1 J YX y JJt •■■ ELEMENTS ANALYTICAL GEOMETRY AND OF THE DIFFERENTIAL AND INTEGRAL CALCULUS. BY ELIAS LOOMIS, LL.D, PEOFESSOE OF NATCEAL PHILOSOPHY AND ASTRONOMY IN TALE COLLEGE, AMD AUTH08 OF A "COURSE OF MATHEMATICS. " NINETEENTH EDITION. NEW YORK: HARPER & BROTHERS, PUBLISHERS, 329 & 331 PEARL STREET FRAN KM N SqU Ali E. 1865. Entered, according to Act of Congress, in the y^ar 1858, by Harper & Brothers, In the Clerk's Office of the Southern District of New York. PREFACE. The following treatise on Analytical Geometry and the Cal- culus constitutes the fourth volume of a course of Mathematics designed for Colleges and High Schools, and is prepared upon substantially the same model as the preceding volumes. It was written, not for mathematicians, nor for those who have a pe- culiar talent or fondness for the mathematics, but rather for the mass of college students of average abilities. I have, there- fore, labored to smooth down the asperities of the road so as not to discourage travelers of moderate strength and courage ; but have purposely left some difficulties, to arouse the energies and strengthen the faculties of the beginner. In a course of liberal education, the primary object in studying the mathe- matics should be the discipline of the mental powers. This discipline is alike important to the physician and the divine, the jurist and the statesman, and it is more effectually secured by mathematical studies than by any other method hitherto proposed. Hence the mathematics should occupy a prominent place in an education preparatory to either of the learned pro fessions. But, in order to secure the desired advantage, it is indispensable that the student should comprehend the reasons of the processes through which he is conducted. How can he be expected to learn the art of reasoning well, unless he see clearly the foundations of the principles which are taught ? This remark applies to every branch of mathematical study, but perhaps to none with the same force as to the Differential and Integral Calculus. The principles of the Calculus are fur- ther removed from the elementary conceptions of the mass of mankind than either Algebra, Geometry, or Trigonometry, and they require to be developed with corresponding care. It is quite possible for a student to learn the rules of the Calculus, B39658 iv Preface. and attain considerable dexterity in applying them to the solu- tion of difficult problems, without having acquired any cleai idea of the meaning of the terms Differential and Differentia! Coefficient. Cases of this kind are not of rare occurrence, and the evil may fairly be ascribed, in some degree, to the imper- fection of the text-books employed. The English press lias foi years teemed with "Elementary treatises on the Calculus,' 1 many of which are wholly occupied with the mechanical pro- cesses of differentiating and integrating, without any attempt to explain the philosophy of these operations. A genuine math- ematician may work his way through such a labyrinth, and solve the difficulties which he encounters without foreign as- sistance ; but the majority of students, if they make any prog- ress, will only proceed blindfolded, and after a time will aban- don the study in disgust. I have accordingly given special attention to the develop- ment of the fundamental principle of the Differential Calcu- lus, and shall feel a proportionate disappointment if my labors shall be pronounced abortive. The principle from which I have aimed to deduce the whole science, appears to me better adapt- ed to the apprehension of common minds than any other ; and although I do not claim for it any originality, it appears to me that I have here developed it in a more elementary manner than I have before seen it presented, except in a small volume by the late Professor Ritchie, of London University. I have de- rived more important suggestions from this little volume, than from all the other works on the Calculus which have fallen under my notice. The exposition of the principles of the Cal- culus contained in the following treatise, appears to me so clear, that I indulge the hope that hereafter this subject may be made a standard study for all the students of our colleges, and not be abandoned entirely to the favored few. While the mental discipline of the majority of students has been the object kept primarily in dew, it is believed that the course here pursued will be found best adapted to develop the taste of genuine mathematicians ; for a clear conception of the fundamental principles of the science must certainly be favor- able to future progress. The student who renders himself fa- miliar with the present treatise will have acquired a degree of Preface. v mental discipline which will prove invaluable in every depart- ment of business ; and he will be enabled, if so inclined, to pursue advantageously any of the standard treatises on the same subject. Every principle in this work is illustrated by examples, and at the close of the volume will be found a large collection of examples for practice, which are to be resorted to whenever the problems which are incorporated in the body of the work are considered insufficient. CONTENTS. ANALYTICAL GEOMETRY. SECTION I. APPLICATION OF ALGEBRA TO GEOMETRY. tagt Geometrical Magnitudes represented by Algebraic Symbols 9 Solution of Problems * * SECTION II. CONSTRUCTION OF EQUATIONS. Construction of the Sum and Difference of two Quantities 13 Product of several Quantities " Fourth Proportional to three Quantities 14 Mean Proportional between two Quantities 15 Sum or Difference of two Squares 15 To inscribe a Square in a given Triangle 16 To draw a Tangent to two Circles 17 SECTION III. ON THE POINT AND STRAIGHT LINE. Methods of denoting the position of a Point 20 Abscissa and Ordinate defined - ~0 Equations of a Point ~1 Equations of a Point in each of the four Angles 22 Equation of a straight Line 23 Four Positions of the proposed Line 24 Equation of the first Degree containing two Variables 26 Equation of a straight Line passing through a given Point 27 Equation of a straight Line passing through two given Points 28 Distance between two given Points 30 Angle included between two Lines 30 Transformation of Co-ordinates 32 Formulas for passing from one System of Axes to a Parallel System 33 Formulas for passing from Rectangular Axes to Rectangular Axes 33 Formulas for passing from Rectangular to Oblique Axes 34 Formulas for passing from Rectangular to Polar Co-ordinates 2s viii Contents. SECTION IV. ON THE CIRCLE. Equation of the Circle when the Origin is at the Center 36 Equation of the Circle when the Origin is on the Circumference 37 Most general form of the Equation 38 Equation of a Tacgent to the Circle 39 Polar Equation of the Circle 42 SECTION V. ON THE PARABOLA. Definitions , 4* Equation of the Parabola ' 45 Equation of a Tangent Line 4G Equation of a Normal Line 48 The Normal bisects the Angle made by the Radius Vector and. Diameter 49 Perpendicular from the Focus upon a Tangent 50 Equation referred to a Tangent and Diameter 50 Parameter of any Diameter 52 Polar Equation of the Parabola 53 Area of a Segment of a Parabola £3 SECTION VI. ON THE ELLirSE. Definitions 50 Equation of the Ellipse referred to its Center and Axes 56 Equation when the Origiu is at the Vertex of the Major Axis CO Squares of two Ordinates as Products of parts of Major Axis 61 Ordiuates of the Circumscribed Circle 61 Every Diameter bisected at the Center 62 Supplementary Chords „ 63 Equation of a Tangent Line 64 Equation of a Normal Line 60 The Normal bisects the Angle formed by two Radius Vectors 67 Supplementary Chords parallel to a Tangent and Diameter 68 Equation of Ellipse referred to Conjugate Diameters 69 Squares of two Ordinates as Products of parts of a Diameter 7 i Sum of Squares of two Conjugate Diameters 71 Parallelogram on two Conjugate Diameters 73 Polar Equation of the Ellipse 74 Area of the Ellipse 75 SECTION VII. ON THE HYPERBOLA. Definitions 77 Equation of the Hyperbola referred to its Center and Axes 78 Equation when the Origin is at the Vertex of the Transverse Axis 81 Squares of two Ordinates as Products of parts of Transverse Axis 82 Contents. i x Page Every Diameter bisected at the Center 8C Supplementary Chords 83 Equation of a Tangent Lin 3 81 Equation of a Normal Line 8.0 The Tangent bisects the Angle contained by two Radius Vectors 8? Supplementary Chords parallel to a Tangent aud Diameter 88 Equation referred to Conjugate Diameters 83 Squares of two Ordinates as the Rectangles of the Segments of a Diameter. . 91 Difference of Squares of Conjugate Diameters 91 Parallelogram on Conjugate Diameters 93 L'olar Equation of the Hyperbola 93 Asymptotes of the Hyperbola 94 Equation of the Hyperbola referred to its Asymptotes ■. 95 Parallelogram contained by Co-ordinates of the Curve 90 Equation of Tangent Line 97 Portion of a Tangent between the Asymptotes 98 SECTION VIII. CLASSIFICATION OF ALGEBRAIC CURVES. Every Equation of the second Degree is the Equation of a Conic Section ... 100 The Term containing the Product of the Variables removed 100 The Terms containing the first Tower of the Variables removed 101 Lines divided into Classes 101 Number of Lines of the different orders 10*1 Family of Curves 105 SECTION IX. TRANSCENDENTAL CURVE3. Cycloid— Defined 106 Equation of the Cycloid 107 Logarithmic Curve — its Properties 107 Spiral of Archimedes — its Equation 109 Hyperbolic Spiral — its 'Equation 110 Logarithmic Spiral— its Equation Ill DIFFERENTIAL CALCULUS. SECTION I. DEFINITIONS AND FIRST PRINCIPLES— DIFFERENTIATION OF ALGEBRAIC FUNC- TIONS. Definitions — Variables and Constants 113 functions — explicit and implicit — increasing and decreasing 114 Limit of a Variable Quantity 115 Rate of Variation of the Area of a Square 118 Rate of Variation of the Solidity of a Cube 119 Differential defined— Differential Coefficient 1211 x Contents. Rule for finding the Differential Coefficient 122 Differential of any power of a Variable 123 Product of a Variable by a Constant 124 Differential of a Constant Term 124 General expression for the second State of a Function 125 Differential of the Sum or Difference of several Functions 126 Differential of the Product of several Functions 127 Differential of a Fraction 129 Differential of a Variable with any Exponent 131 Differential of the Square Koot of a Variable 133 Differential of a Polynomial raised to any Power 133 SECTION II. SUCCESSIVE DIFFERENTIALS— MACLAURIN'S THEOREM— TAYLOR'S THEOREM FUNCTIONS OF SEVERAL INDEPENDENT VARIABLES. Successive Differentials — Second Differential Coefficient 137 Maclaurin's Theorem — Applications 138 Taylor's Theorem — Applications „ 141 Differential Coefficient of the Sum of two Variables 142 Differentiation of Functions of two or more independent Variables 143 SECTION III. SIGNIFICATION OF THE FIRST DIFFERENTIAL COEFFICIENT— MAXIMA AND MIN- IMA OF FUNCTIONS. Signification of the first Differential Coefficient 147 Maxima and Minima of Functions defined 148 Method of finding Maxima and Minima 149 Application of Taylor's Theorem 151 How the Process may be abridged 154 Examples 155 SECTION IV. TRANSCENDENTAL FUNCTIONS. Transcendental Functions 161 Differential of an Exponential Function 161 Differential of a Logarithm 163 Circular Functions 165 Differentials of Sine, Cosine, Tangent, and Cotangent 166 Differentials of Logarithmic Sine, Cosine, Tangent, and Cotangent 170 Differentials of Arc in terms of its Sine, Cosine, etc 172 Development of the Sine and Cosine of an Arc 174 SECTION V. APPLICATION OF THE DIFFERENTIAL CALCULUS TO THE THEORY OF CURVES. Differential Equation of Lines of different Orders 175 Length of Tangent, Subtaugent, Normal, and Subnormal 177 Formulas applied to the Conic Sections 178 Subtangent of the Logarithmic Curve „ ISO Contents. xi P«6» Subtangent and Tangent of Polar Curves J 82 Formulas applied to the Spirals 183 Differential of an Arc, Area, Surface, and Solid of Revolution 184 Differential of the Arc and Area of a Polar Curve 190 Asymptotes of Curves 191 SECTION VI. RADIUS OF CURVATURE— EVOLUTES OF CURVES. Curvature of Circles 184 Radius of Curvature at any Point of a Curve 196 Radius of Curvature of a Conic Section 197 E volutes of Curves defined 199 Equation of the E volute determined 200 Evolute of the common Parabola 201 Properties of the Cycloid 202 Expression for the Tangent, Normal, etc., to the Cycloid 202 Radius of Curvature of the Cycloid 203 Evolute of the Cycloid 204 SECTION VII. ANALYSIS OF CURVE LINES. Singular Points of a Curve 205 Tangent parallel or perpendicular to Axis of Abscissas 205 When a Curve is Convex toward the Axis 206 When a Curve is Concave toward the Axis 208 To determine a Point of Inflection 209 To determine a Multiple Point 211 To determine a Cusp 21£ To determine an isolated Point 215 INTEGRAL CALCULUS. SECTION I. INTEGRATION OF MONOMIAL DIFFERENTIALS— OF BINOMIAL DIFFERENTIALS- OF THE DIFFERENTIALS OF CIRCULAR ARCS. Integral Calculus defined 217 Integral of the Product of a Differential by a Constant 218 Integral of the Sum or Difference of any number of Differentials 219 Constant Term added to the Integral 219 Integration of Monomial Differentials 219 Integration by Logarithms 220 Integral of a Polynomial Differential 221 Integral of a Binomial Differential 223 Definite Integral 224 Integrating between Limits 225 Integration by Series 226 xii Contents. Pag< Integration of the Differentials of Circular Arcs 227 Integration of Binomial Differentials 230 When a Binomial Differential can be integrated 232 Integration by Parts 236 To diminish the Exponent of the Variable without the Parenthesis 237 When the Exponent of the Variable is Negative 242 To diminish the Exponent of the Parenthesis 243 When the Exponent of the Parenthesis is Negative 245 SECTION II. APPLICATIONS OF THE INTEGRAL CALCULUS. Rectification of Plane Curves 247 Quadrature of Curves 253 Area of Spirals 257 Area of Surfaces of Revolution 258 Oubature of Solids of Revolution s. 263 MISCELLANEOUS EXAMPLES 267 ANALYTICAL GEOMETRY. SECTION I. APPLICATION OF ALGEBRA TO GEOMETRV. Article 1. The relations of Geometrical magnitudes may be expressed by means of algebraic symbols, and the demon- strations of Geometrical theorems may thus be exhibited more concisely than is possible in ordinary language. Indeed, so great is the advantage in the use of algebraic symbols, that they are now employed to some extent in all treatises on Ge- ometry. (2.) The algebraic notation may be employed with even greater advantage in the solution of Geometrical problems. For this purpose we first draw a figure which represents all the parts of the problem, both those which are given and those which are required to be found. The usual symbols or let- ters for known and unknown quantities are employed to de- note both the known and unknown parts of the figure, or as many of them as may be necessary. We then observe the re- lations which the several parts of the figure bear to each other, from which, by the aid of the proper theorems in Geometry, we derive as many independent equations as there arc un- known quantities employed. The solution of these equations by the ordinary rules of algebra will determine the value of the unknown quantities. This method will be illustrated by a few examples. Ex. 1. In a right-angled triangle, having given the hus<- and sum of the hypothenuse and perpendicular, to find the perpendicular. Let ABC represent the proposed triangle, right- angled at B. Represent the base A 15 by l>, the perpendicular BC by x, and the sum of the hypoth- snuse and perpendicular by s; then the hypothe- A~ A 10 Analytical Geometry. nuse will be represented by s — x. Then, by Geom., Prop. 11 B. IV., AT? +13(7= AC 2 ; that is, b*+x* = (s—xy=s 2 — 2sx+x\ Taking away x 1 from each side of the equation, we have b"=s 2 — 2sx, or 2sx=s"~ — b 2 ; s*-V Whence x=— - — , 2s from which we see that in any right-angled triangle, the pei- pendicular is equal to the square of the sum of the hypothe- nuse and perpendicular, diminished by the square of the base, and divided by twice the sum of the hypothenuse and perpen dicular. Thus, if the base is 3 feet, and the sum of the hy- pothenuse and perpendicular 9 feet, the expression 9 2 -3 2 2s comes 2X9 =4, the perpendicular. Ex. 2. Having given the base and altitude of any triangle, it is required to find the side of the inscribed square. Let ABC represent the given triangle, in which there are given the base AB and the altitude CH ; it is required to find the side of the inscribed square. Suppose the inscribed square DEFG to be drawn. Represent the base AB by b, the perpendicular CH by h, and the side of the inscribed square by x ; then will CI be represented by 'h—x. Then, because GF is parallel to the base AB, we have, by similar triangles, Geom., Prop. 10, B. IV., AB : GF : : CH : CI ; that is, b :x :: h : h—x; or, since the product of the extremes is equal to that of the means, bh — bx=hx ; bh whence x ~b+h' that is, the side of the inscribed square is equal to the product of the base and height divided by their sum. Thus, if the base of the triangle is 12 feet, and the altitude 6 feet, the side of the inscribed square is found to be 4 feet. AT FLIC AT ION Or A.LGEBHA TO GeOMETBY. 11 H E B Ex. 3. IL/r • n the base and altitudt of any triangle, it is required to inscribe within it Analytical Geometry. Problem X. To construct the equation x=a^Va i —b t . Draw an indefinite line AE, and set c off a distance AB equal to a; from B draw BC perpendicular to AB, and make it equal to b. With C as a center, and a radius equal to a, describe an arc of a A D" - -" E circle cutting AE in D and E. Now the value of Va i —b will be BD or BE. When the radical is positive, its value is to be set off toward the right ; when negative, toward the left. Therefore, AD and AE are the values required ; for AE = AB+BE=a+ v V^F , and AD=AB-BD=«-vV-&\ The preceding values are the roots of the equation x*-2ax=-b*. Problem XI. Having given the base and altitude of any triangle, it is required to find the side of the inscribed square by a geometrical construction. We have found on page 10 the side of the inscribed square to bh be equal to . , . . 1 b+h Hence the side of the inscriDed square is a fourth propor- tional to the three lines b+h, b, and h. Produce the side CA indefinite- c ly, and lay offCK equal to the al- titude //, and KL equal to the base b. Join LH, and draw KI paral- lel to LH ; IH will be equal to a side of the inscribed square. For, by similar triangles, we have " CL : KL : : CH : IH ; /. that is, b+h : b : : h : IH. L- : """ bh -"'H Hence IH= b+li and therefore IH is equal to a side of the inscribed square. Example 3, page 11, maybe constructed in similar manner oy laying off CK equal to nh. Problem XII. It is required to draw a common tangent line to two given circles situated in the same plane. LetCC be the centers of the two circles, CM, CM' their radii Construction of Equations. 17 Let us suppose the problem solved, and that MM' is the common tangent line. Pro- duce this tangent until it meets the line CC, passing through the centers of the circles ; then, drawing the radii CM, CM' to the points of tangency, the an- gles CMT, C'M'T will be right angles, and the triangles CMT, G'M'T will be similar. Hence we shall have the proportion CM : CM' : : CT : CT. Represent CM by r, CM' by r , CC by a, and CT by x. CT will, therefore, be x— a, and the preceding proportion will be- come r '. r' : : x : x — a ; whence rx—ra=r'x, ar and x= -; r—r from which we see that CT or a; is a fourth proportional to the three lines r—r 1 , a, and r. To obtain a: by a geometrical construction, through the cen- ters C, C draw two parallel radii, CN, C'N'. Through N and N' draw the line NN'T, meeting the line CC in T. Through T draw a tangent line to one of the circles, it will also be a tangent to the other. For through N' draw N'D parallel to CT ; then N'D will represent a, ND will repre- sent r—r 1 ; and, since the triangle DNN' is similar to CNT, we have the proportion DN:DN or r—r' : a whence CT which is the value of a; before found. Therefore, a line drawn from T, tangent to one of the circles, will also be tangent to the other ; and, since two tangent lines can be drawn from the point T, we see that the problem proposed admits of two solu- tions. Cor. 1. If we suppose the radius r of the first circle to remain B 18 Analytical Geometry. constant, and the smaller radius r' to increase, the difference r— r' will diminish; and, since the numerator ar remains con- stant, the value of a: will increase ; which shows that the nearer the two circles approach to equality, the more distant is the point of intersection of the tangent line with the line joining the centers. When the two radii r and r' become equal, the de- nominator becomes 0, and the value of ar becomes infinite. Cor. 2. If we suppose r to increase so as to become greater than r, the value of a: becomes negative, which shows that the point T falls to the left of the two circles. Cor. 3. Two other tangent lines may be drawn intersecting each other between the circles. If we represent CT by x, the radii of the circles by r and r', and the dis- tance between their centers by a, we shall have from the similar tri- angles CMT, C'M'T, the proportion CM : CM' : : CT or r : r' : : x ar whence x=- CT, a—x r+r' This expression may be constructed in a manner similar to the former. Through the centers C and C draw two parallel radii CN, CN', lying on different sides / of the line CC; join the points NN', and through T, where this line in- tersects CC, draw a line tangent to one of the circles, it will be a tangent to the other. For through N' draw N'D parallel to CC, and meeting CN pro- duced in D. DN' will then represent a, ND will represent r+r', and the similar triangles NCT, NDN' will furnish the proportion ND : DN' : : NC : CT, or r+r' : a : : r : CT; ar whence CT: r+r which is the value already found for x. (7.) Every Algebraic expression, admitting of geometrical construction, must hare all its terms homogeneous (Algebra Construction of Equations. 