Analytic geometry for colleges, universities schools ebook legaltorrents

ll mmill AfTRONOMY ANALYTIC GEOMETRY FOR COLLEGES, UNIVERSITIES, AND TECHNICAL SCHOOLS BY E VV NICHOLS, PROFESSOR OF MATHEMATICS IN THE VIRGINIA MILITARY INSTITUTE LEACH, SHEWELL, & SANBORN, BOSTON NEW YORK CHICAGO COPYRIGHT, iwi, BY LEACH, SIIKWELL, & SANBOKN C J PETERS & SON, TYPOGRAPHERS AND ELECTROTYPERS PRESS OF BERWICK & SMITH Nsf PEEFACE THIS text-book is designed and Technical Schools to prepare a make it work for The aim for beginners, Colleges, of and the Universities, author has been at the same time to for the requirements of comprehensive For the methods of develop ment of the various principles he has drawn largely upon his sufficiently the usual undergraduate course experience in the class-room all home and authors, In the preparation of the work foreign, have been freely consulted In the first few chapters discussion of each principle whose works were available, elementary examples follow the In the subsequent chapters sets of examples appear at intervals throughout each chapter, and are so arranged as to partake both of the nature of a review and an extension of the At the preceding principles end of each chapter general examples, involving extended application of the are principles deduced, the benefit of those who may a more placed for desire a higher course in the subject The author takes pleasure cussion of Surfaces," Mathematics in appears as the He by in calling attention to a and Lee University, chapter in this work Washington final "Dis A L Nelson, M.A., Professor of which takes pleasure also in acknowledging his indebtedness i PREFACE v Prof C S Venable, LL.D., University of Prof William Cain, C.E., University of to and Virginia, to North Carolina, of Pennsylvania, to Prof E S Crawley, B.S., University for assistance rendered in reading and revising manuscript, and for valuable suggestions given E LEXINGTON, YA January, 1893 W NICHOLS SOLID ANALYTIC GEOMETRY 268 FIG C and Eepreseiiting the semi-axes by a, b, form the to reduced the equation of the surface may be of the surface L + -? If we suppose A=B ? then a c, = &, and the equation duces to the hyperboloid of revolution of two sheets re SURFACES OF THE SECOND ORDER If A=B= C, the equation becomes +f_^= _ x* 269 a* which represents the surface generated by the revolution of an equilateral hyperbola about its transverse axis the surface becomes a cone having an Finally, elliptical base, and the base becomes a circle when B We have, therefore, the following varieties of the if L = 0, A= hyper boloid of two sheets: The hyperboloid proper having three unequal The hyperboloid of revolution The equilateral hyperboloid of revolution The cone We will now examine the second 2 By + C* + G x = axes form, (B) Three cases apparently different present themselves for examination B and C positive (2) B, C, and G (3) B (1) G and negative in the first mem ber CASE positive C and G negative The equation may be written positive and B?/ in which B, + O = Gx G and are essentially positive C, Let the surface be intersected as usual by planes parallel respectively to the co-ordinate planes When x = , By +O = 2 Ga, an ellipse real when a and imaginary when a When y = b, = Gx > 0, < O B6 , a parabola with its axis its axis parallel to the axis of X When z = c, By = Gx - O a parabola with parallel to the axis of X The and principal sections are found by c When When origin making a = 0, b = 0, a b = 0, By + C,? = = C.s = Ga;, a 2 0, a point, the origin its vertex at the parabola with SOLID ANALYTIC GEOMETRY 270 When c, = = G#, a parabola with vertex at the origin 0, B?/ Since every positive value of x gives a real section, and every negative value of x an imaginary section, the surface consists of a single sheet extending indefinitely and contin uously in the direction of positive abscissas, but having no from the origin points in the opposite direction The surface is called the elliptical paraboloid It may be dimensions generated by the motion of an ellipse of variable the same on remains whose centre straight line, constantly and whose plane continues perpendicular to that line, and whose semi-axes are the ordinates of two parabolas having a common transverse axis and the same vertex, but different to each parameters placed with their planes perpendicular other FIG D in the If we suppose G to be positive form the take will the so that equation CASE By +C = Go;, first member SURFACES OF THE SECOND ORDER 271 the sections perpendicular to the X-axis will become imaginary when x > 0, and real when x < In other respects the results are similar to those deduced case in will represent a surface of the same form but turned in the opposite direction from the Thus the equation as in case 1, co-ordinate plane of If B = C, the CASE surface becomes the paraboloid of revolution By YZ -C = a Gx Intersect the surface by planes as before Gz = Ga, a hyperbola with transverse and in the axis in the direction of the Y-axis when a direction of the Z-axis when a When x = a, By > ; < When y = b in the direction of = = Gx B6 a parabola having its the X-axis and extending to the left Cz -\- = , axis Gx -j- Cc a parabola having its axis in When z c B?