0 to a; hence the branch APB. Again, as increases from 45° to 90°, sin. 20 diminishes from 1 to 0; .. r diminishes, and we trace the branch BQA. As increases from 90° to 180°, sin. 20 increases and decreases as before; hence the similar oval in the second quadrant. By following from 180° to 360°, we shall have the ovals in the third and fourth quadrant: and since the sine of an arc advances similarly in each quadrant of the circle, we have the four ovals similar and equal.

In this case we have paid no regard to the algebraical sign of r; we have considered 9 to vary from 0 to 360°, which method we prefer to that of giving all values from 0 to 180°, and then making the sign of r to vary. If the equation had been (x2+ y2)2 = 2ary, we should have found two equal and similar ovals in the first and third quadrant. The locus of the equation r = a (cos.

sin. ) is the same kind of figure differently situated with respect to the lines A X and A C.

The equation to the lemniscata a cos. 20 art. (314), may be similarly traced.

373. In many indeterminate problems we shall find that polar co-crdinates may be very usefully employed. For example,

Let the corner of the page of a book be turned over into the position B C P, and in such a manner that the triangle B C P be constant, to find the locus of P.

Let A Pr, angle PAC 0, and let the area A B C a2; then since the triangles A BE,

PBE are equal, we have A E

B

=

2'

and the

E

or a sin. 20. Hence the locus is an oval APBQ as in the last figure.

If a point be taken in the radius vector S P of a parabola so that its distance from the focus is equal to the perpendicular from the focus on the tangent, the locus of the point is the curve whose equation is r = a sec.

2.

197

PART II.

APPLICATION OF ALGEBRA TO SOLID GEOMETRY.

CHAPTER I.

INTRODUCTION.

374. In the preceding part of this Treatise lines and points have always been considered as situated in one plane, and have been referred to two lines called axes situated in that plane. Now we may readily imagine a curve line, the parts of which are not situated in one plane; also, if we consider a surface, as that of a sphere, for example, we observe immediately that all the points in such a surface cannot be in the same plane; hence the method of considering figures which has been hitherto adopted cannot be applied to such cases, and therefore we must have recourse to some more general method for investigating the properties of figures. 375. We begin by showing how the position of a point in space may

be determined.

Let three planes ZAX, ZAY, and X A Y, be drawn perpendicular to each other, and let the three straight lines A X, A Y, AZ be the intersections of these planes, and A the common point of concourse.

From any point P in space draw the lines PQ, PR, and PS respectively perpendicular to the planes X A Y, Z AX, and ZAY; then the position of the point P is completely determined when these three perpendicular lines are known.

Complete the rectangular parallelopiped A P, then P Q, P R, and PS are respectively equal to A O, A N, and A M.

These three lines A M, A Ñ, and A O, or more commonly their equals A M, M Q, and Q P, are called the co-ordinates of P, and are denoted by the letters x, y, and z respectively.

The point A is called the origin.

The line A X is called the axis of r, the line AY is called the axis of y, and the line A Z is called the axis of z.

The plane X A Y is called the plane of ry, the plane Z A X is called the plane of z x, and the plane Z A Y is called the plane of y*.

From P we have drawn three perpendicular lines, PQ, PR, and PS, on the three co-ordinate planes. The three points, Q, R, and S are called the projections of the point P on the planes of xy, xz, and zy respectively.

The method of projections is so useful in the investigation and description of surfaces, that we proceed to give a few of the principal theorems on the subject so far as may be required in this work.

PROJECTIONS.

376. If several points be situated in a straight line, their projections on any one of the co-ordinate planes are also in a straight line.

For they are all comprised in the plane passing through the given straight line, and drawn perpendicular to the co-ordinate plane; and as the intersection of any two planes is a straight line, the projections of the points must be all in one straight line.

This plane, which contains all the perpendiculars drawn from different points of the straight line, is called the projecting plane; and its intersection with the co-ordinate plane is called the projection of the straight line.

377. To find the length of the projection of a straight line upon a plane.

Let A B be the line to be projected on the plane PQR; produce A B to meet this plane in P; draw A A' and B B' perpendicular to the plane, and meeting it in A' and B'. Join A' B'; then A' B' is the projection of A B.

Since A B and A'B' are in the same plane, they will meet in P. Let the angle B PB' or the angle of the inclination of A B to the plane = 0, and in the projecting plane A B' draw A E parallel to A' B', then

A'B'AE AB cos. BAE AB cos.

=

The same proof will apply to the projection of a straight line upon another straight line, both being in the same plane.

378. To find the length of the projection of a straight line upon another straight line not in the same plane.

Let A B be the line to be projected; CD the line upon which it is to be projected. From A and B draw lines A A' and B B' perpendicular to CD, then A' B' is the projection of A B.

Through A and B draw planes MN and PQ perpendicular to C D. These planes contain the perpendicular lines A A' and B B'.

From A draw A E perpendicular to the plane PQ, and therefore equal and parallel to A'B'; join BE; then the triangle ABE having a right angle at E, we have A' B' = AE = A B cos. BA E, and angle BAE is equal to the angle of inclination between A B and CD; hence

A'B' AB cos. 0.

Also any line equal and parallel to A B has an equal projection A' B' on CD, and the projection of A B on any line parallel to C D is of the same length as A'B'.

379. The projection of the diagonal of a parallelogram on any straight line is equal to the sum of the projections of the two sides upon the same straight line.

Let A B C D be a parallelogram, AZ any straight line through A inclined to the plane of the parallelogram. From C and B draw perpendiculars C E and B F upon A Z, then A E is the projection of A C upon AZ or AE = A C cos. C AZ; and AF is the projection of A B upon

The line A X is called the axis of r, the line A Y is ca y, and the line A Z is called the axis of z.

The plane X AY is called the plane of ry, the plane Z plane of z x, and the plane Z AY is called the plane of

From P we have drawn three perpendicular lines, P on the three co-ordinate planes. The three points, Q, R the projections of the point P on the planes of xy, x tively.

The method of projections is so useful in the investi tion of surfaces, that we proceed to give a few of the on the subject so far as may be required in this work.

PROJECTIONS.

376. If several points be situated in a straight lin any one of the co-ordinate planes are also in a straig For they are all comprised in the plane pass straight line, and drawn perpendicular to the cothe intersection of any two planes is a straight lin points must be all in one straight line.

This plane, which contains all the perpendicu points of the straight line, is called the projecting tion with the co-ordinate plane is called the I line.

377. To find the length of the projection plane.

Α'

Let A B be the line to be projected e A B to meet this plane in P'; draw A A' plane, and meeting it in A' and B'. Jo jection of A B.

*This system of co-ordinate planes may be represented by the sides and floor of a room, the corner being the origin of the axes, the plane X Y is then represented by the floor of the room, and the two remaining planes by the two adjacent sides of the room.

R