19 A.rt. 31) ; that is, each term must contain the same number of literal factors. Thus, each term must either be of one dimen* sion, and so represent a line ; or, secondly, each must be of two dimensions, and represent a surface ; or, thirdly, each must be of three dimensions, and denote a solid ; since dissimilar geometrical magnitudes can neither be added together nor sub- tracted from each other. It may, however, happen that an expression really admitting of geometrical construction appears to be not homogeneous ; but this result arises from the circumstance that the geometrical unit of length, being represented algebraically by 1, disappears from all algebraic expressions in which it is either a factor or a divisor. To render these results homogeneous, it is only necessary to restore this divisor or factor which represents unity. Thus, suppose we have an equation of the form x=ab+c. If we put I to represent the unit of measure for lines, we may change it into the homogeneous equation, lx=ab+cl, ab or x=—+c ; which is easily constructed geometrically. SECTION III. ON THE POINT AND STRAIGHT LINE. (8.) There are two methods of denoting the position of a point in a plane. The first is by means of the distance and direction of the proposed point from a given point. Thus, if A be a known point, and AX be a known direction, the position of the point P will be determined when we a know the distance AP and the angle PAX. The assumed point A is called the pole ; the distance of P from A is called the radius vector ; and the radius vector, to aether with its angle of inclination to the fixed line, are called the polar co-ordinates of the point. (9.) It is, however, generally most convenient to denote the position of a point by means of its distances from two given lines which intersect one another. Thus, let AX, AY be two assumed straight lines which intersect in any angle at A, and let P be a point in the same plane ; then, if we draw PB parallel to AY, and PC parallel to AX, the position of the point P will be de- A. 13 noted by means of the distances PB and PC. The two lines AX, AY, to which the position of the point P is referred, are called axes, and their point of intersection A is called their origin. The distance AB, or its equal CP, is called the abscissa of the point P ; and BP, or its equal AC, is called the ordinate of the same point. Hence the axis AX is called the axis of abscissas, and AY is called the axis of ordinates. The abscissa and ordinate of a point, when spoken of to- gether, are called the co-ordinates of the point, and the two axes are called co-ordinate axes. The axes are called oblique or rectangular, according as YAX is an oblique or a right angle. Rectangular axes are the most simple, and will generally be employed in this treatise. On the Point and straight Line. 2i An abscissa is generally denoted by the letter x, and an or- dinate by the letter y ; and hence the axis of abscissas is often called the axis of X, and tne axis of ordinates the axis of Y. The abscissa of any point is its distance from the axis of or- dinates, measured on a line parallel to the axis of abscissas. The ordinate of any point is its distince from the axis of ab- scissas, measured on a line parallel to ,lhe axis of ordinates. (10.) The position of a point may be determined when its co-ordinates are known. For suppose the abscissa of the point P is equal to a, and its ordinate is equal to b. Then, to determine the position of the point P, from the origin A lay off on the axis of abscissas a distance AB equal to a, and J j through B draw a line parallel to the axis f / ^ of ordinates. On this line lay off a distance -^ B BP equal to b, and P will be the point required. Hence, in order to determine the position of a point, we need only have the two equations x=a, y=b, in which a and b are given. These equations are, therefore, nailed the equations of a point. (11.) It is, however, necessary, in order to determine the position of a point, that not only the absolute values of a and b should be given, but also the signs of these quantities. If the axes are produced through the origin Y to X' and Y', it is obvious that the , / p abscissas reckoned in the direction I 7 7 AX' ought not to have the same sign X- as those reckoned in the opposite di- rection AX ; nor should the ordinates measured in the direction AY' have the same sign as those measured in the opposite direction AY ; for if there were no distinction in this respect, the position of a point as determined by its equations would be ambiguous. Thus the equations of the point P would equally belong to the points P', P", P"', provided the absolute lengths of the co-or- dinates of each were equal to those of P. All this ambiguity is avoided by regarding the co-ordinates which are measured in one direction as plus, and those in the opposite direction minus. It is generally agreed to regard those abscissas which 22 Analytical Geometry. fall on the right of the origin A as positive, and hence those which fall on the left must be considered negative. So, also, it has been agreed to consider those ordinates which are above the origin as positive, and hence those which fall below it must be considered negative. (12.) The angle YAX is called the first angle; YAX' the second angle ; Y'AX' the third angle ; and Y'AX the fourth angle. The following, therefore, are the equations of a point in each of the four angles : For the point P in the first angle, x=+a, y=+b, " P' " second angle, x= — a, y=+b, " P" " third angle, x=—a,y=-b, P'" " fourth angle, x=+a, y=-b. If the point be situated on the axis AX, the equation y=b becomes y=0, so that the equations x=*±a, y=0, characterize a point on the axis of abscissas at the distance a from the origin. If the point be situated on the axis AY, the equation x=a becomes x=0, so that the equations x—0, y=±b, characterize a point on the axis of ordinates at the distance b from the origin. If the point be common to both axes, that is, if it be at the origin, its position will be expressed by the equations x=0, y=0. Ex. 1. Determine the point whose equations are x=+4, y=-3. Ex. 2. Determine the point whose equations are x= — 2 Ex. 3. Determine the point whose equations are x—0 y——b. Ex. 4. Determine the point whose equations are x=— 8, y=0. Definition. — The equation of a line is the equation which ex- presses the relation between the two co-ordinates of every point of the line. On the Point and straight Lin 23 Proposition I. — Theorem. (13.) TJie equation, of a straight line referred to rectangular axes is y=ax-\-b ; where x and y are the co-ordinates of any point of the line, a represents the tangent of the angle which the line makes with the axis of abscissas, and b the distance from the origin at which it intersects the axis of ordinates. Also, a and b may be either positive or negative. Let A be the origin of co-ordinates, AX and AY be rectangular axes, and PC any straight line whose equation is required to be determined. Take any point P in the given line, and draw PB perpendicular to AX ; then -^ will PB be the ordinate, and AB the abscissa of the point P. From A draw AD parallel to CP, meeting the line BP in D. Let AB=£, BP=y, tangent PEX or DAX=a, and AC or DP=6. Then, by Trigonometry, Theorem II., Art. 42, R : AB : : tang. DAX : BD, or R : x : : a : BD. Hence, calling the radius unity, we have BT)=ax. But BP=BD+DP; hence y= ax +b. If the line CP cuts the axis of ordi- nates below the origin, then we shall have BP=BD-DP, or y= ax —b. If x represents a negative line, as AB, then will ax or BD be negative, and the equation y = ax+b will be true, since BP=-BD + DP. (14.) The line PC has been drawn - so as to make an acute angle with the axis of abscissas; but the preceding 24 Analytical Geometry. equation is equally applicable whatever may be this angle, provided proper signs are attributed to each term. The angle which the line makes with, the axis of abscissas is supposed to be measured from the axis AX around the circle by the left. If the angle is obtuse, its tangent will be negative (Trigo- nometry, Art. 70). Thus, if PC be the po- sition of the proposed line with reference to the rectangular axes AX, AY, then, in the proportion E:AB::tang. DAX:BD, the tangent of DAX is negative; BD is therefore negative. The equation of this line may then be written y=—ax-\-b, where it must be observed that the sign — applies only to the quantity a, and not to a?, for the sign of x depends upon its di- rection from the origin A. (15.) There may, therefore, be four positions of the pro- posed line, and these positions are indicated by the signs of a and b in the general equation. 1. Let the line take the position shown in the annexed diagram, cutting the axis of X to the left of the origin, and the axis of Y above it, then a and b are both positive, and the equation is y=+ax+b. 2. If the line cuts the axis of X to the right of the origin, and the axis of Y be- low it, then a will still be positive, but b will be negative, and the equation be- comes y=+ax— b. 3. If the line cuts the axis of X to the right of the origin, and the axis of Y above it, then a becomes negative and b positive. In this case, therefore, the equation is y~— ax+b. 4. If the line cuts the axis of X to the left of the origin, and On the Point and straight Line. 25 .he axis of Y below it, then both a and b will be negative, so that the equation be- comes y— — ax—b. If we suppose the straight line to pass through the origin A, then b will be equal to zero, and the general equation becomes y=ax, which is the equation of a straight line passing through the origin. We here suppose the letters a and b to stand for positive quantities. It is, however, to be borne in mind that they may themselves represent negative quantities, in which case —a and —b will be positive. Ex. 1. Let it be required to draw the line whose equation is y=2x+4. If in this equation we make x=0, the value of y will designate the point in which the line intersects the axis of ordinates, for that is the only point of the line whose abscissa is 0. This supposition will give y=4. Having drawn the co-ordinate axes AX, AY, lay off from the origin A a distance AB equal to 4 ; this will be one point of the required line. Again, if in the proposed equation we make y=0, the value of x, which is found from the equation, will designate the point in which the line intersects the axis of abscissas, for that is tne only point of the line whose ordinate is 0. This supposition will give 2x=— 4, or x= — 2. Lay off from the origin A, toward the left, a distance AC equal to 2 ; this will give a second point of the proposed line, and the line may be drawn through the two points B and C. (1G.) We may determine any number of points in this line by assuming particular values for x or y ; the equation will furnish the corresponding value of the other variable. Making successively £C=1, we find y=6, x=3, we find y=l0, x=2, " y = 8, x=4, " y= 12, etc. In order to represent these values by a figure, we draw two H G 26 Analytical Geo METRr. axes AX, AY at right angles to each other Then, in order to construct the values x—1, y=6, we set off on the axis of abscissas a -y i line AB equal to 1, and erect a perpendicu- lar BG equal to 6 ; this determines one point of the required line. Again, take AC equal to 2, and make the perpendicular CH equal to 8 ; this will determine a sec- ond point of the required line. In the same BCI)E manner we may determine the points K and L, and any num- ber of points. The required line must pass through all the points, G, H, K, L, etc. Any straight line may be constructed by determining two points in that line, and drawing the line through those points. Ex. 2. Construct the line whose equation is y=2x+3. Ex. 3. Construct the line whose equation is y=Sx— 7. Ex. 4. Construct the line whose equation is y— — x-\-2. Ex. 5. Construct the line whose equation is y=— 2x— 5. Ex. 6. Construct the line whose equation is y=ox. Ex. 7. Construct the line whose equation is y—b. Ex. 8. Construct the line whose equation is y=—2. In the equation y=ax+b, the quantities a and b remain the same, while the co-ordinates x and y vary in value for every point in the same line. We, therefore, call a and b constant quantities, and x and y variable quantites. Proposition II. — Theorem. (17.) Every equation of the first degree containing two varia- bles is the equation of a straight line. Every equation of the first degree containing two variables can be reduced to the form Ay=Bx+C, in which. A, B, and C may be positive or negative. Now a straight line may always be constructed of which this shall be the equation. Draw the co-ordinate axes AX, AY at right angles to each On the Point and straight Line. 27 other; make AB equal to C^A, and AC equal to C-hB, and through the points B and C draw the line PBC, it will be the required line. For the equation of this line is AB , ._ but by the construction, AB_C_C_B. AO~A B~A' also, AB=C-A. Therefore the equation of the line PBC is B C X A (2) or, ^ = A* + A ; A*/=Ba; + C. Examples.— Draw the lines of which the following are the equations, 2y=3x-5, 2x=y+7, cc+y=0, y=±-x, x=2y, a=4, x+y=lO, x+y±lO = 0, y=2. What are the values of a and b in these equations? Proposition III. — Theorem. (18.) The equation of a straight line passing through a given point is y-y' = a{x-x'\ where x' and y' denote the co-ordinates of the given point, x and y the co-ordinates of any point of the line, and a the tangent of the angle which the line makes with the axis of abscissas. Known co-ordinates are frequently designated by marking them thus, x',y'; x",y"; x"',y"',etc, which are read x prime, y prime; x second, y second; a; third, y third, etc. T Let P be the given point, and designate its co-ordinates by x' and y'. Then, since the general equation for every point in the required line is y = ax+b; (1) Analytical Geometry. and since P is a point in the line, it follows that y' = ax' + b. (2) By means of equation (2) we may eliminate b from equa- tion (1). Subtracting equation (2) from equation (1), we obtain y-y'=a(x-x'), which is the equation of a line passing through the given point P. Since the tangent a, which fixes the direction of the line, is not determined, there may be an infinite number of straight lines drawn through a given point. This is also apparent from the figure. (19.) If it be required that the line shall pass through a given point, and be parallel to a given line, then the angle which the line makes with the axis of abscissas is determined ; and if we put a' for the tangent of this angle, the equation of the line sought will be y— y'=a'{x— x'). Ex. Draw a line through the point whose abscissa is 5 and ordinate 3, making an angle with the axis of abscissas whose tangent is equal to 2. Proposition IV. — Theorem. (20.) The equation of a straight line which passes through * two given points is y -Hy^-* 1 )- x'—x where x' and y' are the co-ordinates of one of the given points, x" and y" the co-ordinates of the other point, and x and y the general co-ordinates of the line. Let B and C be the two given points, the co-ordinates of B being x' and y', and the co-ordinates of C being x" and y". Then, since the general equation for every point in the required line is y=ax+b, (1) it follows that when the variable abscissa x becomes x', then y will become y' ; hence y'=ax'+b. (2) On the Point and straight Line. 29 Also, when the variable abscissa x becomes x", then y be- comes y", and hence y"~aocf'+b. (3) From equations (2) and (3) we may obtain the values of a and b, and substitute them in the first equation. Or we may accomplish the same object by eliminating a and b from the three equations. If we subtract equation (2) from equation (1), we obtain y-y'=a(x-x'). (4) Also, if we subtract equation (3) from equation (2), we obtain y'—y"=a(x'—x"), y'—y" from which we find a= ,_ „ . Substituting this value of a in equation (4), we have v' — v" which is the equation of the line passing through the two given points B and C. y> y" (21.) We have found a equal to ,_ ,, . This is obvious from the figure. For y'—y" is equal to BD, and x'—x" is equal y' — y" BD to CD ; hence - — ^ is equal to ^rp-, which is the tangent of x'—x ^D the angle BCD, the radius being unity (Trigonometry, Art. 42), If the origin be one of the proposed points, then x"=0, and y"=0, and the equation becomes . y' J x' which is the equation of a straight line passing through the origin and through a given point. Ex. 1. Find the equation to the straight line which passes through the two points whose co-ordinates are x'=l, y'=4, x" = 5, y"=S, and determine the angle which it makes with the axis of abscissas. Ex. 2. Find the equation to the straight lino which passes through the two points x'— 2, y'=3, and x"=4, y"=b. 30 Analytical Geometry. Proposition V. — Theorem. (22.) The distance between two given points is equal to V(x'-z"y+w-y»y, where x' and y' are the co-ordinates of one of the given points, and x" and y" those of the other. Let B and C be the two given points, y Designate the co-ordinates of B by x' and y', and the co-ordinates of C by x" and y". Draw CD parallel to AX. The distance BC is equal to A VCD 2 +BD 2 . But CD=x'—x", and BD=y'—y" ; therefore the expression for the distance between B and C is x V(x'-x"y + (y'-y")\ Proposition VI. — Theorem. (23.) The tangent of the angle included between two straight lines is 1+aa" where a and a' denote the tangents of the angles which the Hvo lines make with the axis of abscissas. Let BC and DE be any two lines intersecting each other in P. Let the equation of the line DE be y=ax+b, and the equation of BC be y=a'x+b' ; then a will be the tangent of angle PEX, and a' the tangent of the angle PCX. Designate the angle PEX by a, and the angle PCX by a'. Now, because PCX is the exterior angle of the triangle PEC, it is equal to the sum of the angles CPE and PEC ; that is, the angle EPC is equal to the difference of the angles PCX and PEX, or EPC = PCX-PEX=a'- a ; whence tang. EPC = tang. (PCX- PEX) = tang, (a' -a). But, by Trigonometry, Art. 77, On THE iOINT AND STRAIGHT LlNE. Si tariff, a' — tangr. a tang. («'-«)= 5 5 — 1+tang. a tang, a.' Therefore, tang. EPC=^-^-. 1+aa' If the angle of intersection of the two lines be a right angle, its tangent must be infinite. But in order that the expression - may become infinite, the denominator 1+aa' must be- 1+aa' come zero; so that in this case we must have aa'= — 1, or a= :. This, then, is the condition by which two straight lines are shown to be at right angles to each other. (24.) This last conclusion might have been derived from the principles of Trigonometry. Thus, let the two lines PC, PE be perpendicu- lar to each other ; then the angle PCE is the complement of PEC. But by (Trig., Art. 28) tang.Xcotang.=R 2 or unity; hence tang. PEC X tang. PCE = 1. 'Now PCX, being the supplement of PCE, has the same tangent (Trig., Art. 27), but with a negative sign (Trig., Art. 70). Hence tang. PEC X tang. PCX=-1. (25.) The equation of a line passing through a given point is y— y' = a(x— x'). If this line be perpendicular to a certain given line, we may for a substitute — -„ where a' is the tangent of the angle which CI this last named, line makes with the axis of abscissas. Hence is the equation ot a line passing through a given point, and per I pendicular to a given line. Proposition VII. — Theorem.* (20.) The equation of a straight line referred to ohlique axes is y—ax+b. 32 Analytical Geometry. where a represents the ratio of the sine of the angle which the line makes with the axis of abscissas, to the sine oi the angle which it makes with the axis of ordinates. Let A be the origin of co-ordinates, and AX, AY oblique axes, and PC any straight line whose equation is required to be determined. Take any point P in the given line, and draw PB parallel to AY ; then will PB be the ordinate, and AB the abscissa of the point P. From E A draw AD parallel to CP, meeting the line BP in D. De- note the angle PEX, or its equal DAX, by a, and the angle YAX by P. Since PB is parallel to AY, the angle ADB is equal to DAY ; that is, equal to (5— a. Let AB=:r, BP=y, and AC or DP=6. Then, by Trigonometry, Theorem I., Art. 49, BD : AB : : sin. a : sin. (0— a), or BD . x :: sin. a : sin. (j3— a). sin. a Hence BD=:r- — -p. r. sin. (j3— a) But BP=BD+DP. sin. a Hence ^ =X sm.(/3-a) +& - The coefficient of x in this equation is equal to the sine of I-he angle winch the line makes with the axis of X, divided by the sine of the angle which it makes with the axis of Y ; and if we represent this factor by a, the equation may be written y=ax+b, which is of the same form as in Theorem L. but the factor a > has a different signification. ON THE TRANSFORMATION OF CO-ORDINATES. (27.) When a line is represented by an equation in reference to any system of axes, we can always transform that equation into another which shall equally represent the line, but in refer- ence to a new system of axes chosen at pleasure. This is z 1 /M' -X' -X On the Transform ation of Co-ordinates. S',i called the transformation of co-ordinates ; and may consist either in altering the relative position of the axes without changing the origin, or changing the origin without disturbing the relative position of the axes ; or we may change both the direction of the axes and the position of the origin. Proposition VIII. — Theorem. (28.) The formulas f oi' passing from one system of co-ordinate axes to another system, respectively parallel to the first, are, x=a+x', y=b+y', m which a and b are the co-ordinates of the new origin. Let AX, AY be the primitive axes, and Y/ ,y' xet A'X', A'Y' be the new axes to which it is proposed to refer the same line. Let AB, A'B, the co-ordinates of the new origin, be represented by a and b : let the co-ordinates of any point P relative a b M to the primitive axes be x and y, and the co-ordinates of the same point referred to the new axes be x' and y'. Then we shall have AM=AB+BM, and PM=MM'+PM'; that is, x=a+x', and y—b+y'i which are the equations required. The new origin A' may be placed in either of the four an- gles of the primitive system, by attributing proper signs to a and b. Proposition IX. — Theorem. (29.) The f oi mulcts for passing from a system of rectangula co-ordinates to another system also rectangular are, x=x' cos. oi—y' sin. a, y—x' sin. a-f-y' cos. a, where a represents the angle included between the two axes of X. Let AX, AY be the primitive axes, and AX', AY' be the new axes, and let us designate the co-ordinates of the point P referred to the primitive axes C 34 Analytical Geometry. by x and y, and its co-ordinates referred to the new axes by x', y . Denote the angle XAX' by a. Through P draw PR perpendicular to AX, and PR' perpendicular to AX'; draw R'C perpendicular, and R'B parallel to AX. Then AR=AC-CR. But AR=x. Also, AC = AR'Xcos. XAX'=a;' cos. a, and CR=BR' = PR' sin. BPR'=?/' sin. a. Hence x=x' cos. a— y' sin. a. Also, PR=BR + PB. But PR=?/; BR=R'C = AR' sin. XAX'=.r' sin. «; and PB = PR' cos. BPR'=y' cos. «. Hence y~x' sin. <*+?/' cos. a. Scholium. If the origin be changed at the same time to a point whose co-ordinates, when referred to the primitive sys tern, are a and b, these equations will become x=a+x' cos. a— y' sin. a, y=b+x' sin. a-fy'cos. a. Proposition X. — Theorem. (30.) The formulas for passing from a system of rectangular, tc a system of oblique co-ordinates, are, x—x' cos. a+y' cos. a', y—x' sin. a+y' sin. a', where a and a' denote the inclination of the new axes to the primitive axis of abscissas. Let AX, AY be the primitive axes, y AX', AY' the new axes. Denote the angle XAX' by a, and the angle XAY' by a'. Through P draw PR parallel to AY, and PF parallel to AY' ; draw, also, P'R' parallel to AY, and P'B par- allel to AX. Then AR =AR'+R'R. But AR =x, AR'=AP' cos. XAX' and Hence -x cos. a, R'R=P'B=PP' cos. BP'P=y cos. a'. x=x' cos. a.-\-y' COS. a'. On the Transformation of Co-ordinates. 