Syllabus -^e in ) T netr t y of ree DJ Ions bv Dr. Gift of and Mrs. A.B. Pier SYLLABUS RSE IN ANALYTICAL GEOMETRY BOSTON: PUBLISHED BY GINN, HEATH )PTBIGHTED BV GlKST, HEATH MATHEMATICS. THE following volumes in Wentvvorth's Mathematical Series arc now ready for delivery : Elements of Algebra . 9| ^B ^1 |H $1.12 Complete Algebra ... 1.40 Plane Geometry ^Hj ^B ^H II ^1 '^ Plane and Solid Geometry . ^H 1-25 Plane and Solid Geometry, and Plane Trigonometry. . 1.40 Plane Trigonometry. Paper Plane Trigonometry and Tables. Paper . ^^1 *^ Plane and Spherical Trigonometry JJHJ ^| Plane and Spherical Trigonometry, Surveying, and Navk Plane and Spherical Trigonometry, and Surveying. Tables ^B HBH ^B B ^H 1 25 Surveying. . P:j_pr- Trigonometric Formulas (Two Charts, each 30 X 40 i:; 1.00 Weutworth & Hill's Five-Place Logarithmic and Trl nometric Tables. (Seven Tai.i. H^ffilM[ Wentworth & Hill's Five-Place Logarithmic and Tri: nometric Tables. Complete Edition . 1.00 Wentworth & Hill's Practical Arithmetic . BK| 1-00 Wentworth & Hill's Examination Manual. I. Arithmetic Wentworth & Hill's Examination Manual. II. Algebra . Wentworth & Hill's Exercise Manual. II. Algebra (The last two may 'be had in one vein ANNOUNCEMENTS. EXEBCISE MANUAL OF AEITHMETI iCISE MANUAL OF GEOMETRY. McLEtLAN's UXIVEIiSF! BRA. WENTWORTII'S GRAMMAR SCHOOL ARJTJIMET. YOUTH'S PRIMARY SCHOOL ARTTTLM1 GINN, HEATH, & CO., Publishers. BOSTON, N SYLLABUS OF A COURSE IN ANALYTIC GEOMETRY OF THREE DIMENSIONS. 1. Explain the method of denoting the position of a point in space by Cartesian Coordinates. State the convention concerning the signs of coordinates. v >\ 2. State the convention concerning positive rotation about r^ any axis. 3. Explain Polar Coordinates in space. Obtain formulas for transforming from rectangular to polar K* coordinates. [1] 2/ = rsin< cos0, z = r sin sin 0. 4. PROBLEM. To find the distance between two points whose coordinates are given. ^ [2] D = \i 5. PROBLEM. To divide a line in any given ratio, m-i : m 2 . f31 x = , v = ni a + nil If the line is bisected, 4G2G19 6. Define the projection of a point on a plane ; of a line on a plane ; of a point on a line ; of a line on a line. Prove that the projection of a line I on a line p is [5] l p = I cos a, where a is the angle between the two lines. 7. Show that the rectangular coordinates of any point are the projections of the radius vector of the point on the three axes ; so that if a, (3, and y are the angles made by r with the axes of X, T", and Z respectively, we have [6] # = rcosa, y = rcos{3, z = and [7] r 2 = a, /?, and 7 are called the Direction Angles of r. 8. Define the Direction Cosines of a line, and prove that the sum of their squares is unity. [8] COS 2 a + COS 2 ft + COS 2 y = 1 . The radius vector of a point and its direction angles may be used as a set of polar coordinates. 9. PROBLEM. To find the angle between two lines when their direction cosines are given. [9] COS 6 = COS a a COS a 2 -}- COS /?! COS @ 2 + COS yj COS y 2 . The lines are perpendicular if [10] = COStt! COSa 2 + COS/?! COS/? 2 + COSyi COSy 2 . They are parallel if [11] a 1 = a 2 , / 8 1 = /? 2 , 71 = 72- TRANSFORMATION OF COORDINATES. 10. PROBLEM. To transform to a new set of axes parallel to the old. [12] x = 11. To transform from one set of axes to a second set hav- ing the same origin. x = x' cos aj + y' cos a 2 + z' cos 03, [13] y = Z = Show that the degree of an equation cannot be altered by either of these transformations. 12. Explain what is meant in space by the locus of an equa- tion or pair of equations. Show that a single equation between cc, y, and z represents a surface. That a pair of such equations represent a line. Show how the form of a surface whose equation is given may be investigated by means of its plane sections. An equation containing only two variables represents a cylin- drical surface. Show how to obtain the equation of a surface formed by re- volving a plane curve about one of the axes. THE PLANE. 13. PROBLEM. To find the equation of a plane in terms of the perpendicular from the origin and its direction cosines. [14] aJCOSa + 2/COS/J + ZCOSy = p. 4 14. Prove that every equation of the first degree, represents a plane, and show how to reduce it to the form [14]. 15. Express the equation of a plane in terms of its intercepts on the axes. 16. PROBLEM. To find the equation of a plane through three given points. 17. PROBLEM. To find the distance from a given point to a given plane. [16] D = XiCOSa + y l COS(3 + Z 1 COSy p. 18. PROBLEM. To find the angle between two planes. [17] cos 6 = A 1 A 2 -{-B 1 B 2 + O 1 They are perpendicular if [18] A,A 2 + B,B 2 + 0, C 2 = 0. They are parallel if [19"| S fS s Jr3. U3.2 X>2 ^2 19. PROBLEM. To find the equation of a plane passing through a given point and parallel to a given plane ; passing through two given points and perpendicular to a given plane. THE STRAIGHT LINE. 20. Show that the equations of a line may always be thrown into the form _,+, y = nz + b. 21. Find the equations of a line in terms of its direction cosines, and the coordinates of a point through which it passes. [21] COS a COS/3 COSy 22. PROBLEM. To throw the equations of any line into the form [21] x a y b zc [22] m n I Vl + m 2 + n 2 Vl + m 2 + n 2 Vl + m 2 -f- w 2 23. PROBLEM. To find the equations of a line passing through two given points. as Xi z Z [23] 2/2 - 24. Problems concerning the angles between lines, or between lines and planes, can be readily solved by the use of the direc- tion cosines of the lines and those of the normals to the planes. THE SPHERE. 