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Fig. 18.

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28. In plane geometry, reckoning the line as a curve finite right line PQ be projected upon any other line 00 of the first order, we have only the point and the curve. by lines perpendicular to 00', then the length of the proIn solid geometry, reckoning a line as a curve of the first jection P Q is equal to the length of PQ into the cusing order, and the plane as a surface of the first order, we of its inclination to PQ-or (in the form in which it is

now convenient to stato the theorem) the perpendicular distance P'Q' of two parallel planes is equal to the inclined distance PQ into the cosine of the inclination. Hence also the algebraical sum of the projections of the sides of a closed polygon upon any line is = 0; or, reversing the signs of certain sides and considering the polygon as made up of two broken lines each extending from the same initial to the same terminal point, the sum of the projections of the one set of lines upon any line is equal to this sum of the projections of the other set of lines upon tho same line. When any of the lines are at right angles tu the given line (or, what is the same thing, in a plane at right angles to the given line) the projections of these lines severally vanish.

31. Consider the skew quadrilateral QMNP, the sides y

QM, MN, NP being respectively parallel to the three rect

angular axes Ox, Oy, Oz; let the lengths of these sides be have the point, the curve, and the surface; but the in- , 7, 8, and that of the side QP berp; and let the cosines crease of complexity is far greater that would hence at of the inclinations (or say the cosine-inclinations) of p to first sight appear. In plane geometry a curve is considered the three axes be an B, y; then projecting successively on in connexion with lines (its tangents); but in solid the three sides and on QP we have geometry the curve is considered in connexion with lines

8, 9, 8-pa, , py, and planes (its tangents and osculating planes), and the

and surface also in connexion with lines and planes (its tan.

p-a$ +By+y5, gert lines and tangent planes); there are surfaces arising whence pe - $ +98 + $?, which is the relation between a out of the line-cones, skew surfaces, developables, doubly distance p and its projections & m; upon three rectand triply infinite systems of lines, and whole classes of angular axes And from the same equations we obtain theories which have nothing analogous to them in plane a' + B2 + y2 - 1, which is a relation connecting the cosine geometry: it is thus a very small part indeed of the sub- | inclinations of a line to three rectangular axes. ject which can be even referred to in the present article. Suppose we have throngh Q any other line QT, and let the

In the case of a surface we have between the coordi. cosine-inclinations of this to the axes be a', 8, 7, and 3 be ite nates (2, y, z) a single, or say a onefold relation, which

cosine-inclination to QP; also let p be the length of ihe projective

of QP upon QT; then projecting on QT we have can be represented by a single relation f(x, y, z) = 0; or we may consider the coordinates expressed each of them

p-a's + B'n+0's, pd. as a given function of two variable parameters y, 9; the

And in the last equation substituting for $, *. ( their values

pa, PB, py we find form z=f(L, y) is a particular case of each of these modes of

8 - aa' + BB'+ny's ropresentation; in other words, we have in the first mode

which is an expression for the mutual cosine-inclination of two f (x, y, z) – 2- f (, y), and in the second mode <= P, y = lines, the cosido-inclinations of which to the axes are a B, you for the oxpression of two of the coordinates in terms of d', s', v respectively, Wo have of course of +88 +99-1, aud the parameters

a " +84 +- 1; and hence also In the case of a curve we have between the coordinates 1 - 89-a +88 +/(a* +852 +93)-(ad' + BB +ne, (2, y, z) a twofold relation: two equations f(x, y, z)= 0,

-(By - 87)*+(gya' - g'a)* + (as - a'B); $14, y, z) = 0 give such a relation; i.e., the curve is here so that the sine of the inclination can only be expressed as a square considered as the intersection of two surfaces (but the root. These formulæ are the foundation of spherical trigonometry. curve is not always the complete intersection of two surfaces, and there are hence difficulties); or, again, the co

The Line, Plune, and Sphere. ordinates may be given each of them as a function of a

32. The foregoing formulæ give at once the equatlons single variable parameter. The form y = pt, 2 = 42, where of these loci. two of the coordinates are given in terms of the third, is a particular case of each of these modes of representation.