35 Also, PR=BR+PB. But PR=y, BR=P'R' = AP' sin. XAX'=z' sin. a, and PB=PP' sin. PP'B=y' sin. a'. Hence . y=x' sin. a+y' sin. a'. Scholium. If the origin be changed at the same time to a point whose co-ordinates, referred to the primitive system, are a and b, these equations will become x=a J rx' cos. a+y' cos. a', y=b+x' sin. a+y' sin. a'. Proposition XL — Theorem. (3 1 .) The formulas for passing from a system of rectangular to a system of polar co-ordinates are, x=a+r cos. v, y=b+r sin. v, where r denotes the radius vector, and v the angle which it makes with the axis of abscissas. Let AX, AY be the primitive axes, A' y the pole, and A'D, parallel to AX, be the line from which the variable angle is to be estimated. Designate the angle PA'D by v, the ra- dius vector A'P by r, the co-ordinates of A D R the point P referred to the primitive axes by x and y, and the co-ordinates of A' by a and b. Now AR=AB+BR. But BR= A'D= A'P cos. PA'D=r cos. v. Hence x=a+r cos. v. Also, PR=DR+PD. But PD=A'P sin. PA'D=r sin. v. Hence y=b+r sin. v. Scholium. If the pole A' be placed at the origin A, these equa- tions will become x=r cos. v, y—r sin. v. SECTION IV. ON THE CIRCLE. (32.) A circle is a plane figure bounded by a line, every point of which is equally distant from a point within called the center. This bounding line is called the circumference of the circle. A radius of a circle is a straight line drawn from the center to the circumference. Proposition I. — Theorem. (33.) The equation of the circle, when the origin of co-ordi- nates is at the center, is a:»+y s =R a ; where R is the radius of the circle, and x and y the co-ordi- nates of any point of the circumference. Let A be the center of the circle ; it is required to find the equation of a curve such that every point of it shall be equal- ly distant from A. Represent this dis- tance by R, and let x and y represent the co-ordinates of any point of the curve, as P. Then, by Geometry, Prop. 11, B. IV., AB 2 +BP 2 =AP 2 ; that is, x* + y 2 =R 2 , which is the equation required. (34.) If we wish to determine the points where the curve cuts the axis of X, we must make y=0; for this is the property of all points situated on the axis of ab- scissas. On this supposition we have x=±R; which shows that the curve cuts the axis of abscissas in two points on different sides of the origin, and at a distance from it equal to the radius of the circle. On the Circle. 37 To determine the points where the curve cuts the axis of ordinates, we make x=0, and we obtain y=±R ; which shows that the curve cuts the axis of ordinates in two points on different sides of the origin, and at a distance from it equal to the radius of the circle. (35.) If we wish to trace the curve through the intermediate points, we reduce the equation to the form y=±VR*-x\ Now, since every value of x furnishes two equal values of y, with contrary signs, it follows that the curve is symmetrical above and below the axis of X. If we suppose x to be positive, the values of?/ continually de crease from x—0, which gives ?/=±R, to x=+K, which gives y = 0. If we make x greater than R, y becomes imaginary, which shows that the curve does not extend on the side of the posi- tive abscissas beyond the value of .r=+R. In the same manner it may be shown that the curve does not extend on the side of the negative abscissas beyond the value of.r= — R. Proposition II. — Theorem. (36.) The equation of the circle, when the origin is on the cti- cumference, and the axis of x passes through the center, is *f=2Rx-x\ where R is the radius of the circle, and x and y the co-ordi- nates of any point of the circumference. Let the origin of co-ordinates be at A, y a point on the circumference of the cir- cle. Draw AX, the axis of abscissas, through the center of the circle. Let P be any point on the circumference, and draw PB perpendicular to AX. V J Denote the line AD by 2R, the distance ^ S AB by x, and the perpendicular BP by y ; then BD will be represented by 2R— x. Now BP is a mean proportional between the segments AB and BD (Geom., Prop. 22, Cor., B. IV.) ; that is. 3S Analytical Geometry. BP=ABxBD, or y*=x(2R-x)=2Rx-x\ which is the equation required. (37.) If we wish to determine where the curve cut,* the axis of X, we make y—0, and we obtain z(2R— x) = 0. This equation is satisfied by supposing x=0, or 2R— x=0, from the last of which equations we derive r=2R. The curve, therefore, cuts the axis of abscissas in two points, one at the origin, and the other at a distance from it equal to 2R. To determine where the curve cuts the axis ofordinates, we make x=0, which gives y=o, which shows that the curve meets the axis of urdinates in but one point, viz., the origin. Proposition III. — Theorem. (38.) The most general equation of the circh is {x-xy+{y-yy=R\ where R denotes the radius of the circle, x' and y' are the co- ordinates of the center, and x and y the co-ordinates of any point of the circumference. Let C be the center of the circle, and Y assume any rectangular axes AX, AY. Let the co-ordinates AB, BC of the cen- ter be denoted by x' and y' ; while the co-ordinates of any point P in the cir- cumference are denoted by x and y. Then, if we draw the radius CP, and Ar~ jj ' X CD parallel to the axis of X, we shall have CD^=x—x', and VD—y—ij'. But CD a +PD s =CP\ Hence we have (x— x'y-\-{y— y'Y—R 2 , which is the equation sought. (39.) To find the points where the curve intersects the axis of X, we must make y=0, which gives, (x-x'Y+y'^K 2 , 0\ the Circle. 30 whence (x-x'Y=R*-y x=x' or j;=z'±VR"-y' 1 , where we see that the values of £ will become imaginary when y' exceeds R ; and it is evident that if the distance of the cen- ter of the circle from the axis of abscissas exceeds the radius of the circle there can be no intersection. To find the point where the curve intersects the axis of Y, we must make x=0, which gives y=y'± VR'-x'"; which becomes imaginary when x' exceeds R, and it is plain that in this case there can be no intersection. Proposition IV. — Theorem. (40.) The equation of a tangent line to the circle is xx'-\-yy'=JV, where R denotes the radius of the circle, x' and y' are the co- ordinates of the point of contact, and x and y are the general co-ordinates of the tangent line. Let BC be a line touching the circle, whose center is A, in the point P. Let the co-ordinates of the point P be x' and y', and draw the radius AP. The equation of the line AP, passing through the origin and through the point x', y', Art. 21, is y' y=x< x - Now a tangent is perpendicular to the radius at the point of contact (Geom., Prop. IX., B. III.). But the equation of a line passing through a given point, and perpendicular to a given line, Art. 25, is y-y'=--(x-x'). The value of a', taken from the equation of the radius, is y' 40 Analytical Geometry. 1 x> cience := :. a' y 1 The equation of the tangent line is, therefore, x' Clearing of fractions and transposing, we obtain xx' -\-yy'= x' 2 + y n . But since the point P is on the circumference, its co-ordinates must satisfy the equation of the circle ; that is, x' 2 + ij'"-=R\ Hence xx'+yy'=~R."\ which is the equation required. (41.) The equation of the tangent may also be obtained, without employing the geometrical property above referred to, by a method which is applicable to all curves whatever. Let us first consider a line BC, meeting. the curve in two points P' and P ;/ ; the co-ordinates of P' be- ing represented by x', y', and those of P" by x", y". The equation of the line BC, Art. 20, is y-2/-fr^-<>; (1) and, since both the points P' and P" are on the circumference, we must have x n +y" =R\ * (2) and x"*+ij"'=R\ (3) Subtracting equation (3) from equation (2), we obtain ij n -y" 2 +x r2 -x"*=0 ; that is, (y'+y") (y'-y")+(x'+x") (x'-x")=o. y'—y" x'+x" whence — -— — — -. x —x y +y Substituting this value in equation (1), we obtain y-y- x'+x" , _ ,, 'y'+y" {X Xh (4) If dow we suppose the secant BC to move toward the point P, tne point P' will approach P" ; and when P' coincides with P", the secant line will become a tanoent to the circumference. On THE C I B C L E. 41 When this takes place, x' will equal x", and y' will equal y'\ and the last equation becomes x' y-y'=--,( x -*'), as before found. (42.) To determine the point in which the tangent intersects the axis of X, we make y=0, which gives xx'—TL"', R 2 or x=— =AC. x' To determine the point in which the tangent intersects the axis of Y, we make x=0, which gives yy'=R>, or R 2 y=— =AB. y Proposition V. — Problem. (43.; Given the base of a triangle, and the sum of the squares of its sides, to determine the triangle. Let AB be the base of the proposed trian- gle. Bisect AB in C ; draw CY perpendicu- .ar to AB, and assume YC, CB as a system of rectangular axes. Let x and y be the co-ordinates of P, the vertex of the triangle, and from P let fall the perpendicular PD. Let a denote AC or CB, and put m for the sum of the squares of the sides AP, BP. Then, by Geom., Prop. XL, B. IV., we shall have PD 2 +AD 2 =AP 2 , and PD 2 +BD 2 =BP 2 ; or y- + (x+ay = AV\ and if + (x-ay=B¥\ Adding these equations together, we obtain 2y 1 -\-2x*+2a'=AV i + BV'=m. 42 Analytical Geometry. Whence . m Comparing this result with Art. 33, we see that this equation represents a circle whose center is the origin C, and the radius V- so that if this circle be described, and lines be drawn from A and B to any point in its circumference, a triangle will be formed which satisfies the proposed conditions. Proposition VI. — Theorem. (44.) The polar equation of the circle, when the origin is on the circumference, is r=2R cos. v, where R represents the radius"of the circle, r the radius vector, and v the variable angle. The equation of the circle referred to rectangular axes, when the origin is on the circumference, Art. 36, is y'=2Rx-x\ (1) Let A be the position of the pole, and AX the line from which the varia- ble angle is estimated. The formulas for passing from a system of rectan- gular to a system of polar co-ordi- nates, the origin remaining the same, Art. 31, are • x=r cos. v, y=r sin. v. Squaring each member of these equations, and substituting the values of a; 3 , ?/ 2 , thus found in equation (1), we obtain r~ sin. 2 u=2Rr cos. v— r 2 cos. 2 v ; or, by transposition, r 2 (sin. 2 u+cos. 2 v)—2Rr cos. v. But sin. 2 u + cos. 2 v is equal to unity. Hence r 2 =2Rr cos. v ; or, dividing by r, we obtain r=2R cos. v, which is the polar equation of the circle. (45.) This equation might have been derived directly from the figure. Thus, by Trig., Art. 41, On the C ni c L E. 43 radius : AB : : cos. BAP : AP, or 1 : 2R : : cos. v : r ; whence r=2R cos. v. (40.) When v=0, the cos. v=l, and we have r=2R=AB. As v increases from to 90°, the radius vector determines all the points in the semicircumference BPA ; and when u=90°, then cos. v=0, and we have r=0. From u=270° to v=360°, the radius vector will determine all the points of the semicircumference below the axis of ab- scissas. Examples. 1. On a circle whose radius is 6 inches, a tangent line is drawn through the point whose ordinate is 4 inches : see figure, Art. 42. Determine where the tangent line meets the two axes. Ans. AC= ; AB = 2. Find the angle which the tangent line in the preceding ex- ample makes with the axis of X. Ans. 3. Find the point on the circumference of a circle whose ra- dius is 5 inches, from which, if a radius and a tangent line be drawn, they will form, with the axis of X, a triangle whose area is 35 inches. Ans. Abscissa = ; ordinate = 4. The radius of a circle is 5 inches, and the variable angle is 36 degrees, the pole being on the circumference ; determine the radius vector. Ans. 5. The radius of a circle is 5 inches, and the radius vector is 8 inches ; determine the variable angle. Ans. 6. The radius vector of a circle is 16 inches, and the variable angle is 42 degrees ; determine the radius of the circle. Ans. , SECTION V. ON THE PARABOLA. (47.) A parabola is a plane curve, every point of which is equally distant from a fixed point and a given straight line. The fixed point is called the focus of the parabola, and the given straight line is called the directrix. Thus, if F be a fixed point, and BC a b given line, and the point P move about F D in such a manner that its distance from F is always equal to the perpendicular distance from BC, the point P will describe a parab- ola of which F is the focus and BC the di- rectrix. The distance of any point of the curve from the focus, is called the radius vector of that point. (48.) From the definition of the parabola, the curve may be described mechanically. Let BC be a ruler laid upon a plane, and let DEG be a square. Take a thread equal in length to DG, and attach one ex- D tremity at G, and the other at some point, as F. Then slide the side of the square DE along the ruler BC, and at the same time keep the thread continually tight by means of the pencil P; the pencil will de- scribe one part of a parabola, of which F is the focus, and BC the directrix. For in every position of the square, PF+PG=PD+PG, and hence PF=PD; that is, the point P is equally distant from the focus F and the directrix BC. If the square be turned over and moved on the other side of On the Parabola. 15 :; ;- the point F, the other part of the same parabola may be de scribed. (49.) A diameter is a straight line drawn through any point of the curve perpendicular to the directrix. The vertex of the diameter is the point in which it cuts the curve. The axis of the parabola is the diameter which passes through the focus. v The parameter of a diameter is the double ordinate which passes through the focus. Proposition I. — Theorem. (50.) The equation of the parabola, referred to rectangular axes whose origin is at the vertex of the axis, is y*=2px, where x and y are the general co-ordinates of the curve, and 2p is the parameter of the axis. Let F be the focus, and DC the di- rectrix. Take AX as the axis of ab- scissas, and let the origin be placed at A, the middle point of BF. Represent BF by p, whence AF will equal — . Let x and y be the co-ordinates of any point P in the curve, and represent FP by r. By the definition of the curve, Also, But that is, PF=PD=AR+AB=x+|. FR=s-f. PR 2 +FR 2 =PF 2 ; y'+ (*-!)'=(-: Whence, by expanding, we obtain y*=2px. (51.) Cor. I. If we make x=0, we have y=0, which shows that the curve passes through the origin A. P If we make x=jr, we shall have 46 Analytical Geometry. y-p< or y— p ; whence 2y=2p ; that is, the constant quantity 2p, called the parameter, is equal to the double ordinate through the focus, conformably to the definition, Art. 49. Cor. 2. From the equation of the parabola we obtain y=± V2px, which shows that for every value of x there will be two equal values of y, with contrary signs. Hence the curve is sym- metrical with respect to the axis of X. Cor. 3. If we convert the equation y 2 =2px into a propor lion, we shall have x : y : : y : 2p ; that is, the parameter of the axis is a third proportional to any abscissa and its corresponding ordinate. Cor. 4. The squares of ordinates to the axis are to each other as their corresponding abscissas. Designate any two ordinates by y'. y", and the correspond ng abscissas by x', x", then we shall have y 12 =2px', and y" 2 =2px". Hence y' 2 : y"* : : 2px' : 2px" : : x' : x". Proposition II. — Theorem. (52.) The equation of a tangent line to the parabola is yy'=p{x+x'). where x', y' are the co-ordinates of the point of contact, and p is half the parameter of the axis. Draw any line P'P", cutting the parabola in the points P', P" ; if this line be moved toward P, it will ap- proach the position of the tangent, and the secant will become a tangent when the points P', P" coincide. Let x', y' be the co-ordinates of the point P', and x", y" the co-ordinates of the point P". The equation of the line passing through these two points, Art. 20, will be On the Pauabola. 4T , v' — y" This is the general equation of a straight line passing through two given points, and has no special reference to the parabola. fn order to make it the equation of a secant line to the parab- ola, we must deduce from the equation of the curve the value of the coefficient of x—x', and substitute it in equation (1). Thus, since the points P' and P" are on the curve, we shall have y" =2px' (2). y"*=2px" (3). Subtracting equation (3) from (2), we nave y l2 -y""-=2p(x'-x"). Whence £z^ = _^_, x'—x y' ' -\-y" Substituting this value in equation (1), the equation of the secant line becomes 2v y-y'=y-r^-^~ x ')- ( 4 ) The secant will become a tangent when the points P', P" coincide, in which case x'=x" and y'—y". Equation (4) in this case becomes which is the equation of a tangent to the parabola at the point P. If we clear this equation of fractions, we have yy'-y"=px-px'. But y"=2px'. Hence yy'=px— px'+2px', or yy'=p{x-\-x'). (53.) Definition. A subtangent is that part of a diameter intercepted between a tangent and ordinate to the point of contact. Cor. 1. To find the point in which the tangent intersects the axis of abscissas, make y=0 in the equation of the tangent, and we have 0=p(x+x') 48 Analytical Geometry. that is, x=—x'. or AT=-AR; that is, the subtangent is bisected at the vertex. Cor. 2. This property enables us to diaw a tangent to the curve through a given point. Let P be the given point; from P draw PR perpendicular to the axis, and make AT=AR. Draw a line through P and T, and it will be a tan- gent to the parabola at P. Schol. In the equation P — represents the trigonometrical tangent of the angle which the tangent line makes with the axis of the parabola. (54.) Definitions. A normal is a line drawn perpendicular to a tangent from the point of contact, and terminated by the axis. A subnormal is the part of the axis intercepted between the normal and the corresponding ordinate. Proposition III. — Theorem. (55.) The equation of a normal line to the parabola is y' y-y'=—-(x-x'), where x', y' are the co-ordinates of the point of intersection with the curve. The equation of a straight line passing through the point whose co- ordinates are x', y', Art. 18, is y-y' = a(x-x')', (1) and, since the normal line is per- T- pendicular to the tangent, we shall have, Art. 23, 1 a= -. a But we have found for the tangent line, Prop II., Schol. On the Parabola. 49 Hence a P y' a— y' V Substituting this value in equation (1), we shall have for the ^quation of the normal line y y-y'=--(x-x'). (2) (56.) Cor. To find the point in which the normal intersects the axis of abscissas, make y=0 in equation (2), and we have, aftei reduction, x— x'=p. But x is equal to the distance AN, and x' to AR ; hence x—x'=p is equal to RN ; that is, the subnormal is constant, and equal to half the parameter of the axis. Proposition IV. — Theorem. (57.) The normal, at any point of the parabola, bisects the an gle made by the radius vector and the diameter passing through that point. Let PT be a tangent to a parabola, PF the radius vector, PN the normal, and PB the diameter to the point P ; the normal PN bisects the angle BPF. Let x' represent the abscissa of the point P. Now FN-AR+RN-AF. But KK=x', RN=p, and AF=-|. Hence But in Prop. I. we found FN=x'+p-j-=x'+±. Hence FN=FP. Therefore tne angle FPN=FNP=the alternate angle BPN, (W.) Cor. FR=AR-AF=rr'-|. But ril=2x', Prop. II., Cor. 1. D so Analytical Geometry. Hence TF=TR-FR= :s'+f=PF; that is, if a tangent to the parabola cut the axis produced, the points of contact and of intersection are equally distant from the focus. Proposition V. — Theorem. (59.) If a perpendicular be drawn from the focus to any tan- gent, the perpendicular will be a mean proportional between the distances of the focus from the vertex and from the point of contact. Let FB be a perpendicular drawn .from the focus to the tangent PT. Join AB, and draw the ordinate PR. Since FT is equal to FP (Prop. IV., Cor.), and FB is drawn perpendicular ^ to PT, PB is equal to BT. But RA is equal to AT, Prop. II., Cor. 1 ; hence TB : BP : : TA : AR, and therefore AB is parallel to PR. But PR is perpendicular to the axis ; hence AB is perpendicular to TF ; and therefore, by similar triangles, FAB, FBT, we have FA : FB : : FB : FT or FP. Proposition VI. — Theorem. (60.) The equation of the parabola referred to a tangent line, and the diameter passing through the point of contact, the origin being the point of contact, is y'=2p'x, where 2p' is the parameter of the diameter passing through the origin. • The formulas for passing from rectangular to oblique axes are (Art. 30, Schol), x=a+x' cos. a+y' cos. a', (1) y=b+x' sin. a-\-y' sin. a 1 . (2) Since the new origin is to be on the curve, its co-ordinates must satisfy the equation of the curve ; that is, l i =2pa, whence a— rr. On the Parabola. 51 Also, 2 f=j;(2Ax'-x") ; or, omitting the accents, R 2 y*=ji(i&x-x-), which is the equation of the ellipse referred to the vertex oi the major axis as the origin of co-ordinaies. On the Ellipse. G\ Proposition III. — Theorkm. (71.) The square of any ordinate is to the product of the parts into which it divides the major axis, as the square of the minor axis is to the square of the major axis. The equation of the ellipse, re- ferred to the vertex A' as the origin of co-ordinates, is, Art. 70, B a y^=j i (2A-x)x. This equation may be resolved into the proportion if : {2k-x)x : : B 2 : A 2 . Now 2A represents the major axis AA', and, since x repre- sents A'R, 2 A— x will represent AR ; therefore (2 A— x)x rep- resents the product of the parts into which the major axis is divided by the ordinate PR. Cor. It is evident that the squares of any two ordinates are as the products of the parts into which they divide the major axis. Scholium. It may be proved in a similar manner that the squares of ordinates to the minor axis are to each other as the products of the parts into which they divide the minor axis. Proposition IV. — Theorem. (72.) If a circle be described on the major axis of an ellipse, then any ordinate in the circle is to the corresponding ordinate in the ellipse, as the major axis is to the minor axis. If we represent the ordinate PR in the ellipse by y', and the ordinate P'R in the circle corresponding to the same abscissa A'R by Y', the equation of the ellipse will give us, a[ by Art. 69, >/"=|(A 2 -* 2 ), and the equation of the circle will give, Art. 33, Y' 2 =(A 3 -.r 2 ). 62 Analytical Geometry. Combining these two equations, we have ?