Page:Encyclopædia Britannica, Ninth Edition, v. 10.djvu/434

420 the surface a set of rectangular axes, the normal and the two principal tangents. Proceeding from the point along a principal tangent to a consecutive point on the surface, and thence (without abrupt change of direction) along the new principal tangent to a consecutive point, and so on, we have on the. surface a curve of curvature; there are, it is clear, two singly inﬁnite series of such curves, cutting each other at right angles at each point of the surface. Passing from the given point in an arbitrary direction to a consecutive point on the surface, the normal at the given point is not intersected by the normal at the con- secutive point ; but passing to the consecutive point along a curve of curvature (or, what is the same thing, along a principal tangent) the normal at the given point is inter- sected by the normal at the consecutive point ; we have thus on the normal two centres of curvature, and the distances of these from the point on the surface are the two principal radii of curvature of the surface at that point; these are also the radii of curvature of the sections of the surface by planes through the normal and the two prin- cipal tangents respectively; or say they are the radii of curvature of the normal sections through the two principal tangents respectively. Take at the point the axis of z in the direction of the normal, and those of x and y in the directions of the principal tangents respectively, then, if if the radii of curvature be a, b (the signs being such that the coordinates of the two centres of curvature are 2:0 and z: b respectively), the surface has in the neigh- bourhood of the point the form of the paraboloid

$ z=\frac{x^2}{2a} + \frac{y^2}{2b}$

and the chief-tangents are determined by the equation. The two centres of curvature may be on the same side of the point or on opposite sides; in the former case a and I; have the same sign, the paraboloid is elliptic, and the chief-tangents are imaginary ; in the latter case a and b have opposite signs, the paraboloid is hyper- bolic, and the chief-tangents are real. The normal sections of the surface and the paraboloid by the same plane have the same radius of curvature ; and it thence readily follows that the radius of curvature of a normal section of the surface by a plane inclined at. an angle 0 to that of 7.x is given by the equation

$\frac{1}{p}=\frac{cos^2\theta}{a}+\frac{sin^2\theta}{b}$

The section in question is that by a plane through the normal and a line in the tangent plane inclined at an angle θ to the principal tangent along the axis of x. To complete the theory, consider the section by a plane having the same trace upon the tangent plane, but inclined to the normal at an angle gb ; then it is shown without difﬁculty (Meunier’s theorem) that the radius of curvature of this inclined section of the surface is = p cos qt).  

 GEORGE I., king of Great Britain and Ireland (George Louis, 1660–1727), born in 1660, was heir through his father Ernest Augustus to the hereditary lay bishoprie of Osnabriick, and to the duchy of Calenburg, which formed one portion of the Hanoverian possessions of the house of Brunswick, whilst he secured the reversion of the other portion, the duchy of Celle or Zell, by his marriage (1682) with the heiress, his cousin Sophia Dorothea. The marriage was not a happy one. The morals of German courts in the end of the 17th century took their tone from the splendid proﬂigacy of Versailles. It became the fashion for a prince to amuse himself with a mistress or more frequently with many mistresses simultaneously, and he was often content that the mistresses whom he favoured should be neither beautiful nor witty. George Louis followed the usual course. Count Konigsmark—a handsome adventurer—seized the opportunity of paying court to the deserted wife. Con- jugal infidelity was held at Hanover to be a privilege of the male sex. Count Kenigsmark was assassinated. Sophia Dorothea was divorced in 1694, and remained in seclu- sion till her death in 1726. When her descendant in the fourth generation attempted in England to call his wife to account for sins of which he was himself notoriously guilty, free-spoken public opinion reprobated the offence in no measured terms. In the Germany of the 17th cen- tury all free-spoken public opinion had been crushed out by the misery of the Thirty Years’ \Var, and it was under- stood that princes were to arrange their domestic life accord- ing to their own pleasure. The prince’s father did much to raise the dignity of his family. By sending help to the emperor when he was struggling against the French and the Turks, he obtained the grant of a ninth electorate in 1692. His marriage with Sophia, the youngest daughter of Elizabeth the daughter of James I. of England, was not one which at ﬁrst seemed likely to confer any prospect of advancement to his family. But though there were many persons whose birth gave them better claims than she had to the English crown, she found herself, upon the death of the duke of Gloucester, the next Protestant heir after Anne. The Act of Settlement in 1701 secured the inheritance to herself and her descend- ants. Being old and unambitious she rather permitted herself to be burthened with the honour than thrust her- self forward to meet it. Her son George took a deeper in- terest in the matter. In his youth he had fought with deter- mined courage in the wars of William III. Succeeding to the electorate on his father’s death in 1698, he had sent a welcome reinforcement of Hanoverians to fight under Marlborough at Blenheim. With prudent persistence hc attached himself closely to the Whigs and to Marlborough, refusing Tory offers of an independent command, and receiv— ing in return for his ﬁdelity a guarantee by the Dutch of his succession to England in the Barrier treaty of 1709. ln 1714 when Anne was growing old, and Bolingbroke and the more reckless Tories were coquetting with the son of James II., the \Vhigs invited George’s eldest. son, who was duke of Cambridge, to visit England in order to be on the spot in case of need. Neither the elector nor his mother approved of a step which was likely to alienate the queen, and which was specially distasteful to himself, as he was on very bad terms with his son. Yet they did not set themselves against the strong wish of the party to which they looked for support, and it is possible that troubles would have arisen from any attempt to carry out the plan, if the deaths, ﬁrst of the electress (May 28) and then of the queen (August 1, 1714), had not laid open George’s way to the succession without further effort of his own. In some respects the position of the new king was not unlike that of William III. a quarter of a century before. Both sovereigns were foreigners, with little knowledge of English politics and little interest in English legislation. Both sovereigns arrived at a time when party spirit had been running high, and when the task before the ruler was to still the waves of contention. In spite of the difference between an intellectually great man and an intellectually