Por first, taking Q to be a fixed point, coordinates (a, b, c) and

the cosine-inclinations («, B, y) to be constant, then P will be a 29. The remarks under plane geometry as to descriptive point in the line through Q in the direction thus determined; or, and metrical propositions, and as to the non-metrical char- taking (2, y, z) for its coordinates, these will be the current coacter of the method of coordinates when used for the ordinates of a joint in the line. The values of t, 7 then ate proof of a descriptive proposition, apply also to solid geo

- a, y-b, 1-C, and we thus have metry; and they might be illustrated in like manner by the

2 - a_y-6_2

4-p)

, instance of the theorem of the radical centre of four

B spheres . The proof is obtained from the consideration that through the point (a, b, c), the cosine-inclinations to the axes being

which (omitting the last equation, - p) are the equations of the line S and S' being each of them a function of the form

ą, B. %, and these quantities being connected by the relation * + y +z2 + ax + hy + cz + d, the difference $-S is a + Be +-1. This equation may be omitted, and then a : 1 mere linear function of the coordinates, and consequently

instead of being equal, will only be proportional to the cosinethat $-S' = 0 is the equation of the plane containing the

inclinations, circle of intersection of the two spheres $ = 0 and S' = 0.

Using the last equation, and writing

x, y, z = a + op, b+ Bp, c+ya, Metricul Theory.

these are expressions for the current coordinates in terms of a persoa 30. The fonndation in solid geometry of the metrical meters, which is in fact the distance from the fixed point (a, heory is in fact the before-mentioned theorem that if a l'any two linear equations, these equations can always be brought

It is easy to see that, if the coordinates (€, 9, +) are connected by

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into the foregoing form, and hence that the two sinear equations Suppose, for instance, that the equations of a line (ucpending on represent a line." Secondly, taking for greator simplicity the point Q to be coin.

the variable parameter 0) are -o(1+3) cident with the origin, and a', 8, 7, P to be constant, then p

ya 1 is the perpendicular distance of a plane from the origin, and then, eliminating , we have

a-2"

aat 3-2-1, d', 8, 9 are the cosine-inclinations of this distance to the axes 12+2+4 -1). P is any point in this plane, and taking its co

the equation of a quadric surface, afterwards called the hyperboloid ordinates to be (x, y, z) then (£, n, ) are (2, y, z), and the fore

of one sheet; this surface is consequently a scroll. It is to be regoing equation p-a's + B 7+ gy's becomes

marked that we have upon the surface a second singly infinite

series of lines; the oquations of a line of this second system (ded'x+By+z=p,

pending on the variable parameter ®) are 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 p, instead of p, the equation is

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

line of the other system. a'(x-a).+B(1-6) + 7 (z - 6)-P1,

Considering any curve (of double curvature) whatever, the tan. and we thence have pı-p-laa'+bB + oy'), which is an expression gent lines of the curve forni a singly infinite system of lines, each for the perpendicular distance of the point (a, b, c) from the plane line intersecting the consecutive line of the system,—that is, they in question.

form a developable, or torse ; the curve and torse are thus in. it is obvious that any linear equation Ax+By+Ca+D-0 separably connected together, forming a single geometrical figure. between the coordinates can always be brought into the foregoing a plane through three consecutive points of the curve (or oscu. form, and hence that such equation represents & plane.

lating plane of the curve) contains two consecutive tangents, that Thirdly, supposing, Q to be a fixed point, coordinates (a, b, c) is, two consecutive lines of the torse, and is thus a tangent plane and the distance QP, -p, to be constant, say this is = d, then, as

of the torse along a generating line. before, the values of & n, s are 2 - a, y-6, 6-4, and the equation + + -p becomes

Transformation of Coordinat. (2-a)* +(y - 0)*+(3-6)* - , which is the station of the sphere, coordinates of the centre - (a, b, c) is for brevity assumed that the origin remains unaltered.

