into the foregoing form, and hence that the two near equations represent a line..

Secondly, taking for greater simplicity the point Q to be coincident with the origin, and a', B, Y, p to be constant, then p is the perpendicular distance of a plane from the origin, and a', B, are the cosine-inclinations of this distance to the axes (a+p2+ y2-1) P is any point in this plane, and taking its coordinates to be (x, y, z) then (§, n, 6) are -(x, y, z), and the foregoing equation p➡at+B'n+y's becomes

dx+By+ýz=p,

which is the equation of the plane in question.

If, more generally, Q is not coincident with the origin, then, taking its coordinates to be (a, b, c), and writing p1 instead of p, the equation is

a'(x − a) + B′( y − b) + y'(z − c ) − P1,

and we thence have p1-p− (aa+b+cy), which is an expression for the perpendicular distance of the point (a, b, c) from the plane in question.

It is obvious that any linear equation Ax+By+C+D-0 between the coordinates can always be brought into the foregoing form, and hence that such equation represents a plane.

Thirdly, supposing Q to be a fixed point, coordinates (a, b, c) and the distance QP,-p, to be constant, say this isd, then, as before, the values of έ, n, are x-a, y-b, x-c, and the equation ++- becomes

Cylinders, Cones, Ruled Surfaces.

33. A singly infinite system of lines or system of lines depending upon one variable parameter forms a surface; and the equation of the surface is obtained by eliminating the parameter between the two equations of the line.

If the lines all pass through a given point, then the surface is a cone; and, in particular, if the lines are all parallel to a given line, then the surface is a cylinder.

Beginning with this last case, suppose the lines are parallel to the line x-mz, y-nz, the equations of a line of the system are z➡mz+a, y=nz+b,-where a, b are supposed to be functions of the variable parameter, or, what is the same thing, there is between them a relation fa, b)=0: we have a-- mz, b--y - nz, and the result of the elimination of the parameter therefore is Kx-mz, y-nx)=0, which is thus the general equation of the cylinder the generating lines whereof are parallel to the line 2-ms, y-nz. The equation of the section by the plane 2-0 is Ax, y)-0, and conversely if the cylinder be determined by means of its curve of intersection with the plane z 0, then, taking the equation of this curve to be f(x, y)-0, the equation of the cylinder is f(x-mz, y-nz)-0. Thus, if the curve of intersection be the circle (x − a) + (y — B)-2, we have (x − mz − a)2 + (y — nz — B)3 — y3 as the equation of an oblique cylinder on this base, and thus also (r− a)2+(y-8)2 = as the equation of the right cylinder.

-

2-0

If the lines all pass through a given point (a, b, c), then the equations of a line are x-a-az- c), y − b - B(z – c), where a, ẞ are functions of the variable parameter, or, what is the same thing, there exists between them an equation fa, B)-0; the elimination v of the parameter gives, therefore, ƒ 2--0; and this equation, or, what is the same thing, any homogeneous equation x-a, y-b, z-c)-0, or, taking f to be a rational and integral function of the order n, say (*)(x-a, y - b, z −c)" — 0, is the general equation of the cone having the point (a, b, c) for its vertex. Taking the vertex to be at the origin, the equation is (*Xx, y, z)"-0; and, in particular, (*)(x, y, z)-0 is the equation of a cone of the second order, or quadricone, having the origin for its vertex.

34. In the general case of a singly infinite system of lines, the locus is a ruled surface (or regulus). If the

system be such that a line does not intersect the consecutive line, then the surface is a skew surface, or scroll; but if it be such that each line intersects the consecutive line then it is a developable, or torse.

It is easily shown that any line of the one system intersects every line of the other system.

Considering any curve (of double curvature) whatever, the tangent lines of the curve form a singly infinite system of lines, each line intersecting the consecutive line of the system,—that is, they form a developable, or torse; the curve and torse are thus inseparably connected together, forming a single geometrical figure. lating plane of the curve) contains two consecutive tangents, that A plane through three consecutive points of the curve (or oscu is, two consecutive lines of the torse, and is thus a tangent plane of the torse along a generating line.

Transformation of Coordinat

is for brevity assumed that the origin remains unaltered. 35. There is no difficulty in changing the origin, and it We have, then, two sets of rectangular axes, Ox, Oy, Oz, and Ox, Oy, Oz, the mutual cosine-inclinations being shown by the diagram

which are obtained by projection, as above explained. Each of these equations is, in fact, nothing else than the before-mentioned equation p=a'§+ẞ'n+y's, adapted to the problem in hand.

But we have to consider the relations between the nine coefficients. By what precedes, or by the consideration that we must have identically + y2+2 = x ̧2 + y ̧2 + z‚3‚ it appears that these satisfy the relationsa2 + B2 + y2 -1, a's + B's +2 a22 + 83 +2

-1 -1,

d'a" + BB" +"=0, a"a + B B+7'7 .0 ad + BB + =0

a2+a+a^ B2 + B22 + B2 g3+y's +g"2

By+By+B"y". ya+ya+y" aß + a'B'+a"B" = 0

either set of six equations being implied in the other set It follows that the square of the determinant

÷(1+λ3−μ3+v3)

The nine coefficients of transformation are the nine functions of the diagram, each divided by 1+x2+μ2+; the expressions contain as they should do the three arbitrary quantities λ, μ, v; and the identity x+y12+2x+y+ can be at once verified. It may be added that the transformation can be expressed in the quaternion form

íx1+j¥1+k1⁄21⁄2➡(1+A)(ix+jy+kz)(1+A)−! where A denotes the vector ix +jμ+kv.

