pass through the origin of the co-ordinates, be represented on Zox, ZOY (Fig. to Art. 142. a.) by

x=az, y=bz;

and if it do not pass through the origin of the co-ordinates, by the equations

145. To find the polar equations of a line in space, the co-ordinate axes being at right angles to one another.

Let the line, as OP (Fig. to Art. 144.), pass through the origin of rectangular co-ordinates, and let it be represented by r; then, the angles which OP makes with the co-ordinate axes being represented as before, we have for the polar equations,

z=r cos, y.

x=r cos. u, y=r cos. ß, But the equations of the projected lines om', on′ being represented by x=az, y=bz (in which a and b are supposed to denote, respectively, tan. zom' and tan. zon'), on substituting these values of x and y in the equation

r2 (=OP2)=x2+ y2+z2,

the latter becomes r2=(a2+b2+1) z2 ;

This value of z being substituted in the same equations for x and y, we obtain

Again, if be the angle which op, the projection of OP, makes with ox, we shall have, since r sin. y=op,

a. If the co-ordinate planes form oblique angles with one another, the polar equations for the line OP may be found thus:

Let op (Fig. to Art. 142. a.) be represented by r as before, and let the angles which it makes with the planes ZOY, ZOX, and XOY be represented by a', ß', and y'; also let the inclinations of the lines Ox, OY, oz to the planes ZOY, ZOX, xoy be denoted by 0, ', and 0”.

Imagine Pp' to be let fall perpendicularly on the plane ZOY; then (Trigon. Art. 30. 7.)

r sin. a'=Pp';

also, if a line be let fall from м perpendicularly on ZoY, such line will be equal to Pp'; hence

146. To find the equations for a line which is to pass through two given points in space, the co-ordinate axes making any angle with one another.

Let the co-ordinates of the two points be a', y', z′, and x", y", z"; it is evident that, with respect to these points, the equations (a) Art. 144. for the projected lines will be

and

x'=az'+h, y=bz'+k,

x"=az"+h, y′′=bz"+k;

whence x'' — x' = a (z'' — z′), or a=

and

y'—y'=b (z′′ —z′), or b =

But subtracting the equations for x' and y' from the equa

tions (a) Art. 144. we have

x-x'=a (z-z'), and y—y'=b (z—z'):

substituting in these the above values of a and b, the required equations are

in which x, y, z represent the co-ordinates of any point whatever in the line.

PROPOSITION V.

147. To find the co-ordinates of a point in which two lines whose equations are given intersect one another.

Let the given equations be

x=az +h, y =bz + k

x' = a'z' + h', y' =b'z' + k'.

At the point of intersection x=x', y=y' and z=z';

substituting the first of these values of z in the equation for , we have

h'-h

+h = α- ·a'

ah'-a'h

and the other in the equation for y, we have

These values of x, y, and z are the co-ordinates of the point of intersection. If a=a' and bb' the values of x, y, and z are infinite; which proves that the lines are then parallel to one another.

PROPOSITION VI.

148. To find the equations for a straight line passing through a given point, and making given angles with three rectangular co-ordinate axes.

Imagine a line, as OP (Fig. to Art. 144.) passing, through the origin, and making with the co-ordinate axes angles equal to those which the given line makes with the same axes, this line will be parallel to the given line.

Let the angles made with ox, or, oz be a, ß, y; then x', y', z' being the co-ordinates of the given point, the equations of the required line, on zox, zoy, will be as in the equations (a) Art. 144., and substituting for a and b their trigonometrical equivalents,

Therefore x, y, z being the co-ordinates of any other point in the line, the required equations for the projections on zox, ZOY, by the substitution of these values of h and k in the equations

149. COR. If it were required to find the equations for a straight line passing through a given point and parallel to a given line:

Through the origin of the co-ordinates a line may be imagined to pass parallel to the given line, consequently making equal angles with the co-ordinate axes; then, the co-ordinates of the given point being x', y', z', the required equations for the line will be, on zox and zoy, the same as those in the proposition.

PROPOSITION VII.

150. To find the angle contained between two lines which intersect one another in space, in terms of the angles which they make with the co-ordinate

axes.

If the intersection should not take place at the origin of the co-ordinates, conceive two lines to be drawn through that point parallel to the given lines; the angle contained between these will be equal to that which is contained between the original lines.

P

Let OA, OP be the two lines drawn from o, and let their lengths be represented by r and r'. Let the rectilinear co-ordinates of A be x, y, z, and those of P be x', y', z'; again, let the inclinations of the lines OA and OP to the axes OX, OY, Oz be represented by a, ß, y, and a', B', y'; and let the required angle be represented by ◊.

Then (Pl. Trig. Art. 57. b.), A, P being joined.

AP22+2-2 rr' cos. ;

also (Art. 142.) A p2 = (x' — x)2+(y'—y)2 + (z' — z)2; expanding the second member of the last equation, we have

2

A P2 = x'2+y'2 + z22 + x2 + y2 + z2 − 2 (x'x + y' y + z' z),

therefore, substituting in the last equation,

cos. cos.a cos. a'+cos. ß cos. B'+cos. y cos. y'. When the two lines are parallel to one another, =0 and cos. 1; therefore the equation becomes

1=cos. a cos. a'+cos. ẞ cos. B' + cos. y cos. y'.

When the two lines are at right angles to one another, 6=90° and cos. =0; therefore

0=cos. a cos. a'+cos. ß cos. B'+cos. y cos. y'.

151. COR. If it were required to express the angle contained between two lines, as OA, OP, in terms of the angles which their projections on two of the three co-ordinate planes make with the line in which those planes intersect one another, let x, y, z, and x', y', z' be co-ordinates of A and P, or of any two points, one in each line. Then the equations for the two lines being,

in which a, a, b, b' (the systems of co-ordinate axes being rectangular) are the tangents of the angles which the projections of the given lines on zox, zoy make with oz.