/' 2 =— -Y' 2 y A a* » or y<: A* ' whence we derive the proportion Y' :y' :: A : B : : 2A : 2B. (73.) Cor. In the same manner, it may be proved that if a circle be described on the minor axis of an ellipse, any ordinate drawn to the minor axis is to the corresponding ordinate in the circle, as the major axis is to the minor axis. If we represent the ordinate PR in the ellipse by x', and the correspond- ing ordinate P'R in the circle by X', we shall have, Prop. I, and X'*=B 2 -y\ Combining these two equations, we have A 2 X — jj- 2 A , or B X' whence we derive the proportion x' : X' : : A : : B : : 2A : 2B. Proposition V. — Theorem. f (74.) Every diameter of an ellipse is bisected at the center. Let PP' be any diameter of an ellipse. Let x', y' be the co-ordi- nates of the point P, 'and x", v" those of the point P'. Then, from the equation of the ellipse, we shall have. Art. 69 y'-|i(A 9 -a: rt ), 3r=4 a (A s ~s" 9 ); whence On the Ellipse. y" _A'-x" 63 But from the similarity of the triangles PCR, P'CR' we have whence y' _ x' x" 2 A'-x" 1 ' Clearing of fractions, we obtain „/2 ~//» . whence, also, Consequently, or that is, x' = X' y'*=y"\ x>*+y"=x" 2 +y"\ CF=CP /a ; CP =CF. Proposition VI. — Theorem. (75.) If from the vertices of the major axis, two lines be drawn to meet on the curve, the product of the tangents of the angles which they form with it, on the same side, will be negative, and equal to the square of the ratio of the semi-axes. The equation of the line AP, passing through the point A, whose co-ordinates are x'=A, y'=0, Art. 18, is y=a(x-A). The equation of A'P, passing through the point A', whose co- ordinates are x" = — A, y"=0, Art. 18, is y—a'{x+A). These lines must pass through the point P in the ellipse. Hence, if we represent the co-ordinates of P by x" and y", we have the three equations y " =a( x "-A) y" =aV'+A) . y'liJ^x-i-A*) Multiplying (1) and (2) together, we have y" 2 = aa'(x" 2 -A 2 ). Hence, comparing with (3), we see that (1) (2) (3) G4 Analytical Geometky. (16.) Scholium. Two lines which are drawn from the same point of a curve to the extremities of a diameter, are called sup- plementary chords. Cor. In the circle, which may be considered an ellipse whose two axes are equal to each other, we have aa'— — 1, which shows that the supplementary chords are perpendicular to each other (Art. 24). Proposition VII. — Theorem. (77.) The equation of a straight line which touches an ellipse is Ayij'+B\xx'=A*B\ where x and y are the general co-ordinates of the tangent line, x' and y' the co-ordinates of the point of contact. Draw any line, P'P", cutting the ellipse in the points P', P" ; if .his line be moved toward P, it will approach the tangent, and the secant will become a tangent when the points P', P" coincide. Let x', y' be the co-ordinates of the point P', and x", y" the co-ordinates of the point P' The equation of the line P'P", passing through these two points Art. 20, will be y'-y r y-y'- 7i( x - x ')- (1) x'-x' Since the points P', P" are on the curve, we shall have A'y" +BV 2 =A 2 B% (2) and Ay 2 +BV"=A ! B 2 . (3) Subtracting equation (3) from (2), we have A*(y'*-ij' / ')+B\x' 2 -x ,r -) = 0, or A'iy'-y") {y'+y") = -B\x'-x n ) (x'+x"). tj'-y" _ B 2 / x'+x" \ x'-x"~~A*\y' + tj")' Substituting this value in equation (1), the equation of the secant line becomes Whence On the Ellitse. B'x'+x" 65 y-y'=- T&-X% (4) The secant P'P" will become a tangent when the points P' f P" coincide, in which case x'=x" and y'=y". Equation (4), in this case, becomes B'x' y-y'=-jt^( x - x ')> which is the equation of a tangent to the ellipse at the point P. If we clear this equation of fractions, we have A a yy'-A*y» = -B\vx'+B'x , ^ 2 , or A 7 yy'+B 2 xx'=Ay"+B\v r ' *». J x ^ hence A'yij'+B'xx^A^B', which is the most simple form of the equation of a tangent line. (78.) Cor. 1. In the equation y-y Wx A*y 0> B 2 x' — -r- 2 — represents the trigonometrical tangent of the angle A-y which the tangent line makes with the major axis. Cor. 2. To find the point in which the tangent intersects the axis of abscissas, make y—0 in the equation of the tangent, and we have A 2 x—— • x' which is equal to CT. If from CT we subtract CR or x', we shall have the sub- tangent RT: A 2 -: x' x' Cor. 3. This expression for the subtangent is independent of the minor axis ; the subtangent is, therefore, the same for all ellipses having the same major axis ; it consequently belongs to the circle described upon the major axis. Cor. 4. Hence we are enabled to draw a tangent to an el° Hpse through a given point. Let P be the given point. On 66 Analytical Geometry. AA' describe a circle, and through P draw the ordinate PR, and produce it to meet the circumference of the circle in P'. Through P' draw the tan- gent P'T, and from T, where it meets the major axis produced, draw PT ; it will be a tangent to the ellipse at P. Cor. 5. Since the co-ordinates of the point P are equal to those of the point P', it follows from Cor. 1 that the tangents at the ex- tremities of a diameter make equal angles with the major axis, and are therefore parallel with each other. Hence, if tangents are drawn through the vertices of any two di- ameters, they will form a parallelogram circumscribing the ellipse. Proposition VIII. — Theorem. (79.) The equation of a normal line to the ellipse is y-y Ay where x and y are the general co-ordinates of the normal line, x' and y' the co-ordinates of the point of in f ovsection with the ellipse. The equation of a straight line passing through the point whose co-ordinates are x', y', Art. 18, is y-y'=a(x-x'); (l) and, since the normal line is per- pendicular to the tangent, we shall have, Art. 23, 1 But we have found for the tangent line, Prop. VII., Cor. 1, BV a '~ ~TT~> Ay Hence On the Ellipse. Ay 67 a— BV Substituting this value in equation (1), we shall have for the equation of the normal line (2) (80.) Cor. 1. To find the point in which the normal inter- sects the axis of abscissas, make y=0 in equation (2), and we have, after reduction, A 2 -B 2 If we subtract this value from CR, which is represented by x', we shall have the subnormal A 2 -B 2 BV NR=z' jt- z-^r- A 2 — B 2 Cor. 2. If we put e 2 for — -£-— , Art. 69, Cor. 6, we shall have CN=eV. If to this we add F'C, which equals c or Ac, Prop. I., Cor. 6, we have F / N=Ae+eV=e(A+ca;') ) which is the distance from the focus to the foot of the normal. Proposition IX. — Theorem. (81.) The normal at any point of the ellipse bisects the angle formed by lines drawn from that point to the foci. Let PT be a tangent line to an ellipse, and PF, PF' two lines drawn to the foci. Draw PN, bi- secting the angle FPF'. Then, by Geometry, Prop. XVII., B. IV., ^~ FP:F'P::FN:F'N; or, by composition, FP+F'P : FF' : : F'P : FN. (1) But FP+F'P=2A. Also, FF' = 2c=2Ae, Prop. I., Cor. 6, and F'P=A+er, Prop. I., Cor. 7. Making these substitutions in proportion (1), we have 68 Analytical Geometry 2A : 2Ae : : A+ex : F'N. Hence F'N=e(A+e.r). But by Prop. VIII., Cor. 2, e(A+ex) represents the distance from the focus F' to the foot of the normal. Hence the line PN, which bisects the angle FPF', is the normal. (82.) Cor. 1. Since PN is perpendicular to TT', and the angle FPN is equal to the angle F'PN, therefore the angle FPT is equal to the angle F'PT' ; that is, the radii vectores are equally inclined to the tangent. Cor. 2. This proposition affords a method of drawing a tan- gent line to an ellipse at a given point of the curve. Let P be the given point; draw the radii vectores PF, PF'; pro- duce PF' to G, making PG equal to PF, and draw FG. Draw PT perpendicular to FG, and it will be the tangent required ; for the angle FPT equals the angle GPT, which equals the vertical angle F'PT'. Proposition X. — Theorem. (83.) If, through one extremity of the major axis, a chord be drawn parallel to a tangent line to the curve, the supplementary chord will be parallel to the diameter which passes through the point of contact, and conversely. Let DT be a tangent to the ellipse, and let the chord AP be drawn parallel to it ; then will A'P be parallel to the W diameter DD', which passes through the point of contact D. Let x', y' designate the co- ordinates of D ; the equation of the line CD will be, Art. 15, y'=pa'x' ; V' whence a'——. x' But, by Prop. VII., Cor. 1, the tangent of the angle which the tangent line makes with the major axis, is BV a ~ TT7- Ay On the Ellipse. G9 Multiplying together the values of a and a', we obtain aa = — 5! A" which represents the product of the tangents of the angles which the lines CD and DT make with CT. But by Prop. VI. the product of the tangents of the angles B a A 2 ' TAT, PA'A is equal to - Hence, if AP is parallel to DT, A'P will be parallel to CD, and conversely. (84.) Cor. Let DD' be any diameter of an ellipse, and DT the tangent drawn through its vertex, and let the chord AP be drawn parallel to DT ; then, by this Proposition, the supple- mentary chord A'P is parallel to DD'. Let another tangent ET' be drawn parallel to A'P, it will also be parallel to DD'. Let the diameter EE' be drawn through the point of contact E; then, by this Proposition, A'P being parallel to T'E, AP (and, of course, DT) will be parallel to EE'. Each of the diameters DD', EE' is therefore parallel to a tangent drawn through the vertex of the other, and they are said to be conjugate to one another. Scholium. Two diameters of an ellipse are said to be con- jugate to one another, when each is parallel to a tangent line drawn through the vertex of the other. If we designate by a and a' the tangents of the angles which two conjugate diameters make with the major axis, then we must have B 2 Proposition XI. — Theorem. (85.) The equation of the ellipse, referred to its center and conjugate diameters, is A'y+B'V=A' 2 B' 2 ; vvhere A' and B' are semi-conjugate diameters. 70 Analytical Geometry. The equation of the ellipse, referred to its center and axes, Art. 68, is Ay+BV=A 2 B\ In order to pass from rectangular to oblique co-ordinates, the origin remaining the same, we must substitute for x and y in the equation of the curve, Art. 30, the values x=x' cos. a+y' cos. a', y=x' sin. ct+y' sin. a'. Squaring these values of x and y, and substituting in the equation of the ellipse, we have A 2 sin. 2 a' B 2 cos. 2 a y /2 +2A 2 sin. a sin. a'lz'y'+A 2 sin. 2 a B 2 cos. 2 a .7j' 2 =A 2 B 2 ; (1) 2B 2 cos. a cos. a' I which is the equation of the ellipse when the oblique co-ordi- nates make any angles a, a' with the major axis. But since the new axes are conjugate diameters, we must have (Art. 84) B 2 aa'= rr, A * B 2 or tang, a tang. a'= — -^ ; whence A 2 tang, t tang. a'+B 2 =0. Multiplying by cc. a cos. a', remembering that cos. a tang.a=sin. a, we have A 2 sin. a sin. a'+B 2 cos. a cos. a'=0. Hence the term containing x'y' in equation (1) disappears, and we have (A 2 sin.V+B 2 cos.V)3/' 2 + (A 2 sin. 2 a +B 2 cos. 2 «)x' 2 =A 2 B\ (2) which is the equation of the ellipse referred to conjugate diam- eters. If in this equation we make y'=0, we shall have A sin. a+B" cos. a • If we make x' — Q, we shall have A 2 B 2 which equals — ri , be- cause DD' and EE' are conjugate diameters, Prop. X., Schol. Hence, by squaring each member of this equation, we have Ayy a =BW /B . (1) But because the points D and E are on the curve, we have Ay 2 =A 2 B 2 -BV 2 , and Ay /2 =A 2 B 2 -BV 2 . Therefore, by multiplication, Ay Y /2 = A 4 B 4 - A 2 B V 2 - A 2 B V 2 + B*x"x " 2 . (2) Comparing equation (1) with equation (2), we see that A 4 B 4 - A 2 BV 2 - A 2 BV' 2 =0 ; or, dividing by A 2 B 4 , we have A 2 -x"-x"'=0, or A 2 =.r' 2 +.r" 2 . (3) In the same manner, we find that B 2 =*/' 2 +y" 2 . (4) Hence, by adding equations (3) and (4), we have A 2 +B 2 =:e ,2 + ?/' 2 +x-" 2 +?/" 2 =A' 2 +B' 2 . (89.) Cor. According to this Proposition, Equation (3), x'^A'-x"*. Also, from the equation of the ellipse, Art. 68, Ay 2 =B 2 (A 2 -a:" 2 ). A 2 Hence X ' 2= ^V"\ or s' = -gy". Tn the same manner, we find B y'=jx". On the Ellipse. 73 Proposition XIV. — Theorem. (90.) If from the vertex of any diamster straight li?ies are drawn to the foci, thei^ product is equal ti the square of half the conjugate diameter. Represent the co-ordinates of the point D, referred to rectangular axes, by x', y'. Then the square of the distance of D from the center of the ellipse is A n =x r -+ij'\ But from the equation of the curve, Art. 68, A 2 Therefore, by substitution, A 2 =B 2 +eV 2 , Art. 69. But, by Prop. XIII., A' 2 +B' 2 =A 2 +B 2 . Therefore, B' 2 =A 2 -e 2 ^ Also, by Prop. I., Cor. 7, :A 2 - e'x' Hence '=B' Proposition XV. — Theorem. (91.) The parallelogram, formed by drawing tangents through the vertices of two conjugate diameters, is equal to the rectargle of the axes. Let DED'E' be a parallelo- gram, formed by drawing tan- gents to the ellipse through the vertices of two conjugate diameters DD ; , EE'; its area is equal to AA'xBB'. Let the co-ordinates of D, referred to rectangular axes, be x', y', and those of E be x", y". The triangle CDE is equal to the trapezoid DEHG, dimin- ished by the two triangles DCG, EHC. That is, 74 Analytical Geometry. 2.CDE = (x'+x")(y'+y")-x ! y'-x"y", =x'y" J rx"y', = *'x +2/ 'tp h y Pr °p- XIIL ' Cor " BV 2 + Ay 2 reducing the fractions to a common A.B ' denominator, A 2 B 2 =—-=- = A.B, Art. G8. A.B Therefore the parallelogram CETD is equa to A.B ; and the parallelogram DED'E' is equal to 4A.B or 2Ax2B=AA'xBB'. Proposition XVI. — Theorem. (92.) The polar equation of the ellipse, when the pole is at one of the foci, is r= E , 1+e cos. v where p is half the parameter, e is the eccentricity, and v is the angle which the radius vector makes with the major axis We have found the distance of any point of the ellipse from the focus, Prop. I., Cor. 7, to be r=FP=A-ex, r'=F'P=A+ex, where the abscissa x is reckoned from the center. In order to transfer the origin from the center to the focus F, we must substitute for x, x'+c; or, putting Ae for c, Art. 69, we have x=x' + Ae. If we represent the angle PFA by v, we shall have x' = r cos. v. Whence x=r cos. v+Ae. Therefore FP=r=A— er cos. v — Ae 2 . By transposition, r(l+e cos. v) = A— Ae 2 =A(l— e a ). • 2B * / A If we put 2p= the parameter of the major axis =— (Art. 87), Ave shall have p=A(l-e 2 ), Prop. L, Cor. 6. P Whence r=— . 1+e cos. v On the Ellipse. Proposition XVII. — Theorem. (93.) The area of an ellipse is a mean proportional between the two circles described on its axes. Let A A' be the major axis of an ellipse ABA'B'. On AA', as a di- ameter, describe a circle ; inscribe in the circle any regular polygon AM'MA', and from the vertices M, M', etc., of the polygon draw per- -^j pendiculars to AA'. Join the points B, P, etc., in which the perpendic- ulars intersect the ellipse, and there will be inscribed in the ellipse a polygon of an equal number of sides. Let Y, Y' be the ordinates of the points M, M', and y, y the ordinates of the points P, B, corresponding to the same ab- scissas x, x'. Y+Y' The area of the trapezoid M'MRC= — (.-b— xO- The area of the trapezoid BPRC= BPRC y+y 1 -— (x-x% Whence But, by Prop. IV Whence consequently, M'MRC" B A y+y' Y+Y' 2/=X Y; y '~ I- Y+Y' BPRC B A ; B = A' M'MRC In the same manner it may be proved that each of ihe trape- zoids composing the polygon inscribed in the ellipse, is to the corresponding trapezoid of the polygon inscribed in the circle, in the ratio of B to A ; hence the entire polygon inscribed in the ellipse, is to the polygon inscribed in the circle, in the same ratio. Hence, if we represent the two polygons by p and P, we shall have p_B P~A* 76 Analytical Geometsy. Since this relation is true whatever be the number of sides of the polygons, it will be true when the number of the sides is indefinitely increased ; that is, it is true for the ellipse and the circle, which are the limits of the surfaces of the polygons. Therefore, if we represent the surfaces of the ellipse and circle by s and S, we shall have • B B S = A' 01 5=S A- But the area of a circle whose radius is A, is represented b) ttA 2 e hence the surface of the ellipse is 7rA 2 ^ = 7rAB, A which is a mean proportional between the two circles de- scribed on the axes. For the area of the circle described on the major axis is ttA 2 ; and the area of that described on the minor axis is 7rB 2 ; and 7rAB is a mean proportional between them. Proposition XVIII. — Theorem. Any chord which passes through the focus is a third propor tional to the major axis and the diameter parallel to that chord Let PP' be a chord of the ellipse passing through the focus F, and let DD / be a diameter parallel to PP'. By Art. 92, PF = r= — -£- . 1+e cos. v H we substitute for v, 180° -\-v, we shall have the value of P / F=r / = V Hence we have PP By Art. 85 CD 3 = -.r+r' = A 3 B 2 1 — e cos. v 2p 1 — e 3 cos. 2 v A 2 B 2 A 2 sin. 2 v + B 2 cos. 2 v ~ A 2 sin. 2 v T (A 2 -AV) cos. V A 2 !^ A 2 (l-e 3 ) _ kp A 2 — A 2 e 2 cos. 2 v 1 — e 2 cos. 2 y~l— e 3 cos. V Hence PP' : CD 3 : : 2 : A ; n, PP' : 2CD : : CD : AC ; and AA' : DD' : : DD' : PP'. This property includes Cor. 5, Prop. I., page 59. SECTION VII. ON THE HYPERBOLA. (94.) An hyperbola is a plane curve in which the difference of the distances of each point from two fixed points is equal to a given line. The two fixed points are called the foci. Thus, if F and F' are two fixed points, and if the point P moves about F in such a manner that the difference of its distances from F and F' is al- ways the same, the point P will de- scribe an hyperbola, of which F and F' are the foci. If the point P' moves about F' in such a manner that P'F— P'F' is always equal to PF' — PF, the point P' will describe a second hyperbola similar to the first. The two curves are called opposite hyperbolas. (95.) This curve may be described by continuous motion as follows : Let F and F' be any two fixed points. Take a ruler longer than the distance FF', and fasten one of its extremities at the point F'. Take a thread shorter than the ruler, and fasten one end of it at F, and the other to the end M of the ruler. Then move the ruler MPF' about the point F'. while the thread is kept constantly stretch- ed by a pencil pressed against the ruler ; the curve described by the point of the pencil will be a portion of an hyperbola. For, in every position of the ruler, the difference of the lines PF, PF' will be the same, viz., the difference between the length of the ruler and the length of the string. If the ruler be turned, and move on the other side of the point F, the other part of the same hyperbola may be described 78 Analytical Geometry. Also, if one end of the ruler be fixed in F, and that of the thread in F', the opposite hyperbola may be described. (96.) The center of the hyperbola is the middle point of the straight line joining the foci. A diameter is a straight line drawn through the center, and terminated by two opposite hyperbolas. The transverse axis is the diameter which, when produced, passes through the foci. The parameter of the transverse axis is the double ordinate which passes through one of the foci. Proposition I. — Theorem. (97.) The equation of the hyperbola, referred to its center and axes, is Ay-BV=-A 2 B 2 , where A and B represent the semi-axes, and x and y are the general co-ordinates of the curve. Let F and F' be the foci, and draw the rectangular axes CX, CY, the origin C being placed at the middle of FF'. Let P be any point of the curve, and draw PR perpen- dicular to CX. Let the difference of the distances of the point P from the foci be represented by 2A. De- note the distance CF or CF by c, FP by r, F'P by r< ; and let x and y represent the co-ordinates of the point P. Then, since FP 2 = PR 2 + RF, we have r=*/ 2 + (r-c)\ (1) Also, F'P 2 =PR 2 +RF' 2 ; that is, r' 2 -?/ 2 +(x+c) 2 . (2) Adding equations (1) and (2), we obtain r °- + r r -= 2 (y+;r+c 2 ) ; (3) and subtracting equation (1) from (2), we obtain r ' 2 — r^—^cx, which may be written (r' + r)(r'-7-) = 4cx. (4) But, from the definition of the hyperbola, we have r' — r=2A. On the Hyperbola. 79 Substituting this value in equation (4), we obtain 2cx r' + r=-r-. A Combining the last two equations, we find r'= A+ x , (o) r=-A+f. (G) Squaring these values and substituting them in equation (3), we obtain c~x~ which may be reduced to Ay + (A J -c s )f=A ! (A ! -c 2 ) (7) which is the equation of the hyperbola. If we put B 2 = c 2 — A 2 , the equation becomes Ay-BV=-A 2 B 2 , which is the equation required. (98.) Scholium 1. The equation of the hyperbola differs from that of the ellipse only in the sign of B 2 , which is positive in the ellipse, and negative in the hyperbola. Transposing, and dividing this equation by A 2 , it may be written y 9 =|kz 9 -A a ). Cor. 1. To determine where the curve intersects the axis of abscissas, make y=0, and we obtain a;=±A=CA orCA', which shows that the curve cuts the axis of X in two points, A and A', at the same distance from the origin, the one being to the right, and the other to the left ; and, since 2CA or AA' is equal to 2A, it follows that the difference of the two lines, drawn from any point of an hyperbola to the foci, is equal to the transverse axis. The line which is perpendicular to the transverse axis at its middle point, and equal to 2B, is called the conjugate axis. Cor. 2. When B is made equal t or the distance from the center to either focus, divided by the semi-transverse axis, is called the eccentricity of the hyperbola. If we represent the eccentricity by e, then -r-=e, or c=Ae. But we have seen that B 2 =c 2 -A 2 . Hence A 2 +B 2 =AV, B 2 , Making this substitution, the equation of the hyperbola be comes ?/ 2 = (e 2 — 1) (a; 2 — A 2 ). Cor. 6. Equations (5) and (6) of the preceding Proposition are ex >"= A+j. cx r=-A +T On the Kvpfebola. 81 c Substituting e for — , these equations become r'=ex + A, r = ex—A, which equations represent the distance of any point of the hyperbola from either focus. Multiplying these values together, we obtain rr'=eV — A 2 , which is the value of the product of the focal distances. Scholium 2. If on BB', as a trans- Verse axis, opposite hyperbolas are described having AA' as their conju- gate axis, these hyperbolas are said to be conjugate to the former. The equation of the conjugate hyper- bolas may be found from the equation Ay-BV=-A 2 B a , by changing A into B and x into y. It then becomes BV-Ay=-A 2 B 2 , which is the equation of the conjugate hyperbolas. PaoposiTioN II. — Theorem. (99.) The equation of the hyperbola, when the origin is a\ the vertex of the transverse axis, is tf=^(x* + 2Ax), where A and B represent the semi-axes, and x and y the gen- eral co-ordinates of the curve. The equation of the hyperbola, when the origin is at the center, is, Art. 97, Ay-BV=-A 5 B 3 . (1) If the origin is placed at A, the or- dinates will have the same value as when the origin was at the center, but the abscissas will be different. If we represent the abscissas reckoned from A by x', then it is plain that we shall have 82 Analytical Geometry CR=AR+AC, or x=x'+A. Substituting this value of a; in equation (1), we have Ay-BV 8 -2B 2 Az'=0, which may be put under the form y*=^(x»+2Ax') ; or, omitting the accents, y*=-^(x>+2Az), which is the equation of the hyperbola referred to the vertex A as the origin of co-ordinates. Proposition III. — Theorem. (100.) The square of any ordinate is to the product of its dis- tances from the vertices of the tranverse axis, as the square of the conjugate axis is to the square of the transverse axis. The equation of the hyperbola, re- ferred to the vertex A as the origin of co-ordinates, is, Art. 99, This equation may be resolved into the proportion f : (.r + 2A)x : : B 2 : A a . Now 2A represents the transverse axis AA', and, since x represents A R, a:+2A will represent A'R ; therefore, (a;+2A)a; represents the product of the distances from the foot of the or dinate PR to the vertices of the transverse axis. Cor. It is evident that the squares of any two ordinates are as the products of the parts into which they divide the trans- verse axis produced. Proposition IV. — Theorem. (101.) Every diameter of an hyperbola, is bisected at the center. Let PP' be any diameter of an hyperbola. Let x', y' be the co-ordinates of the point P, and x", y ' those of the point P' On the Hyperbola. 