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.

The **osculating** plane of a stream line in steady motion contains the resultant acceleration, the direction ratios of which are du du, du d i g d g 2 _ dH dx +v dy + dz - 2v?

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 acm and bbb are actual cylinders, but also when they are the **osculating** cylinders of a pair of cylindroidal surfaces of varying curvature, A and B being the axes of curvature of the parts of those surfaces which are in contact for the instant under consideration.

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.