Quadric Surfaces (Paraboloids, Ellipsoid, Hyperboloids). 37. It appears by a discussion of the general equation of the second order (a, . . Xx, y, z, 1)2=0 that the proper quadric surfaces' represented by such an equation are the following five surfaces (a and b positive) :

elliptic paraboloid.

(1.)2·

" 2a 26

Fig. 22.

of two distinct portions or sheets, may be considered as generated

by a variable ellipse moving parallel to itself along the hyperbolas as directrices.

The hyperbolic paraboloid is such (and it is easy from the figure to understand how this may be the case) that there exist upon it two singly infinite series of right lines. The same is the case with the hyperboloid of one sheet (ruled or skew hyperboloid, as with reference to this property it is termed). If we imagine two equal and parallel circular disks, their points connected by strings of equal length, so that these are the generating lines of a right circular cylinder, then by turning one of the disks about its centre through the same angle in one or the other direction, the strings will in each case

Fig. 25.

-x

generate one and the same hyperboloid, and will in regard to it be the two systems of lines on the surface, or say the two systems of generating lines; and the general configuration is the same when instead

set of three rectangular axes-the tangent, the principal normal, and the binormal.

We have through the point and three consecutive points a sphere of spherical curvature,-the centre and radius thereof being the centre, and radius, of spherical curvature. The sphere is met by the osculating plane in the circle of absolute curvature,-the centre and radius thereof being the centre, and radius, of absolute curvature. The centre of absolute curvature is also the intersection of the principal normal by the normal plane at the consecutive point.

Surfaces; Tangent Lines and Plane, Curvature, &c.

39. It will be convenient to consider the surface as given by an equation f(x, y, z)=0 between the coordinates; taking (x, y, z) for the coordinates of a given point, and (x+dx, y+dy, z+dz) for those of a consecutive point, the increments dx, dy, dz satisfy the condition

In the series of tangent lines there are in general two

22

+

ya 2-3

that the equation -1 is satisfied by each of two pairs of linear relations between the coordinates.

Curves; Tangent, Osculating Plane, Curvature, &c. 38. It will be convenient to consider the coordinates (x, y, z) of the point on the curve as given in terms of a parameter 0, so that dx, dy, dz, d2x, &c., will be propordx dy dz d2x tional to &c. But only a part of the de' de' de' de' analytical formula will be given. rent coordinates.

,, are used as cur

The tangent is the line through the point (x, y, z) and the consecutive point (x + dx, y + dy, z+dz); its equations therefore are

in a second consecutive point, or say it has three-point intersection with the surface; these are called the chieftangents (Haupt-tangenten). The tangent-plane cuts the surface in a curve, having at the point of contact a node (double point), the tangents to the two branches being the chief-tangents.

In the case of a quadric surface the curve of intersection, qua curve of the second order, can only have a node by breaking up into a pair of lines; that is, every tangentplane meets the surface in a pair of lines, or we have on the surface two singly infinite systems of lines; these are real for the hyperbolic paraboloid and the hyperboloid of one sheet, imaginary in other cases.

At each point of a surface the chief-tangents determine two directions; and passing along one of them to a consecutive point, and thence (without abrupt change of direction) along the new chief-tangent to a consecutive point, and so on, we have on the surface a chief-tangent curve; and there are, it is clear, two singly infinite series of such curves. In the case of a quadric surface, the curves are the right lines on the surface.

40. If at the point we draw in the tangent-plane two lines bisecting the angles between the chief-tangents, these lines (which are at right angles to each other) are called the principal tangents. We have thus at each point of

principal tangents become arbitrary; the point is then an umbilicus. 1 The point on the surface may be such that the directions of the It is in the text assumed that the point on the surface is not an

umbilicus.

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 consecutive 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 intersected 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 principal 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 a 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

0:

and the chief-tangents are determined by the equation ས. ya + The two centres of curvature may be on 2a 26" the same side of the point or on opposite sides; in the former case a and b have the same sign, the paraboloid is elliptic, and the chief-tangents are imaginary; in the latter bolic, and the chief-tangents are real. case a and b have opposite signs, the paraboloid is hyper

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 to that of zx is given by the equation

GEORGE L, king of Great Britain and Ireland (George Louis, 1660-1727), born in 1660, was heir through his father Ernest Augustus to the hereditary lay bishopric of Osnabrück, 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 Königsmark-a handsome adventurer-seized the opportunity of paying court to the deserted wife. Conjugal infidelity was held at Hanover to be a privilege of the male sex. Count Königsmark was assassinated. Sophia Dorothea was divorced in 1694, and remained in seclusion 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 century all free-spoken public opinion had been crushed out by the misery of the Thirty Years' War, and it was understood that princes were to arrange their domestic life according 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 Anno. The Act of Settlement in 1701 secured the inheritance to herself and her descendants. Being old and unambitious she rather permitted herself to be burthened with the honour than thrust herself forward to meet it. Her son George took a deeper interest in the matter. In his youth he had fought with determined 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 he 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. In 1714 when Anne was growing old, and Bolingbroke and the more reckless Tories were coquetting with the son of James II., the Whigs 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