8«J Then, from the equation of the curve, we shall have, Art. 9S Scholium 1, B 2 i and y"-= x - 1 (s»»--A'). y> Whence — a;' 2 -A 3 y- V' 2 -A 2 ' But from the similarity of the triangles PCR, P'CR', we have y' x' y" x ' x" a;' 2 -A 2 Whence — jjz = ~ jjz — tz- x" x —A Clearing of fractions, we obtain x"=x"\ Whence, also, 2/' 2 =*/" 2 - Consequently, x»+y'*=x"*+y"\ or CP 2 =CP' 2 ; that is, CP =CP'. Proposition V. — Theorem. (102.) If from the vertices of the transverse axis, two lines be drawn to meet on the curve, the product of the tangents of the an- gles which they form with it, on the same side, will be equal to the square of the ratio of the semi-axes. The equation of the line AP pass- ing through the point A, whose co- ordinates are x'=A, y'=0, Art. 18, is y=a{x-A). The equation of A'P passing through the point A', whose co-ordinates are x'=-A, y' = 0, Art. 18, is y=a'(x+A). Tliese lines must pass through the point P in the hyperbola. Hence, if we represent the co-ordinates of P by a?" and y" we have the three equations y "=:a(x"-A) (1) y" = a'(x" + A) (2) 84 Analytical Geometry. Multiplying (1) and (2) together, we have Hence, comparing with (3), we see that B 2 (3) Cor. In the equilateral hyperbola A=B, and we have * aa'=\; which shows that the angles formed by the supplementary chords, with the transverse axis on the same side, are together equal to a right angle, Art. 24. Proposition VI. — Theorem. (103.) The equation of a straight line which touches an hy perbola is A\jij'-Wxx'=-A?W, where x and y are the general co-ordinates of the tangent line, x' and y' the co-ordinates of the point of contact. Draw any line P'P" cutting the hyper- bola in the points P', P"; if this line be moved toward P it will approach the tan- gent, and the secant will become a tan- gent when the points P', P" coincide. Let x', y' be the co-ordinates of the point P', and x", y" the co-ordinates of the point P". The equation of the line P'P", passing through these two points, will be, Art. 20, V'-v" y-y'= , y u (x-x'). y y x'—x" K ' 0) Since the points P', P" are on the curve, we shall have, Art. 97, Ay 2 -BV 2 =-A 2 B 2 , (2) Ay ,2 -BV 2 =-A 2 B 2 . (3) Subtracting equation (3) from (2), we obtain A*(y'*-y" 2 )-B 2 (x l2 -x" n -) = 0, or A*(y'+y") {y> -y")-W(x'+x") (x'-x")=0. Whence On the Hyperbola. y'-y" W{x'+x) 85 x'-x" A'iy'+y")' Substituting this value in equation (1), the equation of the secant line becomes The secant P'P" will become a tangent when the points l v , P" coincide, in which case x'—x", and y'=y". Equation (4) in this case becomes B*x' y-y =j*-,(?-z'h which is the equation of a tangent to the hyperbola at the point P. If we clear this equation of fractions, we obtain A'yi/ - Ay 2 = B'xx' - B V 2 , or Ay/'-B 2 ;r:c'=-A 2 B 2 , which is the most simple form of the equation of a tangent line. (104.) Cor. 1. In the equation B 2 ~' y-y A 2 */ -,(z-x% B — represents the trigonometrical tangent of the angle which Ay Ihe tangent line makes with the transverse axis. Cor. 2. To find the point in which the tangent intersects the axis of abscissas, make y=0 in the equation of the tangent line, and we have A 2 x=— , x' which is equal to CT. If from CR or x' we subtract CT, we shall have the subtangent x' x' Proposition VII. — Theorem. (105.) The equation of a normal line to the hyperbola is ay, y-!/'=-B^7(*-*')> 86 Analytical Geometry. where x and y are the general co-ordinates of the normal line and x' and y' the co-ordinates of the point of intersection with the curve. The equation of a straight line passing through the point, whose co-ordinates are w\ y' f Art. 18, is y-y'=a(x-x') ; (1) and, since the normal line is perpendicular c T to the tangent, we shall have, Art. 23, 1 a=- -a' But we have found for the tangent line, Prop. VI., Cor. 1, BV a'= Hence a= — AY A> BV Substituting this value in equation (1), we shall have for the equation of the normal line Ay (2) (106.) Cor. 1. To find the point in which the normal inter- sects the axis of abscissas, make y=0 in equation (2), and we have, after reduction, l;N=z= — — — x'. A 2 If we subtract CR, which is represented by x', we shall have he subnormal PAT A * + W , , BV KiM = — — x' — x' = —nr. Cor. 2. If we put e 5 A : A 2 +B : , Art. 98, Cor. 5, we shall have A 2 CN=eV. If to this we add F'C (see next figure), which equals c or Ae, Prop. I., Cor. 5, we have F'N=Ae+eV=e(A+ca;') f which is the distance from the focus to the foot of the normal. On the Hyperbola. 87 Proposition VIII.— Theorem. (107.) A tangent to the hyperbola bisects the angle contained by lines drawn from the point of contact to the foci. Let PT be a tangent line to the hyperbola, and PF, PF' two lines drawn to the foci. Produce F'P to M, and draw PN bisect- ing the exterior angle FPM. Then, by Geom., Prop. XVII., Schol., B. IV., F'P : FP : : F'N : FN ; or, by division, F'P-FP : : F'F : : F'P : FN. (1) But FP-FP-2A, F'F=2c = 2Ae, Prop. I., Cor. 5, and F'P=A +ex, Prop. I., Cor. 6. Making these substitutions in proportion (1), we have 2A : 2Ae : : A+ex : F'N. Hence F'N=e(A+ez). But, by Prop. VII., Cor. 2, e(A+ex) represents the distance trom the focus F to the foot of the normal. Hence the line PN which bisects the angle FPM, is the normal ; that is, it is perpendicular to the tangent TT'. Now, since PN is perpen- dicular to TT', and the angle FPN is equal to the angle MPN, therefore the angle FPT is equal to MPT', or its vertical angle F'PT ; that is, the tangent PT bisects the angle FPF'. (108.) Cor. 1. The normal line PN bisects the exterior an- gle FPM, formed by two lines drawn to the foci. Cor. 2. This Proposition affords a method of drawing a tan- gent line to an hyperbola at a given point of the curve. 4 Let P be the given point ; draw the radius vectors PF, PF'. On PF' take PG equal to PF, and draw FG. Draw PT perpendicular to FG, and it will be the tangent re- quired, for it bisects the angle FPF'. 88 Analytical Geometry. Proposition IX. — Theorem. (109.) If through one extremity of the transverse axis, a chord be dr awn parallel to a tangent line to the curve, the supplement- ary chord will be parallel to the diameter which passes through the point of contact, and conversely. Let DT be a tangent to the hy- perbola, and let a chord AP be drawn parallel to it ; then will A'P be parallel to the diameter DD' which passes through the point of contact D. Let x', y' designate the co-ordi- nates of D; the equation of the line CD will be, Art. 15, y'=a'x' , whence a' = V But, by Prop. VI., Cor. 1, the tangent of the angle which the tangent line makes with the transverse axis, is BV a— Ay- Multiplying together the values of a and a', we obtain B 2 which represents the product of the tangents of the angles DCT and DTX. But, by Prop. V., the product of the tangents of the angles PAX, PA'A is also equal to -t-j. A Hence, if AP is parallel to DT, A'P will be parallel to CD. and conversely. (110.) Let DD' be any diameter of an hyperbola, and DT the^ tan- gent drawn through its vertex, and let the chord AP be drawn parallel to DT ; then, by this Proposition, the supplementary chord A'P is par- allel to DD'. Let another tangent ET' be drawn to the conjugate hy- perbola parallel to A'P, it will also On the Hyperbola. 8& be parallel to DD'. Let the diameter EE' be drawn through the point of contact E ; then, by this Proposition, A'P being parallel to T'E, AP (and, of course, DT) will be parallel to EE'. Each of the diameters DD', EE' is, therefore, parallel to a tangent drawn through the vertex of the other, and they are said to be conjugate to one another. Scholium. Two diameters of an hyperbola are said to be conjugate to one another, when each is parallel to a tangent line drawn through the vertex of the other. If we designate by a and a' the tangents of the angles which two conjugate diameters make with the transverse axis, then we must have B 2 aa'=-jT t . Proposition X. — Theorem. (111.) The equation of the hyperbola, referred to its center and conjugate diameters, is A'y-B'V=-A' ! B' 2 , where A' and B' are semi-conjugate diameters. The equation of the hyperbola, referred to its center and nxes, Art. 97, is Ay-BV=-A 2 B 2 . In order to pass from rectangular to oblique co-ordinates, the origin remaining the same, we must substitute for x and y, in the equation of the curve, Art. 30, the values x—x' cos. a,+y' cos. «,', y=x' sin. a+y' sin. a'. Squaring these values of x and y, and substituting in the equation of the hyperbola, we have A 2 sin. 2 a'|?/' 2 +2A 2 sin. a sin. a'by + A 2 sin.' m /! =- A 2 B 2 , --B 2 COS. 2 a'| -2B 2 COS. a cos. a'| -B 2 COS. 2 a (1) which is the equation of the hyperbola when the oblique co- ordinates make any angles a, a' with the transverse axis. But since the new axes are conjugate 'liameters, we must have, Art. 110, B 2 90 Analytical Geometry. B 2 or tang, a tang. a'=-ra- A. Whence A 2 tang, a, tang, a' — B 2 =0. Multiplying by cos. a cos. a.', remembering that cos. a tang. a=sin. a, we have A 2 sin. a sin. a' — B 2 cos. a cos. a .=0. Hence the term containing x'y' in equation (1) disappears, and we have (A 2 sin. 2 a'-B 2 cos. 2 a')y ,2 + (A 2 sin. 2 a-B 2 cos. 2 a)a;' 2 =-A 2 B 2 , (2) which is the equation of the hyperbola referred to conjugato diameters. If in this equation we make y'=0, we shall have A sin. a— 15- cos. a If we make x'=0, we shall have ____A ! B^__ y A 2 sin. 2 a'-B 2 cos. 2 «' If CD 2 is positive, CE 2 must be negative. For when CD 2 is positive, the numerator of the above ex- pression for its value being negative, its denominator must be negative ; that is, A 2 sin. 2 a— , A or A 2 sin. 2 a' >B 2 cos. 2 a'; that is, the denominator of the expression for CE 2 is positive; hence, since its numerator is negative, CE 2 must be negative. If we represent CD 2 by A' 2 , and CE 2 by -B 2 , equation (2) reduces to V' 2 x" — — A =4-1 • "D/2 ' A/2 ' * » hence A'y 2 -B'V 2 =-A' 2 B' 2 ; or, omitting the accents from x and y, On the Hyperbola. 91 AY-B'V=-A"B", which is the equation of the hyperbola referred to its center and conjugate diameters. Proposition XI. — Theorem. (112.) The square of any diameter is to the square of its con- jugate, as the rectangle of the segments from the vertices of the diameter to the foot of any ordinate, is to the square of that or- dinate. The equation of the hyperbola, re- ferred to conjugate diameters, Art. Ill, is A'y-B'V=-A' 2 B' 2 , which may be put under the form A'y=B' 2 (a; 2 -A' 2 ). This equation may be reduced to the proportion A /2 : B' 2 : : x 2 -A" : y\ or (2A') 2 : (2B') 2 : : (ar+A') (a;- A') : y\ Now 2A' and 2B' represent the conjugate diameters DD', EE' ; and, since x represents CH, x-\-k! will represent D'H, and x— A' will represent DH ; also, GH represents y 2 ; hence DD' 2 : EE' 2 : : DHXHD' : GH 2 . (113.) Cor. It is evident that the squares of any two ordinates to the same diameter, are as the rectangles of the correspond- ing segments from the vertices of the diameter to the foot of the ordinates. Definition. The parameter of any diameter is a third pro- portional to the diameter and its conjugate. The parameter 2B 2 of the transverse axis is equal to — -r— , Art. 98, Cor. 4 • and that 2A 2 of the conjugate axis is equal to —^-. Proposition XII. — Theorem. (114.) The difference of the squares of any two conjugate di- ameters is equal to the difference of the squares of the axes. Let DD', EE' be any two conjugate diameters. Designate 92 Analytical Geometry. the co-ordinates of D by x', y' ; those of E by x", y" ,• the an- gle DCA by a, and the angle ECA by *'. Then y' tang, a =|j, tan I j Therefore, tang, a x tang. aJ~^-j—, x'x' B- which equals -r^, because DD' and EE' are conjugate diame- ters, Prop. IX., Schol. Hence, by squaring each member of this equation, we have Ayy ,2 =BW 2 . (1) But because the point D is on the curve, we have, Art. 97, Ay 2 =-A 2 B 2 +BV 2 ; and because E is on the curve of the conjugate hyperbola, we have, Art. 98, Schol. 2, Ay /2 =+A 2 B 2 +BV /2 . Therefore, by multiplication, Ayy /2 = - A 4 B 4 +A 2 BV 2 - A 2 BV' 2 +BW' 2 . (2) Comparing equation (1) with equation (2), we see that -A 4 B 4 +A 2 BV 2 -A 2 BV /2 =0 ; or, dividing by ~A 2 B 4 , we have A 2 -.-c /2 +.r //2 =0, or A 2 =a;' 2 — x"\ (3) In the same manner, we find that B>=y>'*-y». ( 4 ) Hence, by subtraction, A 2 -B 2 =a;' 2 +z/' 2 -;c" 2 -?/" 2 =A /2 -B' 2 . (115.) Cor. According to this Proposition, Equation (3), x' 2 =A i +x"\ Also, from the equation of the hyperbola, Art. 98, Schol. 2 f Ay 2 =B 2 (A 2 +z" 2 ). . Therefore A 2 On the Hyperbola. 93 or A In the same manner we find B A' Proposition XIII. — Theorem. (1 1 6.) The parallelogram formed by drawing tangents througn the vertices of two conjugate diameters, is equal to the rectangh of the axes. Let DED'E' be a parallelogram formed by drawing tangents to the hyperbola through the vertices of two conjugate diameters DD', EE' ; its area is equal to AA'xBB'. Let the co-ordinates of D, referred to rectangular axes, be x', y' ; and /' those of E be x", y". The triangle CDE is equal to the trapezoid DEHG, plus the triangle ECH, minus the triangle CDG ; that is, 2.CDE=(x'-x")(tj'+y")+x"y"-x>y', ~x'y"—x"y', =a .&L„^! f by Prop. XIL, Cor., A U BV 2 — Ay 2 reducing the fractions to a common — A.B ' denominator, A 2 R 2 =x]r A.B, Art. 97. Therefore the parallelogram CETD is equal to A.B ; and the parallelogram DED'E' is equal to 4 A.B or 2 A X 2B = A A' X BB' Proposition XIV. — Theorem. (117.) The polar equation of the hyperbola, when the pole is at one of the foci, is 1+e cos. v y where p is half the parameter, e is the eccentricity, and v is the angle which the radius vector makes with the transverse axis 94 Analytical Geometry. We have found the distance of any point of the hyperbola from the focus, Prop. L, Cor. 6, to be r=FP=-A+e^, r'=F'P= A+ex, where the abscissa x is reckoned from the center. In order to trans- fer the origin from the center to the focus F, we must substi tute for x, x'+c; or, putting Ae for c, Art. 98, Cor. 5, we have x=x' + Ae. If we represent the angle PFC by v, we shall have, Art. 64, x'=—r cos. v. Whence x= — r cos. v+Ae. Therefore, FP=r=— A— er cos. v+Ae\ By transposition, r(l+e cos. v) = -A+Ae*=A(e*-l). 2B 3 If we put 2p= the parameter of the transverse axis =-r=- A (Art. 113), we shall have p=A(e*-l), Prop. I., Cor. 5. Whence r= , 1+e cos. v where the angle v is estimated from the vertex. ON THE ASYMPTOTES OF THE HYPERBOLA. (118.) If tangents to four conjugate hyperbolas be drawn through the vertices of the axes, the diagonals of the rectangle so formed, supposed to be indefinitely produced, are called asymptotes of the hyperbola. Let AA', BB' be the axes of four conjugate hyperbolas, and -hrough the vertices A, A', B, B', let tangents to the curve be drawn, and let DD', EE' be the diagonals of the rectangle thus formed ; DD', EE' are called asymptotes to the curve. If we represent the angle DCX by a, and the angle E'CX by a', then we shall have On the Asymptotes of the Hyperbola. B 95 tang, a = taner. a' = — A B A* But, since tang, a- sin. a , Trig., Art. 28, we have or COS. a sin. 2 « _B 2 COS. 2 a _ A 2 ' sin. 2 a _B 2 1— sin. 2 a - A 2 ' Whence sin. 2 a= In the same manner, we find B 2 A 2 +B 2 ' A 2 COS. " = J^tf> which equations furnish the value of the angle which the as ymptotes form with the transverse axis. Proposition XV. — Theorem. (119.) The equation of the hyperbola, referred to its center and asymptotes, is A 2 +B 2 *y=— 4— ' where A and B are the semi-axes, and x and y the co ordi- nates of any point of the curve. The equation of the hyperbola, referred to its center and axes, Art. 97, is Ay-BV=-A s B a . (1) The formulas for passing from rectangular to oblique co-ordi- nates, the origin remaining the same, Art. 30, are x=x' cos. a+y' cos. a', y=x' sin. <*+?/' sin. a'. But, since a=— a', these equations become z=(x'+y') cos. a, y=(x —y') sin a OQ Analytical Geometry. Substituting these values in equation (1), we have A:{x'-y'Y sin. 2 a-B'(x'+y'Y cos. 2 ct=--A 2 B 2 . R 2 But sin. 2 a= , Art. 118, A +r> and cos. a A 2 +B 2 ' hence j^(z'-yT-j^{x'+y>y=-tfW 5 4A 2 B 2 that is, ABa a:y =A 9 B 8 , A 2 +B 2 '/,,/. or arv — ? 4 which is the equation of the hyperbola referred to its center and asymptotes. (120.) Cor. The curve of the hyperbola approaches nearer the asymptote the further it is produced, but, being extended ever so fa?', can never meet it. The equation of the hyperbola, referred to its asymptotes, Art. 119, is A 2 + B 2 A 2 +B 2 Put M a for — - — , and we have xy=W, M a or y= — ; J x and, since M 2 is a constant quantity, y will vary inversely as x. Therefore y can not become zero until x becomes infinite ; that is, the curve can not meet its asymptote except at an infinite distance from the center. The asymptotes are, therefore, con- sidered as tangent to the curve at an infinite distance from the center. Proposition XVI. — Theorem. (121.) If from any point of the hyperbola lines be drawn par- allel to and terminating in the asymptotes, the parallelogram so formed will be equal to one eighth the rectangle described on the axes. Designate the co-ordinates of the point P referred to the- On the Asymptotes of the Hyperbola. 97 asymptotes by x', y', and the an- gle DCE', included between the .asymptotes, by fi, we shall have, from the equation of the curve, Art. 119, A 2 +B 2 x'y' sin. (3= — - — sin. |3. The first member of this equa- tion represents the parallelogram PC contained by the co-ordinates of the point P of the curve. But since this equation is true for every point of the curve, it must be true when the point is taken at the vertex A, in which case a/ represents CK, and y' represents AK, and we shall have the parallelogram A 2 4-B 2 CHAK= \ sin./3. Whence the preceding equation becomes x'y' sin. /3 = the parallelogram CHAK. Therefore the parallelogram PC, formed by the co-ordinates of any point of the curve, is equal to the parallelogram HK, which is one fourth of the parallelogram ABA'B', or one eighth of the rectangle described on the axes. Proposition XVII. — Theorem. (122.) The equation of a tangent line to an hyperbola, re ferred to its center and asymptotes, is V 1 y-y'=--,{x-x'), where x', y' are the co-ordinates of the point of contact. The equation of a secant line passing through the points x\ y\ x", y", Art. 20, is y-y , y'-y" , ,* '=— -{x— x'). (1) Since the two given points are on the curve, we must have* Art. 120, , x'y'=W, x"y"=W. Whence x'y'=x r y". G 98 Analytical Geometry. Subtracting x'y" from each member, we have x'y , —x'y"=x"y"—x'y". Whence x'{y'-y")=-y"(x'-x") % y'-y" _ y"_ or x'—x" x' ' Hence, by substitution, equation (1) becomes If we suppose x'=x", and y'=y", the secant will become a tangent, and equation (2) will be y-y'=- v -te- x 'h which is the equation of the tangent line. (123.) Cor. To find the point in which the tangent meets the axis of abscissas, make y=0 in the equation of the tangent line, and we have x=2x' ; that is, the abscissa CT' of the point, where the tangent meets the asymptote CE, is double the abscissa CM of the point of tan- gency. Therefore CM=MT'; and, since the triangles TCT', PMT' are similar, the tangent TT' is bisected in P, the point of contact; that is, if a tangent line he drawn at any point of an hyperbola, the part in- cluded between the asymptotes is bisected at the point oftangency. Proposition XVIII. — Theorem. (124.) If a tangent line be drawn at any point of an hyper- bola, the part included between the asymptotes is equal to the di- ameter which is conjugate to that which passes through the point of contact. Let TT' be a line touching the hyperbola at P. Through P draw the diameter PP', and designate the angle contained by the asymp- totes by /3. Then, by Trigonometry, Art. 78, in the triangle CPM, we have On the Asymptotes of the Hyperbola. 99 CM 2 +MF-CP or cos. CMP= — cos. ]3— 2CMXMF x 2 +y 2 -CY 2 2xy Also, in the triangle PMT', cos. PMT' or cos. (3— PM 2 +MT' 2 -PT /!I 2PM XMT' x 2 +y 2 -VT' 2 2xy Whence we have C? 2 =x 2 +y 2 +2xij cos. (3, TT' 2 =x 2 +y 2 -2xy cos. j3. Whence CP 2 -~PT' 2 =4xy cos. (3. But, since (3=2a, cos. (3=cos. 2 a— sin. 2 a, Trig., Art. 74. A 2 -B 2 Hence, from Art. 118, cos. (3= . 2 p 2 . Also, from the equation of the hyperbola, Art. 119, A 2 +B 2 oi 4^=A 2 +B 2 . Therefore 4xy cos. |3=A 2 -B 2 , and CP 2 -PT' 2 =A 2 -B 2 =A' 2 -B' 2 (Prop. XII.). But CP is equal to A' ; therefore PT'=B' ; that is, the tan- gent TT' is equal to the diameter which is conjugate to PP'. (125.) Cor.. The same is true of a tangent W drawn through the point P' of the opposite hyperbola. Therefore, if we join the points Tt, T't', the figure Ttt'T' will be a par- allelogram whose sides are equal and parallel to 2A', 2B' ; that is, PP', EE'. Hence the asymptotes are the diagonals of all the paral- lelograms which can be formed by drawing tangent lines through the vertices of conjugate diameters. SECTION VIII. CLASSIFICATION OF ALGEBRAIC CURVES. (126.) We have seen that the equations of the circle, the ellipse, parabola, and hyperbola are all of the second degree ; we will now show that every equation of the second degree is geometrically represented by one or other of these curves. The general equation of the second degree, between two variables, is Ay*+Bxy+Cx*+T)y+Ex+F=0, (1) which contains the first and second powers of each variable, their product, and an absolute term. Proposition I. — Theorem. (127.) The term containing the product of the variables in tne general equation of the second degree, can always be made to dis- appear, by changing the directions of the rectangular axes. In order to effect this transformation, substitute for x and y, in equation (1), the values x—x' cos. a— y' sin. a, ) ^) y—x'' sin. a+y' cos. a, ) by which we pass from tne system of rectangular co-ordinates, to another having the same origin, Art. 29. The result of this substitution is '+2Asin.acos.a' +Bcos. 2 a — Bsin. 2 a —2C sin. a cos. a +D cos. a) , ( +D sin. «. — E sin. a ) ( +E cos. a Since the value of a is arbitrary, we may assume it of such value that the second term of the transformed equation may vanish. We shall therefore have 2A sin. a cos. a+B cos. 2 a— B sin. 2 a— 2C sin. a cos. a=0. or (A-C)2 sin. a cos. a+B(cos. 2 a-sin. 2 a)=0. A cos. 2 on — Bsin.acos.aiy /2 + C sin. 2 aj !+Asin. 2 ai +Bsin.a cos. a>:c'* +Ccos. 2 aJ '+F=0. Classification of Algebraic Curves. 103 Bat 2 sin a cos. a=sin. 2a, and cos." a— sin. 2 a=cos. 2a, Trig., Art. 74. Hence (A — C) sin. 2a+B cos. 2a=0 ; or, dividing by cos. 2a, B tang. 2a =-JZIc' If, therefore, in equations (2), we give to the angle a such a value that the tangent of double that angle may be equal to "D — — — p, the term containing xy will disappear from the trans- formed equation. The new equation, therefore, becomes of the form Mt/+Nx 2 +Ry+Sx+F=0. (3) Proposition II. — Theorem. (128.) The terms containing the first power of the variables in the general equation of the second degree, can be made to dis- appear by changing the origin of the co-ordinates. In order to effect this transformation, substitute for x and y, in equation (3), the values x=a+x', y=b+y', by which we pass from one system of axes to another system parallel to the first, Art. 28. The result of this substitution is My ,2 +'Nx r -+2Mb ) y'+2Na ) :c'+M& 2 +Na 2 +R&+Sa+F=0. R I S I In order that the terms containing x' and y' may disappear, ive must have T> 2M6+R=0, or b=-^ t S and 2Na+S=0, or a= — -^, where a and b are the co-ordinates of the new origin. If we employ these values of a and b, and substitute P foi -Mb 2 — Na 2 — Rb— Sa — F, equation (3) reduces to My*+Nx*=F, an equation from which the terms containing the first power of the variables have been removed. . 102 Analytical Geometry. (129.) If one of the terms containing x* or y 2 was wanting from equation (3), this last result would be somewhat modified. If, for example, N=0, the value of a, given above, would re- S duce to — , or infinity. We can, however, in this case cause the term which is independent of the variables to disappear. For this purpose we must put Mb 2 +Rb+Sa+F=0 Mi 2 +R&+F which gives a— ~ . R With this value of a, and the value of b=— —^rr, equation (3) reduces to the form Mj/ 2 + Sx=0; S or, putting Q for -^, 2/*=Qz. Hence every equation of the second degree between two varia- bles may be reduced to one of the forms, My 2 + Nx 2 =P, (4) or y 2 =Qz. (5) (130.) Equation (4) characterizes a circle, an ellipse, or ar hyperbola. First. Suppose M, N, and P are positive. P P Put A 2 =^, and B 2 = M . By substituting these values in equation (4), we obtain Vy 2 Px 2 _ — - — I =P B 2 ^ A 2 ' or Ay+BV=A 9 B 9 , which is the equation of an ellipse, Art. 68. If M=N, this equation characterizes a circle. Secondly. If N and P are both negative, or the equation is oi the form My a -Ns 9 =-P, P P put A2= N' and B = M' and we obtain, by substitution, Classification of Algebraic Curves. 