/ the direction of the X-axis and extending to the right } , Since every value of x, either positive or negative, gives a real section, the surface consists of a single sheet extending This indefinitely to the right and left of the plane of YZ surface is called the Hyperbolic Paraboloid To find its princi make x, y, and z alternately equal to zero pal sections When x = 0, B^/ = Cz two straight lines When y = 0, Cz = Gx, a parabola with axis to the left When = 0, B?/ = Gx, a parabola with axis to the right 2 , 2 The hyperbolic paraboloid admits of no variety Now taking up form (C), By + Cs a + Hy + Iz + K = 0, we see that it is the equation of a cylinder whose elements are perpendicular to the plane of YZ, and whose base in the plane of YZ will be an ellipse or hyperbola according to the signs of B and C + + +K= The fourth form (D), Cz Ga; Hy -f Is sents a cylinder having its bases in the planes XZ repre and YZ SOLID ANALYTIC GEOMETRY 272 \ VA A / \ / \ / FIG E parabolas, and having its right-lined elements parallel to the and to each other, but oblique to the axes of and Y plane X XY The preceding discussion shows that every equation of the second degree between three variables represents one or an other of the following surfaces the ellipsoid The ellipsoid with its varieties, viz : ; the of proper, ellipsoid and the imaginary surface revolution, the sphere, the point, SURFACES OF THE SECOND ORDER The hyperboloid of one or two sheets, 273 with their the hyperboloid proper of one or two sheets, the hyperboloid of revolution of one or two sheets, the varieties, viz : equilateral hyperboloid of revolution of one or two sheets, the cone with an elliptical or circular base The paraboloid, either elliptical or hyperbolic, with the variety, the paraboloid of revolution The cylinder, having its base either an ellipse, hyper bola, or parabola The general equation of surfaces be deduced by a direct method, as follows Surfaces of Revolution of revolution may : Ri FIG F Let the Z-axis be the axis of revolution, and tion x2 of = fz Let P AB, the let the equa generating curve in the plane of XZ, be be the point in this curve which generates the circle SOLID ANALYTIC GEOMETRY 274 PQR, and be the radius of the let r The value circle We will have of r2 may also be expressed in terms of z from the equation of the generatrix in the plane of XZas follows : = CP = r2 (JD =fz Equating these two values of r we have as the general equation of surfaces of revolution It will be observed that the second value of r is the value of x in the equation of the generatrix Hence, to find the equation of the surface of revolution we have only to substi tute x if of the surface for x in the generatrix + Surface of a Sphere Equation of generatrix x Hence the equation of the surface of the sphere is + z2 = K, Ellipsoid of Revolution Generatrix Surface JL_ x* -f- ~^~ J!L ^ = 1, -4- a2 c Similarly, the equation of the hyperboloid of revolution ***-*-* Paraboloid of Revolution x2 4:pz, the generatrix = + IT = 4=pz, the Cone of revolution, z = mx x2 surface of revolution -+- j3 ~T Hence or x x the generatrix, ~ ~~i^~ - m 2 (x + if) = (z - ft) is SURFACES OF THE SECOND ORDER 275 EXAMPLES 9z 7/2 (^ Of ^ + if = r- ? Qf if Determine the nature of the surfaces x +^ _ = 36 ? + =r ? the locus in space of x -f y- is - 16 y = 144 ? + 8x = o? What = -\- ,- y"- -(- z2 Of Of = 25, 79> Find the equation of the surface of revolution about the Z whose generatrix is z = x -\- axis of Find the equation of the cone of revolution whose inter = 9, and whose vertex -f- y section with the plane of XYisx is (0, 0, 5.) Determine the surfaces represented by + y + s = 36 + y _ # = 36 - 36 x + 4y = y + = 36 x 9z* = 36x 2 x* a; a 2 2 14 DAY USE RETURN TO DESK FROM WHICH 4S This book OWED ^Sic dbl below.or date stamped is cfil on tfie last on the date to which renewed recall Renewed books are subject to immediate LD 21-40m-5, 65 (F4308slO)476 General Library University of California Berkeley ^ vo \* t ... Technical Schools to prepare a make it work for The aim for beginners, Colleges, of and the Universities, author has been at the same time to for the requirements of comprehensive For the methods...AfTRONOMY ANALYTIC GEOMETRY FOR COLLEGES, UNIVERSITIES, AND TECHNICAL SCHOOLS BY E VV NICHOLS, PROFESSOR OF MATHEMATICS IN THE VIRGINIA... B.S., University for assistance rendered in reading and revising manuscript, and for valuable suggestions given E LEXINGTON, YA January, 1893 W NICHOLS CONTENTS PAET I PLANE ANALYTIC GEOMETRY CHAPTEK