25. PROBLEM. To find the equation of a sphere in terms of the coordinates of its centre and the length of its radius. [24] ( x -o) t + (y-b) a + (z-c) a =i a . When the centre is at the origin, this becomes [25] 26. Show that the most general form of the equation of a sphere is [26] tf + yt + zt + Gx + Hy+lz + K^Q, and show how to reduce any equation of this form to the form [24], and thus to determine its centre and radius. 27. PROBLEM. To find the equation of a sphere passing through four given points. 28. Show that any two spheres intersect in a circle. 29. Find the equation of the tangent plane at a given point on the surface of the sphere, x 2 + y 2 + 2 = 1 s - [27] 30. PROBLEM. To find the equations of the normal at any point of the sphere. [28] * = */ = *. x, ^ Zi Prove that every normal is a radius. 31. PROBLEM. To find the locus of points dividing harmoni- cally secants drawn from a given point to a sphere. [29] Xt x + y l y + z l z = r i . This is called the polar plane of the given point, and passes through the points of contact of all the tangents that can be drawn from the given point to the sphere. 32. Prove that if several points lie in a plane, their polar planes pass through the pole of the given plane ; and con- versely, that if several planes pass through a point, their poles lie on the polar plane of that point. 33. Prove that the polar plane of a point is perpendicular to the line joining the point with the centre of the sphere, and that the product of the distance of the pole from the centre and the distance of the polar plane from the centre is equal to the square of the radius. 34. PROBLEM. To find the locus of the middle points of a set of parallel chords. [30] X COSa + y COS/3 + 2! COSy = 0. Such a locus is a diametral plane. Define diameter; conjugate diameters. THE CENTRAL QUADRICS. 35. The central quadrics are the ellipsoid^ the bi-parted Jiyper- boloid, the un-parted hyperboloidj and the cone. q& qj2 g2 Investigate their forms. 36. Find the equation of the tangent plane to a central quadric. T321 ^i*^ . y\y . ^i_2 i a? b 2 ~c^ Of the normal line. [33] (x #,) = (y V,) = (z z,}. "- J /v. > *' ni ^" **/ ~ \ ~1/' 4G2619 8 37. Find the equation of the polar plane of a point with respect to a central quadric,. and prove that sections 31 and 32 apply to any central quadric as well as to the sphere. 38. Find the equation of the diametral plane conjugate to a given chord of a central quadric. ,-Q^-, iccosa , ycosB , zcosy n L"J 5 I Ts! "I "2 u * a 2 W c 2 39. The diametral plane conjugate to the diameter through r o /? T \ v i y i /\ L J ~w + '~w + ~^ ;= ^' woo 40. Show that when two diameters are conjugate, their direc- tion cosines are connected by the relation pqr--i COSctj COSa 2 , COS/?! COS (3 2 . COSyi COSy 2 A L" J " o To "1 o 41. Show that the coordinates of any point of a central quadric can be expressed in the form [38] # where X, /*., and v are the direction angles of an auxiliary line. 42. Show that if two diameters are conjugate, the auxiliary lines corresponding to their extremities are mutually perpendic- ular. 43. Show that the sum of the squares of three conjugate diameters is constant. 44. Prove that the paYallelopiped whose edges are three con- jugate diameters has a constant volume. CIRCULAR SECTIONS. 45. Prove that through the centre of every central quadric two planes can be drawn, each of which will cut the quadric in a circle. 46. Show that every plane section of a quadric parallel to a circular section is a circle. 47. Define the umbilics of an ellipsoid, and show how to find them. 48. Show that the circular sections of an hyperboloid and of its asymptotic cone are the same. RULED SURFACES. 49. Show that on the un-parted hyperboloid two sets of right lines can be drawn, lying wholly in the surface of the hyperbo- loid ; and that through every point of the hyperboloid one line of each set will pass. 50. Prove that the two lines passing through a given point of the hyperboloid lie in the tangent plane drawn at the point in question. 51. Show that each line of one system meets all the lines of the other system, and none of the lines of its own system. 10 52. Prove that an un-parted hyperboloid may be generated by the motion of a line which always touches three given lines, no two of which are in the same plane. 53. Show that if a line revolve about another line not in the same plane, it will generate a ruled hyperboloid. 54. Investigate the properties of the ruled paraboloid. W. E. BYERLY, Professor of Mathematics in Harvard University. J. S. GUSHING & Co., PRINTERS, BOSTON. Books on English Literature. Aruo: Church Hurrisou & Hudson & Lamb Hunt ..... Lambert ... 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