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intersecting the axes in A, B, and C; then EA, EB, EC will be co-ordinates of N; let them be represented by a, b, c. Imagine also planes to pass through P parallel to the coordinate planes, intersecting the axes in A', B', c'; then EA', EB', EC' will be the co-ordinates of P; let these be represented by x, y, z. The parts AA', BB', CC' are said to be the co-ordinates of P with respect to N; for if planes were to pass through N parallel to the co-ordinate planes, the planes passing through P, as above mentioned, would cut the axes in points, as a, b, and c, making Na, Nb, Nc, respectively equal to AA', BB', CC': these last co-ordinates are therefore respectively equal to x-a, y —b, z — c.

Let EN be represented by p; then the triangles NEA, NEB, NEC, PEA, &c. being right angled at A, B, C, a', &c., we have (Pl. Trigon., art. 56.)

p cos. NEX = a, p cos. NEY = b, p cos. NEZ = c; also EP cos. PEX = x, EP Cos. PEY=y, EP COS. PEZ = z. Squaring the three first equations, and adding the results together, we have, since cos. NEX + cos. 2 NEY + COS. 2 NEZ = 1,

a2 + b2 + c2 = EN2 (= p2). Squaring the other three equations, and adding the results together, we have in like manner

2
x2 + y2 + z2 = EP2;

also (x − a)2 + (y — b)2 + (z — c)2 = NP2;

2

but EN2 + NP2 = EP2; therefore, developing the first member of NP2 and reducing, we obtain

ax + by + cz = a2 + b2 + c2 = p2,

or pr cos. NEX + py cos. NEY + pz cos. NEZ = p2; or again, x cos. NEX + y cos. NEY + z cos. NEZ = p.

If

any other points P', P", &c. were taken in the plane to which EN is perpendicular, and if the co-ordinates of these points were represented by a', y', z', x', &c., we should have

x'cos. NEX + y' cos. NEY + z' COS. NEZ = p,

x" cos. NEX + y' cos. NEY + z" cos. NEZ = p, &c. ; and hence any one of these equations is called the equation of a plane in space.

184. If it were required to transform the rectangular coordinates of any point in space from one system of planes to another system in which each plane is parallel to the corresponding plane in the former system, the investigation may be made as follows:-the angular point, or origin, of the new system of co-ordinates being, for simplicity, in one of the planes belonging to the former system.

Let EX, EY, EZ, be

three rectangular co-or-
dinate axes, and s any
point in the plane XEY:
let Es be represented by
R and the angle XES by
L; then EA, EB being
the co-ordinates of s, if
these be represented by
X and Y we have
X=R COS. L, Y=R sin. L.

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Let now P be any point in space, м its orthographical projection on the plane XEY; and let its co-ordinates EA', EB', EZ (PM) be represented by x, y, z, respectively: also let EP be represented by r, the angle XEM by 7 and PEM by λ. Then (Pl. Trigo., art. 56.) r cos. λ= EM, and EM cos.l = EA' = x; also EM sin. 7 EB'y; therefore

we have also

xr cos. A cos. 1,

y=r cos. A sin. 7;
z = r sin. λ.

Next, if through s there were made to pass two rectangular co-ordinate planes perpendicular to XEY, cutting it in sx', SY' parallel to EX, EY respectively, and intersecting one another in sz' parallel to Ez; then the co-ordinates SA", SB", and PM being represented by x', y' and z respectively, we should obtain

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Let SP be represented by r'; then, since x2+12 + z2 = p22, on squaring the second members of the equations for x', y',

and z, and adding the results together, we obtain, after putting unity (= rad.) for sin. + cos.2, and cos. (L — 1) for

cos. L cos. l + sin. L sin. 7 (Pl. Trigo., art. 32.),

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From this equation the value of r may be found, the other quantities being given; and such value being substituted in the second members above, there will result the values of x', y', and z.

If a plane oblique to the three co-ordinate planes XEY, XEZ, and YEZ were supposed to pass through P and to cut the plane XEY in some line as A""B_parallel to EX, and the inclination of this plane to XEY, that is, the angle PA"M (which may be represented by ) were given; we should have

PMA"M tan. 0, or z =

y' tan. 0: substituting in this equation the above value of y' the equation becomes

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But, as above, z = r sin. λ; therefore, equating these values of z, there may be obtained the value of r, with which, from the equation for r'2, the value of ' may be found, and subsequently, from their proper equations, the values of x', y'and z.