35. There is no difficulty in changing the origin, and it and radius-d.

A quadric equation wherein the terms of the second order are We have, then, two sets of rectangular axes, Ox, Oy, O2, na + y + 3%, viza, an equatic

and Oxı, Oyy 02, the mutual cosine-inclinations being 202 + y + + Ax+By+Cz+D)-0,

shown by the diagramcan always, it is clear, be brought into the foregoing form; and it thus appears that this is the equation of a sphere, coordinates of the centre - 1A, - 4B, - 1C, and squared radius – 1(A + B + C*) - D.

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Cylinders, Cones, Ruled Surfaces

r 33. A singly infinite system of lines or system of lines

1

a" 8" , quod depending upon one variable parameter forms a surface; and the equation of the surface is obtained by eliminating that is, a B, y are the cosino- inclinations of Ox, to Ox. Oy, the parameter between the two equations of the line.

Oz; a', B, y those of Oy, &c. If the lines all pass through a given point, then the

And this diagram gives also the linear expressions of the surface is a cone; and, in particular, if the lines are all coordinates (41, 11, 2) or (x, y, z) of either set in terms parallel to a given line, then the surface is a cylinder.

of those of tho other set; we thus have Beginning with this last case, suppose the lines are parallel to

2, = a 2+By+go, 2 - ax: +a'yı +a", the line x=ma, y=nz, the equations of a line of the system are

y=dx+8y + 2, y- Bxi +By+8"z, * -mz+a, y-n2+b,—where a, b are supposed to be functions of

2=a"x+Boy+72,

-gocy to'yi+ya the variable parameter, or, what is the same thing, there is between them a relation fia, b)-0: we have a -2- mz, b-y-na,

which are obtained by projection, as above explained. and the result of the elimination of the parameter therefore is

Each of these equations is, in fact, nothing else than the Me- ma, y - 12) = 0, which is thus the general equation of the before-mentioned equation p= a'$+B'n + y' S, adapted to cylinder the generating lines whereof are parallel to the line the problem in hand. * -ms, y«nz. The equation of the section by the plane 2-0 is

But we have to consider the relations between the nine Me, y)-0, and conversely if the cylinder be determined by means of its curve of intersection with the plane 2-0, then, taking the

coefficients. By what precedes, or by the consideration quation of this curve to be fiz, y)-0, the equation of the cylinder that we must have identically &* + y2 +22 = x/2 +972 +37*, is 12 - mz, ! - rz) -0Thus, if the curve of intersection be tho it appears that these satisfy the relations circle (2-a)* +(3-B)-y, we have (c – inz-a) +(y nz -- B) - 78 as the equation of an oblique cylinder on this base, and thus also

al + Bx tgl -1

a+ a ta (ra)*+ly - 8)'-y as the equation of the right cylinder.

+89 +

88 + 88 +8412 1 If the lines all pass through a given point (a, b, c), then the

a'? +81 +gins

ge+q' +'? equations of a line are a-a-alz-c), y-b-B(2-0), where an B aro d'a" +BB'+

By+B'y' +8". fuuctions of the variable parameter, or, what is the same thing, a" a + BB+/"

ga+ay'a + "2" there exists between them an equation fia, B) – 0; the elimination ad + BB tri

+ a'B'+a"B" of the parameter gives, therefore, f

6 cquation, or, what is the same thing, any homogeneous equation

It follows that the square of the determinant viz-a, y-6, 2-0)-0, or, taking 1 to be a rational and integral

B, function of the order n, say (*)(z-a, y-b, 2-0)»-0, is the general

B', equation of the cone having the point (a, b, c) for its vertex. Taking

B", q" the vertex to be at the origin, the equation is (*Xx, y, z)"-0; and, is -1 ; anu hence that the determinant itself is = +1. The dis. in particular, (*)(x, y, z) - 0 is the equation of a cone of the second tinction of the two cases is an important one: if the determinant is onder, or quadricone, having the origin for its vertex.

