equal to the angles between the planes. The theorem, having already been proved for the angles formed by the lines, must therefore be true of the equal angles formed by the planes. But the theorem may also be proved independently as follows: Pass a sphere around the origin as a centre, and let XY, YZ, ZX be the arcs of great circles in which the rectangular planes intersect its surface. X F Z D Y E Pass a plane through the centre parallel to the intersecting plane, and let FD be the great circle in which it intersects the spherical surface. Put Angle YDE = a; Angle XFE = B; Angle XED = y; Join XD by an arc of a the three angles which the in tersecting plane forms with the three rectangular planes. great circle. Then Because XY = XZ = 90°, X is the pole of YZ, and (because FAD = EAD + EAF = EAD + 90°). Taking the sum of the squares, cos' cos y = cos EDA =1 sin'EDA. Because EDA = ADB EDB 90° sin' EDA = cos' α. Hence cos' a + cos2 ß + cos y = 1. Q.E.D. (4) 129. Corollary. In the same way that we proved the corol lary of the last theorem we may show : If a plane intersects a system of three rectangular axes, forming with them the respective angles a, b, c, then sin' a+sin' b + sin' c = 1. EXERCISES. (5) 1. Having a system of three rectangular axes OX, OY, OZ, a line OP is drawn from O, making Angle XOP = 60°; Angle YOP= 50°. Find the angle ZOP and the angles which OP makes with the three planes XOY, YOZ, ZOX. 2. Supposing OP = 32.965 centimetres, and the angles to have the values in the preceding exercise, Ꮓ find the lengths of the perpendiculars dropped from P upon each of the axes OX, OY, and OZ, and the distances of the feet of these perpendiculars from the origin 0. 3. The same thing being supposed, what are the lengths of the respective perpendiculars dropped from P upon the three planes XOY, YOZ, and ZOX? 4. The same thing being supposed, what are the lengths of the projections of OP upon the three planes? P 130. Methods of defining the direction of a line in space. The direction of a line in a plane is defined by the angle which it makes with some fixed line in the plane. For example, if we have a known fixed line OX, and it is required that another line OP through O shall make an angle + 45° with OX, this completely fixes the direction of OP. That' is, there is only one line through O which makes an angle of +45° with OX, when we employ the method of measuring angles defined in Plane Trigonometry, Chap. I. 0 -X But if OP is not confined to one plane, its position is not fixed by the angle XOP, because any number of lines may be drawn through O, some above the plane of the paper and some below it, all making the same angle with OX. The student will see that these lines are all elements of a cone having O as its vertex and OX as its axis. Hence at least two conditions are necessary to define the direction of a line in space. These two quantities may be chosen in various ways, of which the following is the most common. We have (1) a plane of reference, the position of which we suppose fixed, and (2) in this plane we have a fixed line of reference OX. We call the plane of reference the fundamental plane. Let OZ be a line through O, per pendicular to the plane. Let OP be the line of which the direction is to be defined. From any point P of this line drop a perpendicular PQ upon the fundamental plane, and join OQ. X The direction of OP is then defined by the following two angles: (1) The angle POQ which OP forms with its projection OQ; that is, the angle between OP and the plane. (2) The angle XOQ which the projection of OP makes with OX. It will be remarked that the planes of these two angles are perpendicular to each other. To show that these two angles completely fix the direction of OP, we first remark that when the angle XOQ is given the line OQ is fixed. Next, because PQ is perpendicular to the plane, the point P and therefore the line OP must lie in the plane ZOQ, which is fixed because its two lines OZ and QQ are fixed. If the angle QOP in this (vertical) plane is given, there is only one line OP which can form this angle. Hence the direction of the line OP is completely determined by the two angles XOQ and QOP. The plane XOQ is, when used in this way, the fundamental plane. 131. Relation of the preceding system to latitude and longitude. To form another conception of these two angles, pass a sphere around as a centre, and mark on its surface the points and lines in which the lines and planes belonging to the preceding figure intersect it. Then: The fundamental plane OXQ intersects the spherical surface in the great circle MXQN. The line OX intersects it in X. The line OQ intersects it in Q. it in P and Z. Z We therefore have Angle XOQ = arc XQ; Angle QOP = arc QP. If now we imagine this sphere to be the earth, the great circle MN to be its equator, Z to be one of its poles, and P any point on its surface, then The arc QP or the angle QOP is the latitude of P. The arc XQ or angle XOQ is the longitude of P, counted from ZX as a prime meridian. Thus the angles we have been defining may be described under the familiar forms of longitude and latitude. 132. Position of a point. To fix the position of any point relative to a fundamental plane, we must select a point in that plane and a line ОX as a point and line of reference. If P be any point of which we wish to describe the position, we draw the line OP and form the same construction as in article 130. Then the position of P is fixed by the two angles XOQ and QOP, already described, and the distance OP. Hence three quantities are required to fix the position of a point in space. Def. The quantities which fix or describe the position of a point are called the co-ordinates of the point. The angles XOQ and QOP and the length OP are therefore co-ordinates of the point P, and are distinguished as polar coordinates. 133. Polar distance and longitude. In the preceding figure, because MXQN is a great circle, and Z its pole, we have p, the arc ZP, which is called the polar distance of P; P Thus we may define the direction of a line by polar distance and longitude, as well as by latitude and longitude. Applying the same system to positions on the earth's surface, their distance from the north pole of the earth is used instead of their latitude. Thus we should have, R 134. Rectangular co-ordinates. The rectangular co-ordinates of a point are its distances from three rectangular planes. In the figure the lines PQ, PR, and PS are the rectangular co-ordinates of P with respect to the axes OZ, OX, and OY respectively. In other words: The co-ordinate of a point rel W -X ative to each axis is the length of the line parallel to that axis from the point to the plane of the two other axes. |