Art. Page 398. The equations to a line through a given point (x1 y1 Z1), x − x1 = ∞ (z — Z1), Y — Y1 = ß (≈ — ≈1). 399. The equations to a line through two given points (xı yı Zı) (X2 Y2 Z2), 205 400. The equations to a line parallel to xaza, y = ßz + b, are 401. If two straight lines intersect, the relation among the coefficients is 402. The angles which a straight line makes with the co-ordinate axes, aa + B B+ 1 2 √ (1+ α2 + B2) √ (1 + a2 2 + B' 2 405. If the lines are perpendicular to each other, 407. To find the equation to a straight line passing through a given point (x1 y1 21), and meeting a given line at right angles 209 209 or x cos. P, y z + y cos. P, x z + z cos. P, xy= d. 413. The angles which a plane makes with the co-ordinate planes, 210 415. Equations to planes parallel to the co-ordinate planes 416. The traces of a plane are found by putting x, y, or z = 0. 417. The equation to a plane parallel to a given plane, mx + ny + p z = 1, is mx+ny+pz= = or m (x−x1)+n (y—yı)+p (î—zı)=0. 420. If a plane and straight line coincide, the conditions are, manb=1, ma + nß + p = 0. 421. To find the equation to a plane coinciding with two given lines 422. To find the equation to a plane coinciding with one line, and parallel to . 215 215 . 215 423-5 To find the intersection of two planes,-three planes,—four planes 426. The relation among the coefficients of a straight line and perpendicular plane m n 216 428. The equation to a plane passing through a given point, and perpendicular to a given line, 429. The equations to a line through a given point, and perpendicular to a given plane 430. The length of the perpendicular from a given point on a given plane, d= mxi+nyi + p21 1 √ m2 + n2 + på 431. To find the distance of a point from a straight line. If the point be the origin, cos. P, yz cos. P', y z + cos. P, xz cos. P', x z + cos. P, x y cos. P', 435. If the planes are perpendicular, the relation among the coefficients is, mm1 + n n + pp1 = 0. 437. The angle between a straight line and plane, xy. . 218 219 219 CHAPTER IV. THE POINT, STRAIGHT LINE, AND PLANE, REFERRED TO OBLIque axes. 438. The equations to the point remain the same 439. The distance of a point from the origin, 220 & d2 = x2+y2+z2+2xy cos. XY+2 x z cos. XZ + 2y z cos. Y Z. 440. The distance between two points (x y z) (x1 y1 ≈1), d2 = (x − x1)2 + (y − y1)2 + (≈ — ≈1)2 + 2 (x − x1) (y — y1) cos. XY. + 2(x-x1) (≈—≈1) cos. XZ+2 (y—yı) (≈—≈1) cos. Y Z. 444. The relation among the coefficients of a straight line and perpendicular 445. The angle between a plane and straight line,-between two planes 222 CHAPTER V. THE TRANSFORMATION OF CO-ORDINATES. 417. To transform an equation referred to one origin to another, the axes remaining parallel • 223 448. To transform the equation referred to rectangular axes to another also referred to rectangular axes m m1+n n1+p p1=0) mm¿+n no+p p¿=0 3=m X+m1 Y+m12 Z) m2 + n2 + p2=1) y=n X+n1 Y + n ̧Z}, m12+ni2+p12=1}, z = p X+ p1 Y + P2 Z] m22+n,2+P22=1} m1m2+ning+pip2=0] 450. Hence three other systems for X Y Z in terms of x y z 451. The transformation from oblique to other oblique axes. 452. Another method of transformation from rectangular to rectangular axes, x = m a X + my a1 Y + m12 ay Z, 453. The transformation from rectangular to other rectangular axes effected in 454. Formulas of transformation to obtain the section of a surface made by a plane passing through the origin: 225 455. Formulas of transformation when the cutting plane is perpendicular to the CHAPTER VI. THE SPHERE AND SURFACES OF REVOLUTION. Art. Page 457, 8. The equation to a surface is of the form ƒ (x, y, z) = 0 459. Surfaces are divided into orders 228 228 465. The sections on the co-ordinate planes, or the traces, are circles 466. The tangent plane to a sphere, (x1 − a) (x − a) + (yı — b) (y — b) + (≈1 − c) (z − c) = r2, 470. The equation to the spheroid by revolution round the axis major, 471. The equation to the hyperboloid by revolution round the transverse axis, 472. General equation to the above surfaces of revolution round axis of z. x2 + y2 = ƒ (z) 473-6. Section of these surfaces by a plane CHAPTER VII. SURFACES OF THE SECOND ORDER. 477, 8. Reduction of the general equation of the second order to the central form. There is no centre when abc + 2 defa ƒ2 — be2 — c d2 = 0 479. Disappearance of the terms xy, xz, and y z. This transformation always 481. There is only one system of rectangular axes to which, if the equation be referred, it is of the form La+ My2 + N≈2 = 1 483. The central class gives three cases Art. Page 491. This surface has also a conical asymptotic surface 492. The general non-central equation of the second order can be deprived of the terms x y, xz, yz; and then of three other terms; so that its form becomes 240 498. The equations to the Elliptic and Hyperbolic Paraboloids may be obtained from those to the Ellipsoid and Hyperboloid 499. Consideration of the equation z2 = 1x+ly. Cylinder with Parabolic base 242 243 502. To find the surface generated by the motion of a straight line parallel to itself, and passing through a given straight line; a Plane 244 508. The equation to a cylinder, whose base is a parabola on x y 246 246 . 247 217 2 514. A cone with Elliptic base and axis coincident with axis of z 522. The axis of z one directrix, any straight line another, and the generatrix parallel to x y 250 . 250 251 |