- +1, then the axes Ox, Oy, 07 are such that they can by a 34. In the general case of a singly infinite system of | ively; if it is

rotation about O be brought to coincide with Ox, Oy, Oz respect.

-1, then they cannot. But in the latter case, by lines, the locus is & ruled surface (or regulus)... If the measuring 21%, 5 in the opposite directions we change the signs of system be such that a line does not intersect the consecu

all the coefficients and so make the determinant to be - +1; hence tive line, then the surface is a skew surface, or scroll; but this case need alone be considered, and it is accordingly assumed if it be sach that each line intersects the consecutive line have a further set of nine equations, a-82 - 8", &c.;

that is, the

that the determinant is - +1. This being so, it is found that we then it is a developable, or torse.

coefficients arranged as in the diagram have the values

X. 53

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By” – Big I ra” - g" a a'ß" - AB

It is at once seen that these are distinct surfaces; and

the equations also show very readily the general form an! B"y - BY" ga -ga" a" B - "

mode of generation of the several surfaces.

In the elliptic paraboloid (fig.
By - Bing ga - y'a | aß - a'B

20), the sections by the planes

of ac and zy are the parabolas' 36. It is important to express the nine coefficients in terms of three independent quantities. A solution which,

2a' 26'
although unsymmetrical, is very convenient in Astronomy
and Dynamics is to use for the purpose the three angles having the common axes Oz ;
0, 6, 7 of fig. 19; say 0 = longitude of the node; 6 = in and the section by any plané
clinationand 7 =

- longitude of
Hoy from node

2-y parallel to that of ry's the
e'lipse

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Fig. 19. The diagram of transformation then is

so that the surface is generated

Fig. 20.
by a variable ellipse moving parallel to itself along the parabolas e
directrices.

In the hyperbolic
paraboloid (fig. 21) tho
sections by the planes of
xx, xy are the parabolas

2012

hav-
2a' 26
ing the opposite axes
Oz, Oz', and the section
by a plane - parallel
to that of zy is tho
hyperbola ay

29

2a 20
which has its transverse
axis parallel to Ox or
Oy according as y is y
positive or negative.
The surface is thus
generated by a variable

Fig. 21.
hyperbola moving parallel to itself along the parabolas as direz-

1

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* 1+12 - 4* – 18 2(λμ - ν) 2013+w) 2(14 + v) 1-13+ Mama 2(uy-a) *1 2(y1 - A) 2(usta) 1-13 - +

=(1+18 - 4*+10) The nine coefficients of transformation are the nine functinas of the diagram, each divided by 1+1° + +vs; the expressions contain as they should do the three arbitrary quantities a, H,; and the identity ;*+y;*+23-**+ y +m* can be at once verified. It may be added that the transformation can be expressed in the

Fig. 22. quaternion form

trices. The form is best seen from fig. 22, which represents ixi +jyı+kzz -(1+x)(iz+jy+kz)(1+1)-1

the sections by planes parallel to the plane of zy, or sy where A denotes the vector in +34 +kv.

the contour lines; the

continuous lines are Quadric Surfaces (Paraboloids, Ellipsoid, Hyperboloids).

the sections above

the plane of xy, and 37. It appears by a discussion of the general equation the dotted lines the of the second order (a, . . dx, y, 2, 1)2 = 0 that the proper sections below this quadric surfaces represented by such an equation are the plane... The form is, following five surfaces (a and b positive)

in fact, that of a

saddle. (1.) .*

ya
elliptic paraboloid.

In the ellipsoid (fig.

