will be the sum of the projections on ZOY. Let this sum be represented by a', and let the sums of the projections of a, a', a", &c. on zox, XOY be represented respectively by a" and A""; also let the sums of the projections of a, a', a", &c., on Z'OY', Z'OX', X'or' be represented respectively by A, A, A Then, from the last proposition we have

A, A' cos. XOX'+A" cos. YOX'+A"" cos. Zox'
AA' cos. XOY'+A" cos. YOY'+A"" cos. ZOY'
A−A COS. XOZ+A COS. YOZ +A” COS. ZOZ

(1)

(2)

(3)

On squaring both members of each of these equations, and adding the results together, it will be found that the sum of the coefficients of A'2 is the sum of the squares of the cosines of the angles which ox makes with the three rectangular axes ox', or', oz'; therefore that sum will be equivalent to unity. In like manner it will be found that the sums of the coefficients of A2 and A2 are the sums of the squares of the cosines of the angles which oY and oz make with the same axes; and therefore those sums will be severally equal to unity. Again, the sum of the coefficients of 2 A'A" is the sum of the products of the cosines of the angles which ox and OY make with ox', or', oz'; therefore, since ox and OY are at right angles to one another, the sum of those products (=cos. XOY) is zero. In like manner the sums of the coefficients of 2A'Á'" and 2A′′A"" are separately zero.

A‚2+A,,2+A,,,2=A'2+A”2+A'"'2.

And if the co-ordinate planes are so situated that

we should have

A2+A2=0,

A2=A'2+A”2+ A''' 2,

Hence

or the sum of the projections on z'oy' would be a maximum.

CHAP. IV.

TRANSFORMATION OF CO-ORDINATES.

PROPOSITION I.

171. To transform the co-ordinates of a point in

space from one system of axes to another, both systems being rectangular,

and having a common origin.

Let zox, ZoY, XOY be the original co-ordinate planes at right angles to one another, and let

z'ox', z'oy', x'or' be the new planes, at right angles to one another; also let the planes YOX, Y'ox', produced if necessary, intersect one another in ON.

Let м be any point in space; and let its co-ordinates with respect to ox, OY, OZ be x, y, z, while its co-ordinates with respect to ox', or', oz' are x', y', z'. Imagine O N' and ON" to be drawn in the planes XOY, X'oY', respectively perpendicular to ON, that is, in a plane passing through oz and oz'; and let the co-ordinates of M with respect to ON, ON', oz be x, y, z,; also, let the co-ordinates of M with respect to ON, ON", Oz' be x, y, z

119

Imagine o to be the centre of a sphere, and let the different co-ordinate planes be those of great circles; also, let the quadrantal arcs zx, ZY, &c. be drawn as in the figure. Let ox, OY, ON, ON' in the above figure be represented as in that which is annexed, and let m be the projection of м on XOY, also draw the perpendiculars to the co-ordiNnate axes as they are here represented. Then

b

с

d

m

Y

oe (= oa cos. NOx) = x, cos. Nox, ab or ec am sin. amb=y, sin. NOX;

therefore

Also

therefore

and

x=x, cos. Nox+y, sin. NOX (=0c).

bm (
(= am cos.amb) = y, cos. NOX
ae or bc (oa sin. NOX):
sin. NOX;

= x,

y=y, cos. NOX-x, sin. NOX (= cm or od)

z=z, (=Mm) in the above figure.

Next, let ox', OY', ON, ON" in the first figure be represented as in that which is annexed, and let m' be the projection of M on X'OY'; also draw the perpendiculars to the coordinate axes as they are here represented. Then

N

k

a'

Also

therefore

and

N" ok (=oc' cos. NOX')=x' cos. NOX',
c'v or ka'(c'm' sin. c'm'v)=y' sin. NOX';
therefore

x=x' cos. NOX'—y' sin. NOX' (=0a')

ow (=ot cos. Y'ON")=y' cos. NOX',
m's (= m't sin. m'ts)=x' sin. NOX';

y=x' sin. Nox'+y' cos. NOX' (=0g)
z,,=z' (=Mm') in the first figure.

Lastly, let oz', ON', ON" be represented as in the annexed figure, and draw the perpendiculars to the co-ordinate axes; also, let m" be the projection of м on the plane Z'ON'. Then

of(=og cos. N'ON")=y,, cos. N'ON", ghor qf(=m'g sin. gm" k) =z,, sin. N'ON"; therefore

Also

k

y=y, cos. N'ON"-z,, sin. N'ON" (=0q′).

m'h (=m" g cos. gm''h)=z,, cos. N ́ON", gf or hq (og sin. N'ON") =y,, sin. N'ON"; therefore z,z,, cos. N'ON"'+y,, sin. N'ON" (=0k), and

172. In the above values of x, y, z,, substituting the equivalents of x, y, z,, from the next preceding equations, and in the values of x, y, z substituting the resulting values of x,, y, z,, there will be obtained, on representing the angle

N'ON" by , NOx by , and Nox' by 4, the following equations:

x=x' cos. cos. -y' sin. cos. + (x' sin. + y' cos. ) cos. sin. -z' sin. sin. 4,

173. Multiplying both members of each of these equations by the sum of the coefficients of x' in its second member, and adding the results together, the sum of the second members will become x', and we shall have

x'=x (cos. cos. 4+ sin. cos. sin. 4)+(sin. cos. cos. ¥—cos. p sin. ¥) +z' sin. sin. 0.

Multiplying both members of each of the equations for x, y, z by the sum of the coefficients of y' in its second member, and adding the results, we get

y' = x (cos. cos. @ sin. —sin. & cos. ¥) +y (cos. cos. cos. + sin. & sin. 4) +z eos. sin. 0.

In like manner, multiplying both members of each of the equations for x, y, z by the sum of the coefficients of ' in its second member, and adding the results, we have

z=-x sin. sin. -y sin.

cos.+z cos..

174. When 0=0, or the plane x'oY' coincides with XOY, the values of x, y, z, and the corresponding values of x', y', z', will be more simple, since then sin. 0=0, and cos. =1. They will also be more simple if oN be supposed to coincide with ox, in which case NOX=0; whence sin. =0, and cos. =1. If both ON and ox' be supposed to coincide with ox, we shall have =0, and 40. In this last case

x=x',

y=y' cos. -z' sin. 0,

z=y' sin. ' cos. e.

175. In any system of rectangular co-ordinate planes, as ZOX, ZOY, XOY, if OM be represented by r, the angle Mom by μ, and the angle xom by, we shall have

which values may be substituted in any equation relating to

L

x, y, z in rectangular co-ordinates, in order to convert it into an equation with polar co-ordinates.

176. If the system ox', oy', oz' were to move parallel to itself, so that the point of intersection is no longer coincident with the intersection o of the other system; representing this last intersection still by o, and the point of intersection in the translated system by A, the co-ordinates of a with respect to ox, OY, oz being designated a, b, c, the values of x, y, z would differ from those given in Arts. 172. 174. 175. only in having a added to the equivalent of x, b to that of y, and c to that of z.