103 iy_iv B a A 2 ~ ' 01 Ay-BV=-A 2 B 2 , which is the equation of an hyperbola, Art. 97. Thirdly. If N alone is negative, or the equation is of the form My 2 -Nx 2 =~P, we shall obtain, by substitution, as before, Ay-BV=A 2 B 2 , which characterizes the conjugate hyperbola, Art. 98, Schol. 2. (131.) Equation (5) characterizes a parabola, since, by put- ting Q=2p, it becomes if=2px, Art. 50. Hence the only curves whose equations are of the second de- gree, are the circle, parabola, ellipse, and hyperbola. (132.) When the origin of co-ordinates is placed at the ver- tex of the major axis, the equation of the ellipse is, Art. 70, T>2 tf = j;(2Ax-X>). The equation of the parabola for a similar position of the origin is, Art. 50, if=2px; and the equation of the hyperbola is, Art. 99, T>2 rf=-^(2Ax+x-). The equation of the circle is y'=2llx-x\ These equations may all be reduced to the form y*=mx-\-nx*. In the ellipse, m=-r-, and w= — -r. A A In the parabola, m=2p, and n=0. In the hyperbola, m=—r-, and n—-r^. ' l A A 2 In each case m represents the parameter of the curve, and n the square of the ratio of the semi-axes. In the ellipse, n is negative ; in the hyperbola it is positive ; and in the parabola it is zero. 104 Analytical Geometry. (133.) Lines are divided into different orders, according to the degree of their equations. A line of the first order has its equation of the form Ay+Bx+C=0; this class consists of the straight line only. Lines of the second order have their equations of the form Ay 2 + Bxy + Cx* + By + Ex + F = 0. This order comprehends four species, viz., the circle, ellipse, parabola, and hyperbola. (134.) Lines of the third order have their equations of the form Ay" + By'x -f Cyx a + LV + Ey n - + Fyx + Gx* +Hy +Ka;+L=0. Newton has shown that all lines of the third order are com Drehended under some one of these four equations, (1.) xy'+Ey = Ax*+Bx i +Cx+V, (2.) xy = Ax' + Bx* + Cx + D, (3.) t/ 2 =A:c 3 + B.r 2 + Cz+D, (4.) y ^Ax'+Bz-'+Gc+D, in. which A, B, C, D, E may be positive, negative, or evanescent, excepting those cases in which the equation would thus be- come one of an inferior order of curves. He distinguished sixty-five different species of curves com- prehended under the first equation ; four new species were subsequently discovered by Sterling, and four mo e by De Gua. The second equation comprehends only one species of curves, to which Newton has given the name of Trident. The third equation includes five species, each possessing two parabolic branches ; among these is the semi-cubical parabola. The fourth equation comprehends only one species of curves, commonly called the cubical parabola. There are, therefore, eighty different species of lines of the third order. (135.) Lines of the fourth order have their equations of the form At/ 4 + By 3 x + Cy*x°- + Byx 3 + Ex* ' + Fy~ +Gy 2 x +Hyx* + Kx i-Lif +Myx +Nc a J» + Py +Qx +R Classification of Algebraic Curves. 105 Lines of the fourth order are divided by Euler into 146 classes, and these comprise more than 5000 species. As to the fifth and higher orders of lines, their number has precluded any attempt to arrange them in classes. (136.) A family of curves is an assemblage of several curves of different kinds, all defined by the same equation of an inde- terminate degree. Thus, every curve whose abscissas are proportional to any power of the ordinates is called a parabola. Hence the number of parabolas is indefinite. Of these some of the most remarkable have received specific names. The common parabola is sometimes called the quadratic parabola, and its equation is of the form \f=ax. The equation of the cubical parabola is y 3 =ax. The equation of the biquadratical parabola is . . . y i =ax, etc. etc. etc. The equation of the semi-cubical parabola is . . y*=ax The equation of the semi-biquadratical parabola is y~ s =ax etc. etc. etc. All of these parabolas are included it the equation . y , =ax SECTION IX. TRANSCENDENTAL CURVES. (137.) Curves may be divided into two general classes, algebraic and transcendental. When the relation between the ordinate and abscissa of a curve can be expressed entirely in algebraic terms, it is called an algebraic curve ; when this relation can not be expressed without the aid of transcendental quantities, it is called a trans- cendental curve. Among transcendental curves, the cycloid and the logarithmic curve are the most important. The logarithmic curve is use- ful in exhibiting the law of the diminution of the density of the atmosphere ; and the cycloid in investigating the laws of the pendulum, and the descent of heavy bodies toward the center of the earth. The spirals have many curious properties, and are employ- ed in the volutes of the Ionic order of architecture. CYCLOID. (138.) A cycloid is the curve described by a point in the cir- cumference of a circle rolling in a straight line on a plane. AND Thus, if the circle EPN be rolled along a straight line AC, any point P of the circumference will describe a curve which is called the cycloid. The circle EPN is called the generating circle, and P the generating point. When the point P has arrived at C, having described the arc ABC, if it continue to move on, it will describe a second arc similar to the first, a third arc, and so on, indefinitely. As, however, in each revolution of the generating circle, an equal Logarithmic Curve. 107 curve is described, it is only necessary to examine the curve ABC described in one revolution of the generating circle. (139.) After the circle has made one revolution, every point of the circumference will have been in contact with AC, and the generating point will have arrived at C. The line AC will be equal to the' circumference of the generating circle, and is called the base of the cycloid. The line BD, drawn perpen- dicular to the base at its middle point, is called the axis of the cycloid, and is equal to the diameter of the generating circle. Proposition I. — Theorem. (140.) The equation of the cycloid is x = arc whose versed sine is y — V2ry—y\ where r represents the radius of the generating circle. Let us assume the point A as the origin of co-ordinates, and let us suppose that the generating point has described the arc AP. If N designate ^ R the point at which the generating circle touches the base, it is plain that the line AN will be equal to the arc PN. Through N draw the diameter EN, which will be perpendicular to the base. Through P draw PH parallel to the base, and PR per- pendicular to it. Then PR will be equal to HN, which is the versed sine of the arc PN. Let us put EN=2r, AR=x, and PR or HN=?/; we shall then have, by Geom., Prop. XXII., Cor., B. IV., RN=PH= VHNXHE= Vy(2r-ij)=V2ry-y\ and AR=AN-RN=arc PN-PH. Also, PN is the arc whose versed sine is HN or y. Substituting the values of AR, AN, and RN, we have re = arc whose versed sine is y— <\ equal to one third of PB, PI equal to one fourth of PB, etc., the curve pass- ing through the points B, G, H, I, etc., will be a hy- perbolic spiral, because the radius vectors are inversely pro- portional to the correspoiding arcs estimated from A. LOGARITHMIC SPIRAL. (154.) While the line PA revolves uniformly about P, let the geneiating point move along PA in such a manner thai 112 Analytical Geometry. the logarithm of the radius vector may be proportional to the measuring arcs, it will describe the logarithmic spiral. Proposition IV. — Theorem. (155.) The equation of the logarithmic spiral is t = a log. r, where r represents the radius vector, and t the measuring arc. For this equation is but an expression of the definition. (156.) The logarithmic spiral may be constructed as follows : Divide the arc of a circle ACE into any number of equal parts, AB, BC, CD, etc., arid upon the radii drawn to the points of divi- sion, take PL,PM, PN, etc., in geo- metrical progression. The curve passing through the points L, M, N, etc., will be the logarithmic spiral ; for it is evident that AB, AC, etc., being in arithmetical progression, are as the loga- rithms of PL, PM, etc., which are in geometrical progression. See Algebra, Art. 315. DIFFERENTIAL CALCULUS. SECTION I. DEFINITIONS AND FIRST PRINCIPLES-DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. Article (157.) In the Differential Calculus, as in Analytical Geometry, there are two classes of quantities to be considered, viz., variables and constants. Variable quantities are generally represented by the last let- ters of the alphabet, x, y, z, etc., and any values may be assign- ed to them which will satisfy the equations into which they enter. Constant quantities are generally represented by the first letters of the alphabet, a, b, c, etc., and these always retain the same values throughout the same investigation. Thus, in the equation of a straight line, y=ax + b, the quantities a and b have but one value for the same line, while x and y vary in value for every point of the line. (158.) One variable is said to be a function of another varia- ble, when the first is equal to a certain algebraic expression containing the second. Thus, in the equation of a straight line y=ax-\-b, y is a function of x. So, also, in the equation of a circle, and in the equation of the ellipse, B y=—V'2Ax — x\ (159.) When we wish merely to denote that y is dependent upon x for its value, without giving the particular expression which shows the value of x, we employ the notation H 114 Differential Calculus. y=F(x), or y=f(x), or x=F(y) f ovx=f(y), which expressions are read, y is a function of a;, or x is a func- tion of y. To denote a function containing two independent variables, as x and y, we inclose the variables in a parenthesis, and place the sign of function before them. Thus, the equation u=ay+bx' 2 may be expressed generally by u=f{x, y), which is read, u is a function of x and y, and simply shows that u is dependent for its value upon both x and y. (160.) An explicit or expressed function is one in which the value of the function is directly expressed in terms of the varia ble and constants, as in the equation y=ax i . + b. An implicit or implied function is one in which the value of the function is not directly expressed in terms of the variable and constants, as in the equation y s — 3ayx-\-x s =0, where the form of the function that y is of x can be ascertain ed only by solving the equation. (161.) An increasing function is one which is increased when the variable is increased, or decreased when the varia- ble is decreased. Thus, in the equation of a straight line, y—ax+b, if the value of a; is increased, the value of y will also increase; or, if x is diminished, the value of y will diminish. A decreasing function is one which is decreased when the variable is increased, and increased when the variable is de- creased. Thus, in the equation of the circle, ?/= VR'-x% the value of y increases when x is diminished, and decreases when x is increased. In the equation y= VIV— x\ x is called the independent variable, and y the dependent variable, because arbitrary values are supposed to be assigned to.r, and the corresponding values of y are deduced from the equation. Definitions and first Principles. 115 (1G2.) The limit of a variable quantity is that value which it continually approaches, so as, at last, to differ from it by less than any assignable quantity. Thus, if we have a regular polygon inscribed in a circle, and if we inscribe another polygon having twice the number of sides, the area of the second will come nearer to the area of the circle than that of the first. By continuing to double the num- ber of sides, the area of the polygon will approach nearer and nearer to that of the circle, and may be made to differ from it by a quantity less than any finite quantity. Hence the circle is said to be the limit of all its inscribed polygons. So, also, in the equation of a circle, the value of y increases as the point P ad- vances from A to B, at which point it be- comes equal to the radius of the circle. As the point P advances from B to C, the J value of y diminishes until at C it is re- duced to zero. The radius of the circle is, therefore, the limit which the value of y can never exceed. So, also, in the same equation, the radius of the circle is the limit which the value of a; can never exceed. If we convert } into a decimal fraction, it becomes .1111, etc., or -'- + -!- + ' 4- i 4- etr Hence the sum of the terms of this series approaches to the value of i, but can never equal it while the number of terms is finite. The limit of the sum of the terms of this series is there- fore i. So, also, the sum of the series, 1+i+i+i+rV. etc., approaches nearer to 2, the greater the number of terms we employ ; and, by taking a sufficient number of terms, the sum of the series may be made to differ from 2 by less than any quantity we may please to assign. The limit of the sum of the terms of this series is therefore 2. (1G3.) When two magnitudes decrease simultaneously, they may approach continually toward a ratio of equality, or to- ward some other definite ratio. Thus, let a point P be sup- 1 1G Differential Calculus. posed to move on the circumference of a circle toward a fixed point A. The arc AP will diminish at r> the same time with the chord AP, and, by bringing the point P sufficiently near to A, we may obtain an arc and its chord, each of which shall be smaller than any given line, and the arc and the chord con- tinually approach to a ratio of equality. But the ratio of two magnitudes does not necessarily ap proach to equality, because the magnitudes are indefinitely diminished. Thus, take the two series, *» 3' 6» 10' I 5» 21' 2B> Llv "! 1 I 1 •_»_ _"_ JL JL pic X > 4' 9' 1 6> 2 5' 36' 4 9> ClU The ratio of the corresponding terms is, 1 4 9 16 25 36 49 pf n x > 3' 6' 10' 15» 21' 2 8' Ctl " The ratio here increases at every step, but not without limit However far the two series may be continued, the ratio of the corresponding terms is never so great as 2, though it may be made to differ from 2 by less than any assignable quantity. The limit of the ratio of the corresponding terms of the two series is therefore 2. (164.) If a variable quantity increase uniformly, then other quantities, depending on this and constant quantities, may either vary uniformly, or according to any law whatever. Thus, in the equation of a straight line, y=2x-\-3, if we make x=l, we find y=5, x=2, " y=l, x=3, " y=9, etc. etc. ; that is, when x increases uniformly, y increases uniformly. Again, take the equation of the parabola, y— V4x, if we make x=l, we find y=2.000, x=2, " y=2.828, z=3, " y=3AQ4, x =4, " y=4.000, etc. etc., \ Differentiation of Algebraic Functions. 117 where, although x increases uniformly, y does not increase uniformly. (165.) If the side of a square increases uniformly, the area does not increase uniformly. Thus, let AB be the side of a square, and let it increase uniformly by the additions Ba, ab, be, etc., DF and let squares be described on these lines, as in the annexed figure ; then it is obvious that the square on the side Aa exceeds that described on the side AB, by twice the rect- A Bab c angle ABxBa, together with the square on Ba. The square described on Ab has received a further increment of two equal rectangles, together with three times the square on Ba ; the square on AC has received a further increment of two equal rectangles and five times the square on Ba. Hence, when the side of the square varies uniformly, the area does not vary uniformly. Thus, suppose the side of a square is equal to 10 inches, and let it increase uniformly one inch per minute, so as to become successively 11, 12, etc., inches. While the side increases from 10 to 11 inches, the area in- creases from 100 to 121 inches=21 inches. While the side increases from 11 to 12 inches, the area in- creases from 121 to 144 inches=23 inches. While the side increases from 12 to 13 inches, the area in- creases from 144 to 109 inches=25 inches. etc., etc., etc. Hence the rate of increase of the area depends upon the length of the side. When the side is 11 inches, the area is in- creasing more rapidly than when the side was 10 inches. (1G6.) There is an important distinction between the abso- >ute increase of a variable quantity, and its rate of increase. By the rate of increase at any instant we understand what would have been the absolute increase if this increase had been uniform. Thus, while the side of a square increases from 11 to 12 inches in one minute, the area increases from 121 to 144 inches. The absolute increase of the area is 23 inches; but the rate of increase of the area when '.he side was 11 inches was such as would have given an increase of less than 23 inches per minute ; and when the side was 12 inches the 118 Differential Calculus. rate of increase was such as would have given an increase? or more than 23 inches per minute. While, therefore, the rate of increase of the side of a square is uniform, the rate of increase of its area is continually changing. The object of the Differential Calculus is to determine the ratio between the rate of variation of the independent variable and that of the function into which it enters. Proposition I. — Theorem. (167.) The rate of variation of the side of a square is to thai of its area, in the ratio of unity to twice the side of the square. If the side of a square be represented by x, its area will be represented by x*. When the side of the square is increased by h and becomes x+h, the area will become (x-t-h) 9 , which is equal to x'+2xh+h\ While the side has increased by //, the area has increased by 2xh-{h\ If, then, we employ h to denote the rate at which x increases, 2xh-\-h* would have denoted the rate at which the area increased had that rate been uniform ; in which case we should have had the following proportion : rateof increase of the side : rate of 'increase of the area ::h: 2x11+1? or as l : 2x +h. But since the area of the square increases each instant more and more rapidly, the quantity 2x-\-h is greater than the incre- ment which would have resulted had the rate at which the square was increasing when its side became x continued uni- form ; and the smaller h is supposed to be, the nearer does the increment 2x + h approach to that which would have result- ed had the rate at which the square was increasing when its side became x continued uniform. When h is equal to zero, this ratio becomes that of 1 to 2x, which is, therefore, the ratio of the rate of increase of the side tc that of the area of the square, when the side is equal to x. (168.) Illustration. If the side of a square be 10 feet, its area will be 100 feet If the side be mcreased to 11 feet, its Differentiation of Algebraic Functions. 119 area will become 121 feet ; the area has increased 21 feet ; and the ratio of the increment of the side to that of the area is as 1 to 21. When the length of the side was 10.1 feet, its area was 102.01 ; the area had increased 2.01 feet ; and the ratio of the increment of the side to that of the area, was as 0.1 to 2.01, or 1 to 20.1. When the length of the side was 10.01 feet, its area was 100.2001 ; and the ratio of the increment of the side to that of the area, was as 1 to 20.01. When the length of the side was 10.001 feet, the ratio was as 1 to 20.001. Hence we see that the smaller is the increment of the side of the square, the nearer does the ratio of the increments of the side and area approach to the ratio of 1 to 20. This, there- fore, was the ratio of the rates of increase at the instant the side was equal to 10 feet ; and this ratio is that of one to twice ten, or twice the side of the square. We have here another illustration of the principle of Art. 163, that two magnitudes which decrease simultaneously may continually approach toward some finite ratio. However small we suppose the increment of the side of the square or the increment of the area to become, the ratio of the two in- crements continually approaches to that of 1 to 2x. Proposition II. — Theorem. (169.) The rate of variation of the edge of a cube is to that of its solidity, in the ratio of unity to three times the square of the edge. If the edge of a cube be represented by x, and its solidity by u, then u— x 3 . If the edge of the cube be increased by h so as to become z+h, and the corresponding solidity be represented by u', then we shall have u'= (x + h) *=z 3 + 3x*h + 3xh* + h\ The increment of the cube is u' -u=3x'h+Sxh' , +h\ Hence, if the solidity of the cube had increased uniformly when 120 Differential Calculus. the edge increased uniformly, we should have had the propoi tion, rate of increase of the edge : rate of increase of the solidity ::h: 3x' 2 h + 3xh 2 +h\ or as 1 : 3x" + 3xh +h\ But since the solidity at each instant increases more and more rapidly, the increment 3x 2 + 3xh + If is greater than that which would have resulted had the rate of increase when its edge became x continued uniform. Now the smaller h be- comes, the nearer does the increment 3x^+3x11+)^ approach to that which would have resulted had the rate at which the cube was increasing when its edge became x continued uni- form. When h is equal to zero, this ratio becomes that of 1 to 3x", which is, therefore, the ratio of the rate of increase of the edge to that of the solidity, when the edge is equal to x. (170.) The rate of variation of a function or of any variable quantity is called its differential, and is denoted by the lettei d placed before it. Thus, if u=x 3 , then dx : du : : 1 : 3x*. The expressions dx, du are read differential of a:, differential of u, and denote the rates of variation of a; and u. If we multiply together the extremes and the means of the preceding proportion, we have du=3x' i dx, which signifies that the rate of increase of the function u is 3x' times that of the variable x. If we divide each member of the last equation by dx, we have du dx which expresses the ratio of the rate of variation of the func- tion to that of the independent variable, and is called the dif- ferential coefficient of u regarded as a function of x. (171.) Illustration. If the edge of a cube be 10 feet, its solidity will be 1000 feet. If the edge be increased to 11 feet, its aoliiity will be 1331 feet; the solidity has increased 331 Differentiation of Algebraic Functions. 