23) the sections by the (2.) a hyperbolic paraboloid.

planes of zz, zy, and 2a 28

xy are each of them an (3.)

ellipse, and the section 1, ellipsoid.

by any parallel plane

is also an ellipse. The 1, hyperboloid of one sheet.

surface may be con

sidered as generated (5.) -1, hyperboloid of two sheets.

by an ellipse moving parallel to itself along two ellipses 13

directrices. 1 The improper quadric surfaces represented

by the general equation planes of ex; zy are the hyperbolas of the second order are (1) the pair of planes or plane-pair, including as

In the hyperboloid of one sheet (fig. 24), the sections by the & special case the twice repeated plane, and (2) the cone, including as a special case the cylinder. There is but one form of cone; but the

1,

-1, cylinder may be parabolic, elliptic, or hyperbolic.

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cy, and that by any parallel plane, is an ellipse ; and the surface set of three rectangular axes—the tangent, the principal may be considered as generated by a variable ellipse moving normal, and the binormal

. parallel to itself along the two hyperbolas AS

We have through the point and three consecutive points directrices.

a sphere of spherical curvature,—the centre and radius In the hyperboloid of

thereof being the centre, and radius, of spherical curvature. two sheets (fig. 25), the sec

The sphere is met by the osculating plane in the circle of tions by the planes of 2

absolute curvature,—the centre and radius thereof being and zy are the byperbolas

the centre, and radius, of absolute curvature. The centre of - en

absolute curvature is also the intersection of the principal

normal by the normal plane at the consecutive point. having the common transverse axis &Od'; the section by any plane z = y par

Surfaces; Tangent Lines and Plane, Curvature, &c. allel to that of xy, y being in absoluto magnitude >c,

39. It will be convenient to consider the surface as is the ellipse

given by an equation f(x, y, z)=0 between the coordi

nates ; taking (2, y, z) for the coordinates of a given point,

and (x + dx, y+dy, z + dz) for those of a consecutive Fig. 24

point, the increments dx, dy, dz satisfy the condition and the surface, consisting of two distinct portions or sheets, may be considered as generated

a for

df df
+ I a

dz -1, by a variable ellipse moving

dy

dz parallel to itself along the

but the ratio of two of the increments, suppose dx : dy, hyperbolas as directrices. The hyperbolic paraboloid is

may be regarded as arbitrary. Only a part of the analysuch (and it is easy from the

tical formulæ will be given. , n, Ś are used as current figure to understand how this

coordinates. may be the case) that there

We have through the paint a singly infinite series of exist upon it two singly infinite series of right lines. The same

right lines, each meeting the surface in a consecutivo is the case with the hyperboloid

point, or say having each of them two-point intersection of one sheet (ruled or skew

with the surface. These lines lie all of them in a plane hyperboloid, as with reference

which is the tangent plane; its equation is
to this property it is termed).
If we imagine two equal and

e - x
E – 2) + (n - y) +

df

(10 – 9) + (5-3) = 0, parallel circular disks, their points connected by strings of

as is at once verified by observing that this equation is equal length, so that these are

satisfied (irrespectively of the value of du: dy) on writing the generating lines of a right circular cylinder, then by turn

therein , n, $ = x + dx, y+dy, z+dz. ing one of the disks about its

The line through the point at right angles to the tancentre through the same anglo

gent plane is called the normal; its equations are in one or the other direction, the strings will in each case Fig. 25.

