Greatly exaggerated here, the small difference ? "'r "'( blue ) between the osculating, unperturbed orbit ( black ) and the perturbed orbit ( red ), is numerically integrated starting from the initial position ( the " epoch of osculation " ).
12.
Further work from the 1930s onwards was carried out by J . Kanitani, Shiing-Shen Chern, A . P . Norden, G . Bol, S . P . Finikov and G . F . Laptev . Even the basic results on osculation of curves, a manifestly projective-invariant topic, lack any comprehensive theory.
13.
For arithmetic, Bhrat + K [ c G a gives several algorithms for whole number multiplication and division, ( flag or straight ) division, fraction conversion to repeating decimal numbers, calculations with measures of mixed units, summation of a series, squares and square roots ( duplex method ), cubes and cube roots ( with expressions for a digit schedule ), and divisibility ( by osculation ).
14.
There are three different schools of logical thought on the very simple subject of fraternal anogenital copulation ( AKA; homosexuality, gay sex, androphilia, sodomy, buggery, anal sex, or what ever one may prefer to call this phenomenon . [ There is often orogenital copulation and osculation involved as well, but for brevity s sake I shall not attempt to address those under this subject . ] At its simplest and most basic level, it is a conduct that a significant number of human beings engage in and indicate that they enjoy with great fervor and as a result, many of these same humans also endorse it as an enjoyable practice for others to join in on either with them personally or with a partner of their own choosing.
15.
One speaks also of curves and geometric objects having " k "-th order contact at a point : this is also called " osculation " ( i . e . kissing ), generalising the property of being tangent . ( Here the derivatives are considered with respect to arc length . ) An osculating curve from a given family of curves is a curve that has the highest possible order of contact with a given curve at a given point; for instance a tangent line is an osculating curve from the family of lines, and has first-order contact with the given curve; an osculating circle is an osculating curve from the family of circles, and has second-order contact ( same tangent angle and curvature ), etc.
