Art. Page 143 143 297. Perpendiculars are drawn from a point P to two given lines, and the distance between the feet of the perpendiculars is constant, to find the locus of P. 298. A given line moves between two given lines, to find the locus of a given point in the moving line 299. To find a point P without a given line, such that the lines drawn from P to the extremities of the given line shall make one angle double of the other 144 300. Four problems producing loci of the second order, not worked 301. From the extremities of the axis major of an ellipse, lines are drawn to the ends of an ordinate, to find the locus of their intersection 302. To find the locus of the centres of all the circles drawn tangential to a given line, and passing through a given point 305. To trace the locus of the equation y = ±(6-x) √√√.. 306. The Witch of Agnesi, y=±2a 308. To trace the locus of the equation a y2=x3+mx2 + n x+p. The semi 311. To trace the locus of the equation x y2+ a2y=nx+p 312, 3. The conchoid of Nicomedes, 314. The Lemniscata, (x2 + y2)2 = a2 (x2 — y2); r2 = a2 cos. 20 315. Another Lemniscata 316. To trace the locus of the equation y2-bxx√ b2 — x2 317. To find a point P, such that the rectangle of the distances from P to two given points shall be constant 318. To trace the locus of the equation y + x2 y2+2y3+x3=0 by the introduction of a third variable u 319. To trace the locus of the equation y5 — 5 a x2 y2+x3=0 324. Example of a conic section passing through four given points . CHAPTER XIII. ON THE INTERSECTION OF ALGEBRAIC CURVES. Art. Page 331. There may be n intersections between a straight line and a line of the nth order 175 332. There may be m n intersections between two lines of the mth and яth orders; exceptions 333. Method of drawing a curve to pass through the points of intersection, and thereby to avoid elimination 334. Example. From a given point without an ellipse, to draw a tangent to it. Generally to any conic section 336-7. To draw a normal to a parabola from any point 340. To construct the roots of the equation +83 + 23 x2 + 3 2 x + 16 = 0 344. To find any number of mean proportionals between two given lines CHAPTER XIV. TRANSCENDENTAL CURVES. 348, 9. Definition of Transcendental curves; Mechanical curves 350. The Logarithmic curve, y = u* 351. The Catenary, y=}( e* +e=") 352. Trace the locus of the equation y=a*. 353. Trace the locus of the equation y=x. the letter B should be placed on the axis A Y, where the curve to this curve ; cuts that axis 354. The curve of sines, y sin. x . 355. The locus of the equation y=x tan. x. The figure belongs to Art. 352. The correct figure is given in the Errata + √2a x − x2 a . 185 Art. 360. The Epitrochoid, which becomes the Epicycloid when m = 1 Page 363. The involute of the circle; the figure is not correct. See Errata. 368, 9. Spirals approaching to Asymptotic circles, (~ — b) 0 = a ; 0 √ a r — r2=b. 194 370. Spirals formed by twisting a curve round a circle :a 371. The Logarithmic Spiral, ra 194 . 195 195 . 196 PART II. APPLICATION OF ALGEBRA TO SOLID GEOMETRY. CHAPTER I. INTRODUCTION. Art. Page 197 197 If A B be 198 374. The system of co-ordinates in one plane not sufficient for surfaces 198 200 CHAPTER II. THE POINT AND STRAIGHT LINE. 381. The equations to a point, x = a, y = b, z = c; or (x − a)2 + (y − b)2 + (≈ − c)2 = 0 382, 3. The algebraical signs of the co-ordinates determined. Equations corre sponding to various positions of points 384. Two of the projections of a point being given, the third is known 385. To find the distance of a point from the origin, 386, 7. If a, ß, y, be the three angles which a straight line through the origin makes with the co-ordinate axis, 389. The equations to the straight line, x = a z + a, y = ßz + b, y 392-5. Equations to the line corresponding to various values of a, ß, a, b. 396. To find the point where a straight line meets the co-ordinate planes Art. 398. The equations to a line through a given point (x1 Y1 Z1), 399. The equations to a line through two given points (x1 Y1 Z1) (X2 Y2 Z2), Page 205 400. The equations to a line parallel to x = az + a, y = ßz+ b, are x = az + a', y = ßz + b'. 401. If two straight lines intersect, the relation among the coefficients is - α), (al- a) (B' B) = (bl - b) (al 402. The angles which a straight line makes with the co-ordinate axes, 207 207 (cos. lx)2 + (cos. l y)2 + (cos. 1 z)2 = 1. 403. The cosine of the angle between two straight lines, aa + B B+ 1 2 √ (1+ a2+ B2) √ (1 + a2 2 + ßl 2 cos. lx cos. lx + cos. ly cos. l'y + cos. Iz cos. l'z. 405. If the lines are perpendicular to each other, 207 407. To find the equation to a straight line passing through a given point (x1 y1 ≈1), and meeting a given line at right angles CHAPTER III. THE PLANE. 408. The equation to a plane, xX1+yyı + z z1 = d2, or mx + ny + pz = 1, or x cos. dx + y cos. d y + z cos. d z = d, or x sin. Px + y sin. Py + z sin. P z = =d, or a 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, 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. 212 417. The equation to a plane parallel to a given plane, m x + ny + p z = 1, is mx+ny+pz= =2, or m (x-x1)+n (y—yı)+p (2−zı)=0. |