1-3 generate ono and the same hyperboloid, and will in regard to it be the

df df two systems of lines on the surface, or say the two systems of generat

dy ing lines; and the general configuration is the same when instead

In the series of tangent lines there are in general two of circles we havo ellipses. It has been already shown analytically that the equation

ya

=1 is satisfied by each of two pairs (real or imaginary) lines, each of which meets the surface az

in a second consecutive point, or say it has three-point of linear relations between the coordinates.

intersection with the surface; these are called the chief

tangents (Haupt-tangenten). The tangent-plane cuts the Curves; Tangent, Osculating Plane, Curvature, &c.

surface in a curve, having at the point of contact a node 38. It will be convenient to consider the coordinates (double point), the tangents to the two branches being the (, y, z) of the point on the curve as given in terms of a chief-tangents. parameter 0, so that dx, dy, dz, d2x, &c., will be propor- In the case of a quadric surface the curve of intersecda

d? tional to do' dodo' dd2" &c. But only a part of the tion, qua curve of the second order, can only have a node

by breaking up into a pair of lines; that is, every tangentanalytical formula will be given É; , % are used as cur- plane meets the surface in a pair of lines, or we have on rent coordinates.

the surface two singly infinite systems of lines; these are The tangent is the line through the point (-0, y, z) and real for the hyperbolic paraboloid and the hyperboloid of the consecutive point (x + dx, y+dy, z + dz); its equations one sheet, imaginary in other cases,

At each point of a surface the chief-tangents determine 3-2-1-1_$-.

two directions; and passing along one of them to a condy dz

secutive point, and thence (without abrupt change of The osculating plane is the plane through the point direction) along the new chief-tangent to a consecutive and two consecutive points, and contains therefore the point, and so on, we have on the surface a chief-tangent tangent; its equation is

curvè ; and there are, it is clear, two singly infinite series --, 1-4, 6-2

of such curves. In the case of a quadric surface, the dac

curves are the right lines on the surface. daz

40. If at the point we draw in the tangent-plano two or, what is the same thing,

lines bisecting the angles between the chief-tangents, these (4-x)(dyd-z-dzdạy)+(7-y)(dzdz-ded2)+(5-2)(didydydou) - 0. lines (which are at right angles to each other) are called

The normal plane is the plane through the point at the principal tangents. We have thus at each point of right angles to the tangent. It meets the osculating plane in a line called the principal normal; and drawing through principal tangents become arbitrary; tho point is then an umbilicus.

1 The point on the surfaco may be such that the directions of the the point a line at right angles to the osculating plane, It is in the toxt assumed that the point on the surface is not an this is called the binormal We have thus at the point a umbilicus.

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the surface a set of rectangular axes, the normal and the 2=a and = b respectively), the surface has in the neightwo principal tangents.

bourhood of the point the form of the paraboloid Proceeding from the point along a principal tangent to a consecutive point on the surface, and thence

2a 20 (without abrupt change of direction) along the new

and the chief-tangents are determined by the equation principal tangent to a consecutive point, and so on, we have on the surface a curve of curvature; there are, 0

The two centres of corvature may be on

2a 26 it is clear, two singly infinite series of such curves, the same side of the point or on opposite sides; in the cutting each other at right angles at each point of the former case a and 6 have the same sign, the paraboloid is surface.

Passing from the given point in an arbitrary direction elliptic, and the chief-tangents are imaginary ; in the latter to a consecutive point on the surface, the normal at the holic, and the chief-tangents are real

case a and I have opposite signs, the paraboloid is hypergiven point is not intersected by the normal at the consecutive point; bat passing to the consecutive point along by the same plane have the same radius of curvature ; and

The normal sections of the surface and the paraboloid a curve of curvature (or, what is the same thing, along a principal tangent) the normal at the given point is inter- normal section of the surface by a plane inclined at an

it thence readily follows that the radius of curvature of a sected by the normal at the consecutive point; we have thus on the normal two centres of curvature

, and the angle 8 to that of ze is given by the equation distances of these from the point on the surface are the

i cosa sin'e two principal radii of curvature of the surface at that point; these are also the radi of curvature of the sections of the The section in question is that by a plane through the surface by planes through the normal and the two prin- normal and a line in the tangent plans inclined at an cipal tangents respectively; or say they are the radii of angle o to the principal tangent along the axis of z. curvature of the normal sections through the two principal To complete the theory, consider the section by a plane tangents respectively. Take at the point the axis of , in having the same trace upon the tangent plane, but the direction of the normal, and those of u and y in the inclined to the normal at an angle ; then it is shown directions of the principal tangents respectively, then, if without difficulty (Meunier's theorem) that the radius if the radii of curvature be a, b (the signs being such that of curvature of this inclined section of the surface is a the coordinates of the two centres of curvature are p Cos g.