121 feet ; and the ratio of the increment of the edge to that of the solidity is as 1 to 331. When the length of the edge was 10.1 feet, its solidity was 1030.301 ; the solidity had increased 30.301 feet; and the ratio of the increment of the edge to that of the solidity was as 0.1 to 30.301, or 1 to 303.01. When the length of the edge was 10.01 feet, its solidity wag "003.003001 ; and the ratio of the increment of the edge to hat of the solidity was as 1 to 300.3001. When the length of the edge was 10.001 feet, the ratio was as 1 to 300.030001. Hence we see that the smaller is the increment of the edge of the cube, the nearer does the ratio of the increments of the edge and solidity approach to the ratio of 1 to 300. This, therefore, was the ratio of the rates of increase at the instant the edge was equal to 10 feet ; and this ratio is that of one to three times the square often. (172.) It will be seen from these examples that, in order to discover the rate of variation of a function, we ascribe a small increment to the independent variable, and learn the corre- sponding increment of the function. We then observe toward what limit the ratio of these increments approaches, as the in- crement of the variable is diminished, which limit can only be attained when the increment of the variable is supposed to be- come zero. This limit expresses the ratio of the rates of variation of the function and the independent variables, at the instant when the variable was equal to x. But because, in order to find the value of the differential co- efficient, we make h equal to zero, it must not be inferred that dx and du are therefore equal to zero, du denotes the rate of variation of the function u, and dx the rate of variation of the variable x; and since only their ratio is determined, either of them may have any value whatever, dx may, there 'ore, be supposed to have a very small or a very large value at r leasure. Proposition III. — Theorem. (173.) The differential coefficient of the function u=x* is 4x\ If we suppose x to be increased by any quantity h, and des J 22 Differential Calculus. ignate by u' the new value of the function under this supposi tion, we shall have u'=(x+hy ; or, expanding the second member of the equation, we have u'=x i +4x a h+6xVf+4xh s +h\ If we subtract from this the original equation, we obtain u'-u= 4x*h + Gx 7i 2 + 4xh 3 + h*. Hence we see that if the variable x is increased by h, the func- tion u will be increased by 4x 9 h+6x*h*+4xh a +h l . If both members of the last equation be divided by h, we shall have ■■4x*+6xVi+4xh*+h 3 , h which expresses the ratio of the increment of the function u to that of the variable x. The first term 4x 3 of this ratio is in- dependent of h, so that, however we vary the value of h, this first term will remain unchanged, but the subsequent terms are dependent on h. If we suppose h to diminish continually, the value of this ratio will approach to that of 4x 3 , to which it will become equal when h equals zero. This, therefore, is the ratio of the rate of increase of the independent variable to that of the func tion. at the instant the variable was equal to.r. Hence, Art. 172, du — =4x . dx (174.) The method here exemplified is applicable to the de- termination of the differential coefficient of any function of a single variable, and is expressed in the following Rule. Give to the variable any arbitrary increment h, and find the corresponding value of the function ; from which subtract its primitive value. Divide the remainder by the increment h, and find the limit of this ratio, by making the increment equal to zero ; the result will be the differential coefficient. Ex. 1. If vb increase uniformly at the rate of 2 inches per Differentiation* of Algebraic Functions. 123 second, at what rate does the value of the expression 2x* in- crease when x equals 6 inches ? Ans. 48 inches per second. Ex. 2. If x increase uniformly at the rate of 3 inches per second, at what rate does the value of the expression 4x 3 in- crease when x equals 10 inches ? Ans. Ex. 3. If x increase uniformly at the rate of 5 inches per second, at what rate does the value of the expression 2x* in- crease when x equals 4 inches ? Ans. Proposition IV. — Theorem. (175.) To obtain the differential of any power of a variable, we must diminish the exponent of the power by unity, and then multiply by the primitive exponent, and by the differential of the variable. To prove this proposition, let us take the function u=x n , and suppose x to become x + h, then u'=(x+h) n . Developing the second member of this equation by the Bi- nomial theorem, we have u'=x n + 7^.^■ n -'^+ 7^ ~ -a r"" 3 /* 3 + , etc. Subtracting from this the original equation, we have ,_ n(n — 1) u' — u=nx n ~'/H — --^ x n h-+, etc. Dividing both members by h, we have U '- U n-l , rc(n-l) — : — =nx -\ x n -h+, etc. h 2 which expresses the ratio of the increment of the function to that of the variable. If now we make h equal to zero, Art. 174, the second term of the second member of this equation reduces to zero, and also all the subsequent 'terms pf the development, since they contain powers of h. Hence 24 Differential Calculus. du dx- =n ? > or du=nx n ~ 1 dx, which conforms to the proposition abov r e enunciated. Proposition V. — Theorem. (17G.) The differential of the product of a variable quantity by a constant, is equal to the constant multiplied by the differ- ential of the variable. Suppose we have the function u^ax*. When x becomes x-\-h, we have u'=ax i +4ax*h + 6ax'*h' 2 +, etc. Also, u' — u=4ax*h+6axVf+, etc. tt u' — u Hence — -. — =4ax +Gax'h-{-, etc. If now we make h equal to zero, Art. 174, all the terms id the second member of this equation except the first disappear and we have du -j-=4ax . dx or du=4ax ? dx ; that is, the differential of ax* is equal to the differential of x* multiplied by a. Proposition VI. — Theorem. (177.) The differential of a constant term is zero; hence d constant quantity connected with a variable by the sign plus or minus, will disappear in differentiation. Suppose we have the function u=b+x\ When x becomes x+h, we have u' = b+x* + 4x*h + GxVi 2 -f , etc., and u' — u=4x 3 h+6x 2 /f+, etc. u' — u Hence — r— =4x' + Gx*h+. etc. ; h and, making h equal to zero, Art. 174, we have Differentiation of Algebraic Functions. 12D du — — Ax dx ' or du=4x*dx, where the constant term b has disappeared in differentiation. Ex. 1. What is the differential of 5ax 5 l Ans. 1 5ax*dx. Ex. 2. What is the differential of %x° + b? Ans. Ex. 3. What is the differential of 3x b 1 Ans. Ex. 4. What is the differential of 7aV + 6 3 ? Ex. 5. What is the differential of 4a¥x t — c? Ex. 6. What is the differential of 3a 3 cx°-d? Proposition VII. — Theorem. (178.) If u represents any function of x, and we change x into x-\-h, the new value of the function will consist of three parts : \st. The primitive function u. 2d. The differential coefficient of the function multiplied by the first power of the increment h. 3d. A function of x and h multiplied by the second power of the increment h. We have seen in the preceding Propositions that when u is a function of x, and we change x into x+h, the new value of the function consists of a series of terms which may be ar- ranged in the order of the ascending powers of h, and the de- velopment is of the following form u'=A + Bh+Ch 2 +'Dh*+, etc. Now when we suppose h equal to zero, the second member of this equation reduces to A, and u 1 on this supposition be- comes u; hence A=u, and the development may be written u'—u+Bh+Ch a +Bh 9 +, etc., or u' = u+Bh+h\C+Dh+, etc.). If now we represent C + D/t-f- , etc., by C, where C is a function both of x and h, we have u'^u + Bh+C'h*. (1) 126 DIFFERENTIAL CALCULUS. Transposing and dividing by h, we find u' — u B+C'A, zero :=B; h and when we make h equal to zero, we have du dx that is, B is the differential coefficient of the function. We see from equation (1) that u', the new value of the func- tion, consists of the primitive function u, plus the differential coefficient of the function multiplied by h, plus a function of x and h multiplied by h\ This new value of the function will be frequently referred to hereafter under the form u'=u + Ah + Bh\ (2) Proposition VIII. — Theorem. (179.) The differential of the sum or difference of any num- ber of functions dependent on the same variable, is equal to the sum or difference of their differentials taken separately. Let us suppose the function u to be composed of several variable terms, as, for example, u—y+z— u, where y, z, and u are functions of a:. If we change x into x-\-h, we shall have u'— u = (y '—y) + (z' — z) — (u' — u). But by the preceding Proposition y'—y may be put under the form of Ah-rBh*. So, also z' — z may be put under the form of A'/i + B'A', and u' — u may be represented by A'7i + B'7i 8 ; that is, u'-u=(Ah+Bh*)+(A'h+B'h*)-(A"h+B"h*). Dividing each member by h, we have ^p=(A+BA)+(A'+B70-(A"+B"A); and making h equal to zero, Art. 174, we have du -5-=A+A'-A", dx or du=Adx+A'dx — A"dx. Differentiation of Algebraic Functions. 127 But kdx is the differential of y; A'dx is the differential of 8 ; and A"dx is the differential of v. Hence du=dy+dz — do. Ex. 1. What is the differential of Gx i -5x i -2x? Ans. (24^ 8 -l5^-2)^. Ex. 2. What is the differential of ax*— ex? Ans. Ex. 3. What is the differential of 3ax 3 -bx* ? Ans. Ex. 4. What is the differential of a 6 +3aV+3aV+^ 6 ? Ans. Ex. 5. What is the differential of 5.r 3 -3x 2 +6x+2? Ans. Ex. 6. What is the differential of 7.;r 5 +6r , -5«:r + 3.r-6 ? J. 715'. Proposition IX. — Theorem. (180.) 77*e differential of the product of two functions de- pendent on the same variable, is equal to the sum of the products obtained by multiplying each by the differential of the other. Let us designate two functions by u and v, and suppose them to depend on the same variable x ; then, when x is in- creased so as to become x+h, the new functions may be writ- ten, Art. 178, u'=u+Ah + B/r, v'=v+A'h + B'h\ If we multiply together the corresponding members of these equations, we shall have u'v' = uv + Avh +Bvh\ + A'uh + AA'h 2 +, etc., + B'uh* +, etc., where, it will be observed, the terms omitted contain powers of the increment higher than h\ Transposing, and dividing by //, w'e have u'v'—uv t =Av+A'u+ other terms involving h. When we make h equal to zero, Art. 174, the terms involving h disappear, and we have 128 Differential Calculus. d(uv) -±- ! - = Av+A l u; ax or, multiplying by dx, d(u v) = vAdx + uA'dx. But Adx is equal to du, and A'dx is equal to dv. Hence d(uv) = vdu+udv, (1) which was the proposition to be demonstrated. (181.) Cor. If we divide both members of equation (1) by uv, we shall have d(uv) du dv UV U V that is, the differential of the product of two functions, divided by their product, is equal to the sum of the quotients obtained by dividing the differential of each function by the function itself. Ex. 1. What is the differential ofxy'l Ans. y*dx + 2xydy. Ex. 2. What is the differential ofx'if! Ans. Ex. 3. What is the differential of ax 2 y* ? Ans. Ex. 4. What is the differential of ax*(x*+2b) ? Ajis. Ex. 5. What is the differential of (.r 3 +«) (2x+b) ? Ans. Ex. G. What is the differential of (x 3 +a) (3x'+b) ? Ans. Proposition X. — Theorem. (1S2.) The differential of the product of any number of func- tions of the same variable, is equal to the sum of the products obtained by multiplying the differential of each function by the product of the others. Let us designate three functions by u, v, and z, and suppose them to depend on tre same variable x. Substitute y for vz, and we shall have Differentiation of Algebraic Functions. 129 uvz=uy, ind d(uvz) = d{uy). But, by the preceding Proposition, d(uy)—ydu+udy ; (1) and since y=vz, we have, by the same Proposition, dy=zdv+vdz. Substituting. these values of y and dy in equation (1), it be- comes d(iivz) = vzdu+uzdv+uvdz. (2) The same method is applicable to the product of four or more functions. (183.) Cor. If we divide both members of equation (2). by uvz, we shall have d'(uvz)_du dv dz uvz uvz which is an extension of Art. 181. Ex. 1. What is the differential of xy 2 z? Ans. y 2 zdx+2xyzdy+xy'dz. Ex. 2. What is the differential ofan/V? Ans. ■ Ex. 3. What is the differential of axy'z 3 ? Ans. Ex. 4. What is the differential of x{x^a) (x+2b) ? Ans. Ex. 5. What is the differential of ax\x*+a) (x+3b) ? Ans. Proposition XI. — Theorem. (184.) The differential of a fraction is equal to the denom- inator into the differential of the* numerator, minus the numer- ator into the differential of the denominator, divided by the square of the denominator. Let us designate the fraction by -, ard suppose '--=y, (1) • v then u = v y- I 130 Differential Calculus. Therefore, by Prop. IX., du=ydv + vdy ; whence vdy=du—ydv. (2) Substituting in the second member of equation (2) the value :>f y from equation (1), we have udv vdy=du . Dividing by v, we obtain vdu—udv dy that is, d(-) u\ vdu—udv v which was the proposition to be demonstrated. (185.) Cor. If the numerator u is constant, its differentia will be zero, Art. 177, and we shall have c\ —cdv v)~ d 2 x Ex. 1. What is the differential of — ? y ■2xi/ s dx—3x~y*dy 2xydx — 3x*dw Ans. — — ; — Z—Z-, or — - ; -, Ex. 2. What is the differential of — ? x a Ans. Ex. 3. What is the differential of — 3 ? ax Ans. Ex. 4. What is the differential of -^— ? l—x Ans. 1+x* Ex. 5. What is the differential of — ^? l—x Ans. a*+x' Ex. 6. What is the differential of ti s? —x Ans. \ Differentiation of Algebraic Functions.INI Proposition XII. — Theorem. (186.) To obtain the differential of a variable affected with any exponent whatever, we must diminish the exponent of the power by unity, and then multiply by the primitive exponent and by the differential of the variable. This is the same as Prop. IV., and the demonstration there given, being founded on the binomial theorem, may be con- sidered sufficiently general, since the binomial theorem is true, whether the exponent of the power be positive or negative, integral or fractional. This theorem may, however, be de- duced directly from Prop. X. Let it be required to find the differential of re", where the ex- ponent n may be either positive or negative, integral or frac tional. Case first. When n is a positive and whole number. x n may be considered as the product of n factors each equai to x. Hence, by Prop. X., Cor., d(x n ) dixxxx . . . .) dx dx dx dx X xxxx .... X X X X and since there are n equal factors in the first member of the equation, there will be n equal terms in the second ; hence d{x n ) ndx or d(x n ) = nx n ~ x dx. Case second. When n is a positive fraction. T Represent the fraction by -, and let r u—x 3 . Raising both members to the power s, we shall have u°=x T , and, since r and s are supposed to represent entire numbers, we shall hctve, by the first Case, s u*~ l du = 7*x T ~ i dx ; rx r ~ 1 rx r ~ l whence we find du= — —rdx—-^ — dx. SU -(.-1) sx" which may be reduced to 132 Differential Calculus. du — -x s dx, s f wiiich is of the same form as nx*~ l dx, substituting - for n. ° s Case third. When n is negative, either integral or fractional Suppose u=x~", which may be written u= — . J x a Differentiating by Prop. XL, Cor., we have x 2a ' and differentiating the numerator by Case first, or by Case second, if n represents a fraction, we have — nx n ~ 1 dx du= -^ ; or, subtracting the exponent 2n from n— 1, we have du=—nx~ D ~ 1 dx, which is of the same form as nx n ~ l dx, by substituting — n for -f?i, Proposition XII. may, therefore, be considered general, what ever be the exponent of a\ Ex. 1. What is the differential of ax** 1 ? Ans. a(n+l)x n dx. a - Ex. 2. What is the differential of -x n +c? b Ans. Ex. 3. What is the differential of aVx^l Ans. Ex. 4. What is theMifferential of far*? Ans. Ex. 5. What is the differential of cx~ z 1 Ans. Ex. 6. What is the differential of x' 2 y 2 z a ? Ans. Ex. 7. If the area of a square increase uniformly at the rate of r \ of a square inch per second, at what rate is the side in- creasing when the area is 100 square inches? Ans. Differentiation of Algebraic Functions. 133 Ex 8. If the solidity of a cube increase uniformly at the rate of a cubic inch per second, at what rate is the edge in- creasing when the solid becomes a cubic foot ? Ans. Proposition XIII. — Theorem. (187 ) The differential of the square root of a variable quan- tity, is equal to the differential of that quantity divided by twice the radical. Let it be required to find the differential of y/x, or x 2 . According to the preceding Proposition, i i-i d(x 2 ) = \x 2 dx, i = \x 2 dx, dx which may be written ~%Jx Ex. 1. What is the differential of Vax*1 Sax^dx . i f . Ans. — =, or %a~x-dx. 2Vax* Ex. 2. What is the differential of Vabx 7 1 Ans. Ex. 3. What is the differential of yTax'l Ans. — x Ex. 4. What is the differential of aVx— - I Ans. Ex. 5. What is the differential of V~ax+ -/cV? Ans. Proposition XIV. — Theorems (188 ) To obtain the differential of a polynomial raised to any power, we must diminish the exponent of the power by unity, and then multiply by the primitive exponent and by the differ- ential of the polynomial. Let it be required to differentiate the function u=(ax+xy. 134 Differential Calculus, Substitute y for ax+x*, and we have u=y\ Whence, by Prop. XII., du=ny n ~ 1 dy. Restoring the value of y, we have du — n {ax -\-x"Y~ l d{ax +£ a ) , which is conformable to the Proposition. The differentiation of ax+x* is here only indicated. Ii «ve nctually perform it, we shall have du — n {ax + x 2 ) n_I (a + 2x) dx. Ex. 1. What is the differential of Va+bx 2 ? bxdx Ans. 'a+bx' Ex. 2. What is the differential of {ax*+x*) 3 ? Ans. Ex. 3. What is the differential of Vax+bx^+cx 3 ! Ans. Ex. 4. What is the differential of {ax-xyi Ans. Ex. 5. What is the differential of {a+bxrf ? Ans. i Ex. 6. What is the differential of {a+x*)~l Aiis. Ex. 7. If x increase uniformly at the rate of T ^ of an inch per second, at what rate is the expression (l+#) 3 increasing when x equals 9 inches ? Ans. (189.) By the application of the preceding principles, com- plicated algebraic functions may be differentiated. a~\-x Ex. 8. What is the differential of the function u= 5 ? a+x According to Prop. XI., (a + x*) dx — 2x {a + x) dx {a+xy which may be reduced to {a— 2ax— x*)dx du- {a+xj Ex. 9. Differentiate the function u— Vx^+a^/x. Ans. Differentiation of Algebraic Functions. 133 (b+xY Ex. 10. Differentiate the function u = - Ex. 11. Differentiate the function u= x Ans. x 1 (a+xj Ans. Ex. 12. Differentiate the function u=- (a+x 2 ) a ' Ans. Ex. 13. If the side of an equilateral triangle increase uni- formly at the rate of half an inch per second, at what rate is its perpendicular increasing when its side is equal to 8 inches? Ans. — — inch per second. 4 Ex. 14. If the side of an equilateral triangle increase uni- formly at the rate of half an inch per second, at what rate is the area increasing when the side becomes 8 inches ? Ans. 2v/3 inches per second. Ex. 15. If a circular plate of metal expand by heat so that its diameter increases uniformly at the rate of T \-$ of an inch per second, at what rate is its surface increasing when the di- ameter is exactly two inches ? Ans. — — inch per second Ex. 16. If a circular plate expand so that its area increases uniformly at the rate of T \ of a square inch per second, at what rate is its diameter increasing when the area of the circle is exactly a square inch? Ans. — inch per second. 50-v/TT r Ex. 17. If the diameter of a spherical soap bubble increases uniformly at the rate of T \ of an inch per second, at what rate is its capacity increasing at the moment the diameter becomes two inches 1 Ans. - inch per second. o Ex. 18. If the capacity of a spherical soap bubble inci eases uniformly at the rate of two cubic inches per second, at what 136 Differential Calculus. rate is the diameter increasing at the moment it becomes two inches? Ans. - inch per second. IT Ex. 19. A boy standing on the top of a tower whose height is 60 feet, observed another boy running toward the foot of the tower at the rate of five miles an hour on the horizontal plane ; at what rate is he approaching the first when he is 80 feet from the foot of the tower ? Ans. 4 miles an hour. Ex. 20. If the diameter of a circular plate expand uniformly at the rate of T \ of an inch per second, what is the diameter of the circle when its area is expanding at the rate of a square inch per second 1 Ans. — inches. 7T Ex. 21. If the diameter of a sphere increase uniformly at the rate of r \ of ah inch per second, what is its diameter when the capacity is increasing at the rate of five cubic inches per second ? Ans. inches. Ex. 22. If the diameter of the base of a cone increase uni- formly at the rate of T y inch per second, at what rate is its solidity increasing when the diameter of the base becomes 10 inches, the height being constantly one foot? Ans. 2tt inches per second. SECTION II. OF SUCCESSIVE DIFFERENTIALS — MACLAURTN'S THEOREM — TAYLOR'S THEOREM — FUNCTIONS OF SEVERAL INDEPEND- ENT VARIABLES. (190.) Since the differentials of all expressions which con- tain x raised to any power, also contain x raised to the next inferior power, Art. 186, we may consider the differential co- efficient of a function as a new function, and determine its dif- ferential accordingly. We thus obtain the second differential coefficient. For example, if tc=ax 3 , du -rr-=2ax . dx Now since 3a« 2 contains x, we may differentiate it as a new function, and we obtain d[~~r) =Gaxdx. But, since dx is supposed to be a constant, ydu\_d(du) d 2 u \dxJ dx dx ' the symbol d*u (which is read second differential of u) being used to indicate that the function u has been differentiated twice, or that we have taken the differential of the differential oftt. Hence * d*u -7— =6axdx : dx or, dividing each side by dx, d 2 u dV =Gax > wnere dx" 2 represents the square of the differential of x, and noi the differential of z\ The expression Gax being the differential coefficient of the -6a, 138 Differential Jalculus. first differential coefficient, is called the second differential co- efficient. Again, since Gax contains x, we may differentiate it as a new function, and we obtain d 3 u — =6adx ; or, dividing each side by dx, dSi dx 3 which is the differential coefficient of the second differential coefficient, and is called the third differential coefficient. d 3 u . The third differential coefficient -=-; is read third differential dx 3 of m, divided by the cube of the differential of x. As the expression 6a does not contain x, the differentiation can be carried no further, and we find the function u=ax 3 has three differential coefficients. Other functions may have a greater number of differential' coefficients. The learner must not confound d 2 u with oV, the former de- noting the differential of the differential of u, and the latter the square of the differential of u. Ex. 1. Determine the successive differentials of ax 4 . Ex. 2. Determine the successive differentials of (a+.r 2 ) 3 . MACLAURIN'S THEOREM. (191.) Maclaurin's theorem explains the method of develop- ing into a series any function of a single variable. Proposition I. — Theorem of Maclaurin. If u represent a function of x which it is possible Jp develop in a series of positive ascending powers of that variable, then will that developinent be , v (du\ (d?u\x* (d 3 u\ x 3 ^^ + U)^(avJ2 + (av)^ + ' etC " where the brackets indicate the values which the inclosed func- tions assume when x equals zero. Let u represent any function of x, as, for example, (a+x) n , and let us suppose that this function, when expanded, will con- Maclauein's Theorem. 