She stopped suddenly, her expression **osculating** from concerned to humor and then back again.

For the **osculating** circle at any point includes the whole of the y curve which lies beyond; and the successive convolutions envelop one another without intersection.

It follows that the **osculating** plane of the curve formed by the string must contain the normal to the surface, i.

The whole of the foregoing reasonings are applicable, not merely when A and B are actual regular cones, but also when they are the **osculating** regular cones of a pair of irregular conical surfaces, having a common apex at 0.

Such a curve may be considered as described by a point, moving in a line which at the same time rotates about the point in a plane which at the same time rotates about the line; the point is a point, the line a tangent, and the plane an **osculating** plane, of the curve; moreover the line is a generating line, and the plane a tangent plane, of a developable surface or torse, having the curve for its edge of regression.

This is called the **osculating** orbit: The essential principle of Lagrange's elegant method consists in determining the variations of this **osculating** ellipse, the co-ordinates and velocities of the planet being ignored in the determination.

The position and velocity being given in all three co-ordinates, a certain **osculating** plane is determined for each instant in which the planet is moving at that instant.