185. If a plane oblique to the three co-ordinate planes XEY, XEZ, YEZ were to pass through P, and in it were taken two other points P', P", whose co-ordinates with respect to SX', SY', sz, are represented by x", y', z" and x"", y"", z""; there might be found in a similar manner the values of those co-ordinates. Again, if a line as SN represented by p were let fall from s perpendicularly on the plane we should have (art. 183.)

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(1), (2),

x'cos. NSX' + y' cos. NSY' + z' cos. NSZ' = p x'cos. NSX' + y' cos. NSY' + z' cos. NSZ' = p x'"' cos. NSX' + y'"' cos. NSY' + z''' cos. NSZ' then dividing all the terms by cos. NSZ we get, on sub

tracting (2) from (1) and (3) from (1),

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(3)

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from these equations the values of Cos. NSX'

COS. NSZ

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may be found.

Now, the co-ordinates of N with respect to s being SA"", SB"", and NM', we have

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But, again, the angle which a plane perpendicular to SN makes with a plane perpendicular to sz' is equal to the angle which those perpendiculars make with each other; thus SNM' expresses the inclination of the former to the latter plane; and consequently such angle of inclination is found.

186. The position of the line in which a plane passing through two points as P, P', besides the point s, intersects the plane XEY, may be found in the following manner:

Let a, ẞ, y represent, with respect to s X', SY', sz', the coordinates of any fourth point in the plane: it is evident that when this fourth point is any where in the required line of section, the co-ordinate y will be zero; and since the plane is to pass through s, the perpendicular p will also be zero: therefore the equation corresponding to (1), (2) or (3) becomes, for a line in the plane XEY,

α

a cos. NSX' + B cos. NSY' = 0; hence- =

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COS. NSY'

cos. NSX'

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and this last fraction expresses the tangent of the angle A""'SM';

α
therefore =tan. A"SM'.
B

The line sm, drawn perpendicular to SM', will be the required line of section; and the complement of A""'S M' will be equal to the angle made by that line with the axis X'S B, or with EX which is parallel to it.

187. At any one station on the earth the horizon makes constantly with the equator an angle equal to the co-latitude of the station; but the horizon and its vertical or azimuthal circles necessarily change their positions as the spectator changes his place on the earth's surface. Therefore if, at any station, it were required to transform the vertical and horizontal co-ordinates into others; for example, if it were required to compute, from an observed altitude and azimuth,

the right ascension and declination, or the longitude and latitude of a celestial body, it would be necessary that there should be known, besides the elements first mentioned, which are supposed to have been observed, the latitude of the station and the solar time of the observation. From this last, the right ascension of the mid-heaven, or the distance of the meridian from the true point of the vernal equinox, may (art. 312.) be found: the time shown at any instant by a sidereal clock is, however (art. 173.) the right ascension of the midheaven, or meridian, at that instant. With these data, the required co-ordinates can be obtained by the solution of spherical triangles or by the formulæ in art. 181., in the following

manner.

Let the figure represent a hemisphere of the heavens projected stereographically on the horizon wOE of the station: let z represent the zenith of

the observer, PZO the meridian
and P the pole of the equator
WME: let MY on the equator
be the right ascension of the
mid-heaven, then will be w
the equinoctial point: let
represent the ecliptic, making
with the equator the angle
CTE equal to the obliquity of
those circles to one another,
and p its pole. Lets be the

NC

K

C

S

E

L

N

A

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R

M

T

celestial body, through which are drawn the vertical circle ZA, the horary or declination circle PR, and the circle of celestial longitude pL.

Here zs is the observed zenith distance of s, the immediate result of the observation for the altitude being corrected on account of the effects of refraction and parallax, and OA, or OZA, or its supplement PZS is the observed azimuth, also PZ is the colatitude of the station. These three terms being known, we can, in the triangle PZS, find PS and the angle ZPS: the arc PS is the complement of SR which is the declination of s; and the angle ZPS is measured by MR, which being added to yм (the right ascension of the mid-heaven) gives R the right ascension of s.

Or let w and E be the eastern and western points of the horizon respectively; then OA or the angle OZA being the observed azimuth, by adding it (in the case represented by the figure) to a quadrant or 90 degrees, we have the value of the angle WZA or the arc wOA: let this be represented by a, also let As, the observed altitude of the celestial body, be re

L

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