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GEORGE I., king of Great Britain and Ireland (George she found herself, upon the death of the duke of Gloucester, Louis, 1660-1727), born in 1660, was heir through his the next Protestant heir after Anno. The Act of Settlement father Ernest Augustus to the hereditary lay bishopric of in 1701 secured the inheritance to herself and her descendOsnabrück, and to the duchy of Calenburg, which formed ants. Being old and unambitious she rather permitted one portion of the Hanoverian possessions of the house of herself to be burthened with the honour than thrust herBrunswick, whilst he secured the reversion of the other selt forward to meet it. Her son George took a deeper inportion, the duchy of Celle or Zell, by his marriage (1682) terest in the matter. In his youth he had fought with deterwith the heiress, his cousin Sophia Dorothea. The marriage mined courage in the wars of William III. Succeeding to was not a happy one. The morals of German courts in the the electorate on his father's death in 1698, he had sent end of the 17th century took their tone from the splendid a welcome reinforcement of Hanoverians to fight under profligacy of Versailles. It became the fashion for a prince Marlborough at Blenheim. With prudent persistence he to amuse himself with a mistress or more frequently with attached himself closely to the Whigs and to Marlborough, many mistresses simultaneously, and he was often content refusing Tory offers of an independent command, and receivthat the mistresses whom he favoured should be neither ing in return for his fidelity a guarantee by the Dutch of his beautiful nor witty. George Louis followed the usual course. succession to England in the Barrier treaty of 1709. In Count Königsmark—a handsome adventurer-Beized the 1714 when Anne was growing old, and Bolingbroke and opportunity of paying court to the deserted wife. Con the more reckless Tories were coquetting with the son of jugal infidelity was held at Hanover to be a privilege of James II., the Whigs invited George's eldest son, who was the male sex. "Count Königsmark was assassinated. Sophia duke of Cambridge, to visit England in order to be on the Dorothea was divorced in 1694, and remained in seclu- spot in case of need. Neither the elector nor his mother sion till her death in 1726. When her descendant in the approved of a step which was likely to alienate the queen, fourth generation attempted in England to call his wife and which was specially distasteful to himself, as be wa to account for sins of which he was himself notoriously on very bad terms with his son. Yet they did not set guilty, freo-spoken public opinion reprobated the offence themselves against the strong wish of the party to which in no measured terms. In the Germany of the 17th cen they looked for support, and it is possible that troubles tury all free-spoken public opinion

had been crashed out would have arisen from any attempt to carry out the by the misery of the Thirty Years' War, and it was under plan, if the deaths, first of the electress (May 28) and stood that princes were to arrange their domestic life accord then of the queen (August 1, 1714), bad not laid ope! ing to their own pleasure

George's way to the succession without further effort of his The prince's father did much to raise the dignity of his family. By sending help to the emperor when he was In some respects the position of the new king was not struggling against the French and the Turks, he obtained unlike that of William LİL a quarter of a. the grant of a ninth electorate in 1692. His marriage Both sovereigns were foreigners, with little knowledge of with Sophia, the youngest daughter of Elizabeth the English politics and little interest in English legislation daughter of James I. of England, was not one

which at first Both sovereigns arrived at a time when party spirit bad seemed likeiy to confer any

prospect of advancement to his been running high, and when the task before the ruler was family. But though there were many persons whose birth to still

the waves of contention. In spite of the difference gave them botter claims than she had to the English crown, between an intellectually great man and an intellectually

own.

8. century before

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