given points will be divided at these points into two arcs, of which one is greater and the other is less than a semicircle, so the points will be joined by two different geodesics of which one is a maximum, and the other is a minimum. This illustrates the remark in Art. 337, that if a geodesic joins two given points we cannot assert absolutely that the length is either a maximum or a minimum.

Ex. 2. Determine the geodesic joining two given points on a right circular cylinder.

Let the equation to the cylinder be x2+y2 = a2; and let the two points be (a, 0, 0), (a cos a, a sin a, c). Then from (66) we have

dx

dy

dz

d.

d.

d.

ds

ds

ds

(89)

and this shews that the geodesic cuts all the generating lines of the cylinder at a constant angle; consequently the geodesic is

the helix, the inclination of the line of which to the plane of (x, y) = B. The equations to it may be found as follows. From

And if x = a cos 0, y = a sin 0, then z = ake, if k = tan ß; and

these are the equations to the helix. To determine the quantity ß, we superior limit, c = aa tan ß;

.. B =tan

See Art. 347, Vol. I. have, by the values of the

And as the number of the values of ß which satisfy this equation is infinite, so the number of geodesics which can be drawn joining the two points is also infinite. This is also evident from the geometry, inasmuch as the geodesic may be drawn from the first point round the cylinder any number of times before it falls into the second point; and the greater the number of times that it is drawn round the cylinder, the smaller is ß, since c=s sin ß, wheres the length of the geodesic.

344.] Determine the equations of the geodesic drawn on the surface of an ellipsoid.

Let the equation to the ellipsoid be

and the equations to a geodesic which are given by (68) give

where k is a constant introduced in integration, and may be