Page:Encyclopædia Britannica, Ninth Edition, v. 10.djvu/429

Rh SOLID A1'ALYTICAL.] the radius vector 7' is given as an algebraical or exponential function of the inclination 0. Trilinear C'o01'clz'naI‘es. 24. Consider a fixed triangle ABC, and (regarding the sides as iiideliiiite lines) suppose for a moment that p, q, 1' denote the distances of a point P from the sides BC, CA, AB respcetivcly,—these distances being measured either perpendicularly to the several sides, or each of them in a given direction. To ﬁx the ideas each distance may be considered as positive for a point inside the triangle, and the sign is thus ﬁxed for any point whatever. There is then an identical relation between 22, (1, 7': if a, I), c are the lengths of the sides, and the distances are measured perpendiculai'ly thereto, the relation is ap+ bq+cr= twice the, area of triangle. But taking .1‘, 3/, z proportional to p, q, 7', or if we please proportional to given multiples of 2;, q, 1', then only the ratios of .4‘, 3/, z are determined 3 their abso- lute values remain arbitrary. But the ratios of 2), g, 1', and consequently also the ratios of 2', 3/, 2 determine, and that uniquely, the point; and it being understood that only the ratios are attended to, we say that (x, 1/, 2) are the coordinates of the point. The equation of a line has thus the form ax + by + c2 = 0, and generally that of a curve of the nth order is a homogeneous equation of this order between the coordinates, (* Qt, 3/, 2')": 0. The advantage over C-artesiaii coordinates is in the greater symmetry of the analytical forms, and in the more convenient treat- ment of the line inﬁnity and of points at inﬁnity. The method includes that of Cartesian coordinates, the lionio- gr-neous equation in .c', 3/, is in fact an equation in 7, I , which two quantities may be regarded as denoting Car- tesian coordinates; or, what is the same thing, we may in the equation write .2: It may be added that if the tri- linear coordinates (.c, g/, 2) are regarded as the Cartesian coordinates of a point of space, then the equation is that of a cone having the origin for its vertex 3 and conversely that such equation of a cone may be regarded as the equation in trilinear coordinates of a plane curve. (lcncral Poi/at (.-'00/'dz'nuIes. I.1'nc— Ooordi-21 ates. 25. All the coordinates considered thus far are poiiit- coordinates. More generally, any two quantities (or the ratios of three quantities) serving to determine the position of a point in the plane may be regarded as the coordinates of the point 3 or, if instead of a single point they determine a system of two or more points, then as the coordinates of the system of points. But, as noticed under CURVE, there are also 1iiie—coordiiiates serving to determine the position of a line 3 the ordinary case is when the line is determined by means of the ratios of three quantities 5,-:7, §(corre— lzitive to the trilinear coordinates .2‘, 3/, z). A linear equa- tiun uE+?ny+cZ=0 represents then the system of lines such that the coordinates of each of them satisfy this relation, in fact, all the lines which pass through a given point; and it is thus regarded as the line—equatioii of this point 3 and generally a homogeneous equation ( * §£,77,§)" = 0 represents the curve which is the envelope of all the lines the eoordinatcs of which satisfy this equation, and it is thus regarded as the liiie—eqiiatioii of this curve. II. SOLID ANALYTICAL GEOMETRY (§§26—40). '26. 'e ai'e here concerned with points in spacc,—tlie position of a point being determined by its three coordi- nates rr, 3/, “Ye consider three coordinate planes, at right angles to each other, dividing the whole of space into eight portions called octaiits, the coordinates of a point being the perpendicular distances of the point from GEOMETRY 415 the three planes respectively, each distance being considered 33 P0SitiVe 01‘ _I10gative according as it lies on the one or the other side of the plane. Thus the coordinates in the eight octants Z have respectively the signs I’ 37 5 3/ 7 '7 + + -t M ,,. + — + — + + — — + + + — + — — N ._ + _. _ _ _ 3/, Fig. 16. The positive parts of the axes are usually drawn as in ﬁg. 16, which represents a point P, the coordinates of which have the positive values OM, MN, X P. 27. It may be remarked, as regards the delineation of such solid ﬁgures, that if we have in space three lines at right angles to each other, say Oa, Ob, Oc, of equal lengths, then it is possible to project these by parallel lines upon a plane in such wise that the projections Oa’, Ob’, O0’ shall be at given inclinations to each other, and that these lengths shall be to each other in given ratios: in particular the two lines Oa’, Oc' may be at right angles to each other, and their lengths equal, the direction of Ob’, and its proportion to the two equal lengths Oct’, Oc' being arbitrary. It thus appears that we may as in the ﬁgure draw Ox, Oz at right angles to each other, and Oy in an arbitrary direction; and moreover represent the coordi- nates .1', 2 on equal scales, and the remaining coordinate 3/ on an arbitrary scale (which may be that of the other two coordinates 9.‘, 2, but is in practice usually smaller). The advantage, of course, is that a ﬁgure in one of the co- ordinate planes ws is represented in its proper form with- out distortion 3 but it may be in some cases preferable to employ the isometrical projection, wherein the three axes are represented by lines inclined to each other at angles of 120°, and the scales for the coordinates are equal (ﬁg. 1 7). For the delineation of a sur- face of a tolerably simple form, it is frequently suﬁicient to draw (according to the fore- going projection) the sections 0 by the coordinate planes 3 and in particular when the surface M is symmetrical in regard to -7 '9" the coordinate planes, it is _ _3‘7 sufficient to draw the quarter- F‘<‘=>'- 1" sections belonging to a single octant of the surface 3 thus ﬁg. 18 is a" convenient representation of an octant of the wave surface. Or a surface may be delineated by means of a series of parallel sections, or (taking these to be the sections by a series of horizontal planes) say by a series of contour lines. Of course, other sections may be drawn or indicated, if necessary. For the delineation of a. curve, a convenient method is to represent, as above, a series of the points P thereof, each point P being accompanied by the ordinates PN, which serves to refer the point to the plane of x_z/ 3 this is in effect a representa- tion of each point P of the curve, by means of two points P, N such that the line PN has a ﬁxed direction. Both as regards curves and surfaces, the employment of stereo- grapliic representations is very interesting. 07 4..