139 tain the ascending powers of*, and coefficients not containing x, which are to be determined. Let these coefficients be rep- resented by A, B, C, etc., then we shall have u=A+Bx+Cx-+T>x'+'&x i +, etc. (1) If we differentiate this equation, and divide both sides by dx, we obtain — =B+2Cx+3T)x 2 +4Vx i +, etc. dx If we continue to differentiate, and divide by dx, it is ob vious that the coefficients A, B, C, etc., will disappear in suc- cession, and the result will be as follows : ^=2C+2.3D.r+ 3.4Ez 2 +, etc., dx 2 —= 2.3D +2.3.4E.r +, etc., dx 3 etc., etc. Represent by (u) what u becomes when x=0. Represent by (^) what ^ becomes when x=0. Represent by (^) what -^ becomes when x=0, and so on ; the preceding equations furnish us (u) =A, (§) = B ' (S)=-° • whence we see Substituting these values in equation (1), it becomes which is Maclaurin's theorem. (192.) Ex. 1. Expand (a+x) n into a series. When .r=0, this function reduces to a n . Hence (u) = a\ 140 Differential Calculus. By differentiation, we obtain du , . , s =«(«+x)- f which becomes, when £=0, Hence (JLj— na »-i m Also, ^i=n(n-l) («+*)*-, which becomes, when x=0, w(t7.— l)a n_2 . ai d s u ' d? =n{ - n ~ l) (n_2) ( a + x ) n ~ 3 > which becomes, when x=0, n(n-l)(n-2)a n -\ Substituting these values in Maclaurin's theorem, we have (a+xy=a n +na n - l x+ n{n ~ a"-y+, etc., which is the same as found by the Binomial theorem. Ex. 2. Develop into a series the function u= . a+x By differentiation, we find d u _ 1 dx (a+xy' (£u_ 2 dx*~ (a+x) 3 ' d 3 u _ 2.3 dx 3 ^~{a+xy' Making x=0, in the values of u, of -z-, of -r-r, of -r-„ etc., we ax ax ax find , . 1 fdu\ 1 fd 2 u\ 2 fd 3 u\ 2.3 (M)= v UJ = -^' U?J=^ WvH— » etc - Substituting these values in Maclaurin's theorem, we obtain 1 1 x x 2 x 3 : ~~^+7i - ^T+' etC - a-\x a a* a 3 a* Taylor's Theorem. 141 Ex. 3. Develop into a series the function . r 1—x Ans. l+x+x*+x s +x*+, etc. Ex. 4. Develop into a series the function Va+x. i 1 -i 1-3 1.3 -5 .Ans. a 2 +-a 2 x— — a 2 xM a 2 x 3 — , etc. 2 2.4 2.4.6 ' Ex. 5. Expand into a series (a — x)~ 2 . Ans. Ex. 6. Expand into a series (a+x)~ 3 . Ans. a -3 — 3a _4 a;+6a-V— 10a - V+15a-V-, etc. (193.) When in the application of Maclaurin's theorem the variable x is made equal to 0, the function u, or some of its differential coefficients, may become infinite. Such functions can not be developed by Maclaurin's theorem. Thus, if we have 1 w=log. x, u=coi. x, or u—-, ° X when we make x=0, u becomes equal to infinity. Also, if we have u=ax 2 , the first differential coefficient is du a dx = V ax 2x 2 vvhich becomes infinite when x is made equal to zero. Hence neither of these functions can be developed by Miio laurin's theorem. TAYLOR'S THEOREM. (194.) Taylor's theorem explains the method of developing into a series a function of the sum or difference of two variables. The following principle is assumed in the demonstration of Tavlor's theorem. Proposition II. — Theorem. If we have a function of the sum or difference of two varia- bles, x and y, the differential coefficient will be the same, if we uppose x to vary and y to remain constant, as when we suppose y to vary and x to remain constant. Thus, let u = (x+y)\ f 42 Differential Calculus. If we suppose x to vary and y to remain constant, we have du , . . -=n(x+y)-; and if we suppose y to vary and x to remain constant, we have du : . the same as under the first supposition. Proposition III. — Taylor's Theorem. (195.) .Any function of the sum of two variables may be de- veloped into a series of the following form : „, . du d 2 u v 2 d?u v 3 ^+y)=*+ 5 y+ 5? t2+te-UT S + ' etc - where u represents the value of the function when y=0. Let u' be a function of x+y, which we will suppose to be developed into a series, and arranged according to the powers of y, so t-hat we have it'=F(x+y)=A+By+Cy*+T)y 3 +, etc., (1) where A, B, C, D, etc., are independent of y, but dependent upon x and upon the constants which enter the primitive func- tion. It is now required to find such values for A, B. C, D, etc., as shall" render the development true for all possible values which may be attributed to x and y. If we differentiate under the supposition that x varies and y remains constant, we shall have dtuJ__dA dB dC a dB 3 dx dx dx J dx dx If we differentiate under the supposition that y varies and x remains constant, we shall have du' _ ' _ _ . -3-=B+2Cy+3D2/ 2 +, etc. But, by the preceding Proposition, du' _du' dx dy' Hence we must have dA dB dC ~dx~ + ~dx~ y+ dx~ !/ '' i ~> etC " =B+2Cy+3D — =7i(n-l)(n-2)x a -\ etc. These values being substituted in the formula, give s , n(n— 1) n(n— 1) (n — 2) etc. the same as found by the Binomial theorem. Ex. 2. Required the development of the function %'= Vx+y. i i 1 -1 1 _Ji 1.3 _i Ans. u , =(z+y)*=x z +-x 2 y~^ y+^-^-c y-,etc Ex. 3. Required the development of the function u'=Vx-\-y. i a 1 _a 2-a 2.5 _i Ans. u'=(x+y) 3 =x 3 +-x z y~^ Y+gg-g* Y". etc. (197.) Although the genera*! development of every function of x+y is correctly given by Taylor's theorem, particular values may sometimes be assigned to the variables which shall render this form of development impossible; and this impos- sibility will be indicated by some of the coefficients in Taylor's theorem becoming infinite. Thus, if we have i u'=a-\-(b— x+ y) 2 , u =a-jr(b—x) 3 . Therefore, — = dx 2{b-xf and u =a, where u' and u are expressed under different forms ; and, in general, when the proposed function changes its form by at- tributing particular values to the variables, the development can not be made by Taylor's theorem. DIFFERENTIATION OF FUNCTIONS OF TWO OR MORE INDEPEND- ENT VARIABLES. (198.) Let u be a function of two independent variables x and y ; then, since in consequence of this independence, how- ever either be supposed to vary, the other will remain un- changed, the function ought to furnish two differential coeffi- cients ; the one arising from ascribing a variation to x, and the other from ascribing a variation toy; y entering the first co- efficient as if it were a constant, and x entering the second as if it were a constant. If we suppose y to remain constant and x to vary, the dif- ferential coefficient will be du dx ' and if we suppose x to remain constant and y to vary, the de- ferential coefficient will be du dy (199.) The differential coefficients which are obtained under these suppositions are called partial differential coefficients. The first is the partial differential coefficient with respect to x, and the second with respect to y. If wc multiply the several partial differential coefficients by dx and dy, we obtain du du dx**' iy d y> which are called partial differentials ; the first is a partial dif- ferential with respect to x, and the second a partial differential with respect to y. The differential which is obtained under the supposition that both the variables have changed their values, is called the total differential of the function ; that is, * K 146 Differential Calculus. du dh, du = —dx H — —d v. dx dy J (200.) If we have a function of three variables, x, y, and z, we should necessarily have as many independent differentials, of which the aggregate would be the total differential of the function ; that is, . die du du . du = -j-dx ■\-- r dii+ -r-dz. dx dy J dz Hence, whether the variables are dependent or independent, we conclude that the total differential of a function of any num- ber of variables is the sum of the several partial differentials, arising from differentiating the function relatively to each varia- ble in succession, as if all the others were constants. Ex. 1. If one side of a rectangle increase at the rate of 1 inch per second, and the other at the rate of 2 inches, at what rate is the area increasing when the first side becomes 8 inches, and the last 12 inches? vlns. 28 inches per second. Ex. 2. If one side of a rectangle increase at the rate of 2 inches per second, and the other diminish at the rate of 3 inches per second, at what rate is the area increasing or diminishing, when the first side becomes 10 inches, and the second 8? Ans. Ex. 3. If the major axis of an ellipse increase uniformly at the rate of 2 inches per second, and the minor axis at the rate of 3 inches, at what rate is the area increasing when the major axis becomes 20 inches, the minor axis at the same instant be ing 12 inches ? Ans. 21tt inches. Ex. 4. If the altitude of a cone diminishes at the rate of 3 inches per second, and the diameter of the base increases at the rate of 1 inch per second, at what rate does the solidity vary when the altitude becomes 18 inches, the diameter of the base at the same instant being 10 inches? Ans. R' Then SECTION III. SIGNIFICATION OF THE FIRST DIFFERENTIAL COEFFICIENT- MAXIMA AND MINIMA OF FUNCTIONS. Proposition I. — Theorem. (201.) The tangent of the angle which a tangent line at any point of a curve makes with the axis of abscissas, is equal to the first differential coefficient of the ordinate of the curve. Let CPP' be a curve, and P any point of it whose co-ordinates are x and y. Increase the abscissa CR or x by the arbitrary incre- ment RR', which we will repre- sent by h ; denote the correspond- ing ordinate PR' by y', and draw the secant line SPP. PD=P , R'-PR=3/'-y- But from the triangle PDP' we have PD :PD:: 1 : tang. S=p^; and, substituting for PD and PD their values, we have ^-^=tang. S, ft 3 which expresses the ratio of the increment of y to that of x. In order to find the differential coefficient of y with respect to t, we must find the limit of this ratio by making the incremen equal to zero, Art. 174. Now if ft be diminished, the point P' approaches P, the secant SP approaches the tangent TP ; and, finally, when ft=0, the point P' coincides with P, and the secant with the tangent. In this case we have -p = tang. 1. ax (202.) If it is required to find the point of a given curve at 14S Differential Calculus. which the tangent line makes a given angle with the axis of X, we must put the first differential coefficient equal to the tangent of the given angle. If we represent this tangent by a, we must have * dy ax and this, combined with the equation of the curve, will give the values of a; and y for the required point. ■ Ex. It is required to find the point on a parabola, at which the tangent line makes an angle of 45° with the axis. The equation of the parabola, Art. 50, is y 2 =2px. Differentiating, we obtain 2ydy—2pdx, dy p or -r=—. dx y But since tang. 45° equals radius or unity, we have P_ y~ Whence, from the equation y"~=2px, we find -,-P X ~2' Hence the required tangent passes through the extremity ot the ordinate drawn from the focus. 1, or p=y. OF THE MAXIMA AND MINIMA OF FUNCTIONS OF A SINGLE VARIABLE. (203.) If a variable quantity gradually increase, and, after it has reached a certain magnitude, gradually decrease, at the end of its increase it is called a maximum. Thus, if a line P'R', moving from A along AB so as to be always at right angles to AB, gradually increases until it comes into the position PR, and after that gradually decreases, the line is said to be a maximum, or at its greatest " R' R R" value, when it comes into the position PR. (204.) If a variable quantity gradually decrease and, after Maxima and Minima of Functions. 149 it has attained a certain magnitude, gradually increases, at the end of its decrease it is called a minimum. Thus, if a line F'R', moving from A along AB, gradually decreases until it comes into the position PR, and after that gradually increases, the line is said to be a minimum, or at its least value, when it comes into the position PR. R R R" (205.) If u be a function of x, and if we represent by u the value which u assumes when x is decreased by an indefinitely small quantity, and by u" the value which u assumes when x is increased by an indefinitely small quantity ; then, if u be greater than both u' and u", it will be a maxiiium ; if u be less than both u' and u", it will be a minimum. Hence the maximum value of a variable function exceeds those values which immediately precede and follow it, and the mini- mum value of a variable function is less than those values which immediately precede and follow it. (206.) We have seen, Art. 201, that if?/ represents the or- dinate, and x the abscissa of any curve, the tangent of the an- gle which the tangent line forms with the axis of abscissas, will be represented by dy dx' If PR becomes a maximum, the tan- gent TP, being then parallel with the axis T- of abscissas, makes no angle with this axis, and we have dy dx If PR becomes a minimum, the tangent TP, being then parallel with the axis of abscissas, makes no angle with this axis, and we have dy R dx = 0. R dy Thus, the equation -f-=0 simply expresses the condition that the tangent at P is parallel with the axis of abscissas ; and, 150 Differential Calculus. consequently, the ordinate to that point of the curve may be either at its maximum or minimum value. (207.) In order, therefore, to determine whether a function has a maximum or a minimum value, we make its first differ du ential coefficient, — , equal to zero, and find the value of a: ir this equation. Represent this root by a. Then substitute suc- cessively for x in the given function, a+A and a — h. If both he results are less than the one obtained by substituting a, this value will be a maximum ; if both results are greater, this value will be a minimum. Ex. 1. Find the value of £ which will render u a maximum in the equation u—\0x — x\ Differentiating, we obtain du - r =10-2x. ax Putting this differential coefficient equal to zero, we have 10-2.r=0. Whence x=5. Let us now substitute for x in the given function 5, 5—1 and 5 + 1 successively. Substituting 4 for x, we have u' =40 — 16 = 24. 5 " x, " u =50-25=25. " 6 " x, " w" = 60-36 = 24. The results of the substitution of 5—1 and 5 + 1 for x are both less than that obtained by substituting 5. Hence the function u is a maximum when x=5. Ex. 2. Find the value of x which will render u a minimum in the equation M=.r 2 - 16^ + 70. Differentiating, we obtain du - r -=2x-16. dx Putting this equal to zero, we have 2x-16=0. Whence x=8. Maxima ant Minima or Functions. 151 Let us now substitute for x in the given function 8, 8—1 and 8+1 successively. Substituting 7 for x, u' =19-112 + 70 = 7. 8 " x, u =64-128 + 70 = 6. 9 " x, u"=81 — 144+70=7. The results of the substitution of 8 — 1 and 8 + 1 for x are both greater than that obtained by substituting 8. Hence the function u is a minimum when x=8. (208.) A general method of determining maxima and mini- ma of functions of a single variable, may be deduced from Taylor's theorem. Art. 195. Suppose we have u=F(x), and let the variable x be first increased by h, and then dimin- ished by h ; and let u'=F(x-h), u"=F(x+h) ; then, by Taylor's theorem, we shall have du cPu Ji 2 (Tu h* u ~ u=: dx h+ d?U + d?Y^^ etc - du. d 2 u h 2 cPu h 3 dx dx 1.2 dx 1.2.3 Now, in order that u may be a maximum, it must be greater than either v! or u" ; that is, the second members of both the above equations, for an infinitely small value of h, must be neg- ative ; and in order that u may be a minimum, it must be less than either vf or u" ; that is, the second members of the above equations, for an infinitely small value of h, must be positive. Now when h is infinitely small, the sum of each series in the above equations will have the same sign as the first term, be- cause the first term will be greater than the sum of all the suc- ceeding ones. But the first terms have contrary signs ; hence the function u can have neither a maximum nor a minimum, un- less the first term of each series be zero, which requires that - = 0, dx and the roots of this equation will give all the values of x which can render the function u either a maximum or a minimum. Having made the first differential coefficient equal to zero, the sign of the sum of each series will be the same as the sign of the second differential coefficient. 152 Differential Calculus. If the second differential coefficient is negative, the function is a maximum ; if positive, a minimum. If the second differential coefficient reduces to zero, the sierns of the series will again be opposite, and there can be neither a maximum nor a minimum unless the third differential coefficient reduces to zero, in which case the sign of the sum of each series will be the same as that of the fourth differential coefficient. (209.) Hence, in order to find the values of x which will ren- der the proposed function a maximum or a minimum, we have the following Rule. Find the first differential coefficient of the function ; place it equal to zero, and find the roots of the equation. Substitute each of these roots in the second differential coeffi- cient. Each one which gives a negative result will, when substi- tuted in the function, make it a maximum, and each which gives a positive result will make it a minimum. If either root reduces the second differential coefficient to zero, substitute in the third, fourth, etc., until one is found which does not reduce to zero. If the differential coefficient which does not reduce to zero be of an odd order, this root will not render the function either a maximum or a minimum. But if it be of an even order and negative, the function will be a maximum; if positive, a minimum. Ex. 1. Find the values of a; which will render u a maximum or a minimum in the equation z^=.t 3 -3.c 2 -24z+85. Differentiating, we obtain du -r = 3x 2 -C)x—24. dx Placing this equal to zero, we have 3.z 2 -6x-24=0, or x * — 2x— 8 = 0, the roots of which are + 4 and —2. The second differential coefficient is d'u -j- 2 =6x-6. ax' Substituting 4 for x in the second differential coefficient, the result is 4- IS, which, being positive, indicates a minimum ; sub- Maxima and Minima of Functions If) 3 stituting -2 for x, the result is -18, which, being negative, in- dicates a maximum. Hence the proposed function has a maximum value when x= — 2, and a minimum value when x=4. This result may be illustrated by assuming a series of values for x, and computing the corresponding values of u. Thus, If?— -4, u= 69, x—— 3, m=103, x=— 2 , u=113 ?n aximum. x— — l, u=l05, x= 0, u— 85, x= + l, u= 59, a:=+2, w= 33, x= + 3, u= 13, a:=+4, u= 5 minimum. x=-{-5, u= 15, z= + 6, M = 49. Thus it is seen that the value of the function increases, while x increases from —4 to —2; it then decreases till x=4, and after that it increases again uninterruptedly, and will continue to do so till x equals infinity. This peculiarity may be illus- trated by a figure. If we assume the different values of x to represent the abscissas of a curve, and erect ordinates equal to the corre- sponding values of u, the curve line which passes through the extremities of all the ordinates, will be of the form represented in the annexed figure, where it is evident that the ordinates attain a maximum corresponding to the abscissa —2, and a minimum corresponding to the abscissa 4. Ex. 2. Find the values of a: which will render u a maximum or a minimum in the equation tt=x 3 -18a; 2 +96.r-20. Ans. x=4 renders the function a maximum, and £=8 renders it o minimum. 154 J) 1FFERKNTIAL Cal CULTS. Ex. 3. Find the values of x which will render it a maximum or a minimum in the equation w=x 3 -18x 2 + 105.r. Ans. This tunction has a maximum value when x=5, and a minimum value when x=l. Illustrate these results by a figure, as in the preceding ex- ample. Ex. 4. Find the values of x which will render u a maximum or a minimum in the equation u=x i -l Gx 3 + 88.r - 1 92x + 1 50. Ans. This function has a maximum value when x = 4. and a minimum value when x=2 or G. If we assume a series of values for x, we shall obtain the corresponding values of u as follows : If x=l, u= 31, .t=2, m= 6 minimum. x—3, u= 15, x=4, u= 22 maximum. x=5, u— 15. x=6, u= 6 minimum. x=l, u= 31, £=8, m=150. The curve representing these values has the form represented in the annexed figure, where two minima are seen corresponding to the abscissas 2 and 6, and a maximum corresponding to the abscissa 4. Ex. 5. Find the values of x which will render u a maximum or a minimum in the equation 700 u = x* - 25x 4 + — -x 3 - 1 000.r a + 1 920.T - 1 1 00. o Ans. This function has two maximum values corresponding to x=2 and x=Q. and two minimum values corresponding to x=4 and x=8. (210.) The following principles will often enable us to abridge the process of finding maxima and minima: 6 18 Maxima and Minima of Functions. 155 1. If the proposed function is multiplied or divided by a con- stant quantity, the same values of a; which render the function a maximum or a minimum, will also render the product or quotient a maximum or minimum ; hence a constant factor may be omitted. 2. Whatever value of x renders a function a maximum or a minimum, must obviously render its square, cube, and every other power a maximum or a minimum ; and hence, if a func- tion is under a radical, the radical may be omitted. In the solution of problems of maxima and minima, we must obtain an algebraic expression for the quantity whose maxi- mum or minimum state is required, find its first differential co- efficient, and place this equal to zero ; from which equation the value of the variable x, corresponding to a maximum or a minimum, will be obtained. (211.) The following examples will illustrate these principles. Ex. 1. It is required to find the maximum rectangle which can be inscribed in a given triangle. Let b represent the base of the trian- gle ABC, h its altitude, and x the alti- tude of the inscribed rectangle. Then, by similar triangles, we have CD : CG : : AB : EF, or h : h-x : : b : EF. Hence EF= S (A-*). Therefore the area of the rectangle is equal to EFxGD, or j{hx-x*), which is to be a maximum. But since j is a constant, the quantity hx—x* will also be a maximum, Art. 210. du , —=h-2x=0, ax h Hence or x=: Hence the altitude of the rectangle is equal to half the alti tude of the triangle. 56 Differential Calcui'js. Ex. 2. What is the altitude of a cylinder inscribed in a given right cone when the solidity of the cylinder is a maximum ? EeT a represent the height of the cone, b the radius of its base, and x the altitude of the inscribed cylinder. Then, by similar triangles, we have AD: BD a : b or Hence AE :EF, a-x : EF. EF=-(a-x), Now the area of a circle whose radius is R is ttR 2 (Geom., Prop. XIII., Cor. 3, B. VI.). Hence the area of a circle whose radius is EF is nb* — j-(a— a:)*. Multiplying this surface by DE, the height of the cylinder, we obtain its solidity, —x{a-x)\ which is to be a maximum. 7Tb' Neglecting the constant factor — -, we have 2i—x(a— z)*=a*x— 2ax 2 -}-x\ a maximum. Differentiating, we have du -7-=a'—4ax+ 3x~ = 0, where x may equal a or \a. The second differential coefficient is d 2 u ~r^— — 4a+6x. ax The value x~a reduces the second differential coefficient to a positive quantity, indicating a minimum; the value x=\a reduces this coefficient to a negative quantity, indicating a maximum ; that is, the height of the greatest cylinder is one third the altitude of the cone. Ex. 3. What is the altitude of the maximum rectangle which f.an be inscribed in a given parabola ? Maxima and Minima of Functions Put AT) = a and AE=;c ; then, by A the equation of the parabola y*=2px, we have Hence GE= V2px, and GH=2 ^/2px. Therefore the area of GHK1 is 2 V2px(a—x), which is a maximum, or s/x{a—x) is a maximum. Hence du dx And Hence u— ax-— x~, —i i = \ax 2 — #.r 2 =0. a x 2 or a=3.r, and x—\a. Consequently the altitude of the maximum rectangle is two thirds of the axis of the parabola. There is a parabola whose abscissa is 9, and double ordi- nate 16; required the sides of the greatest rectangle which can be inscribed in it. Ans. Ex. 4. What is the length of the axis of the maximum parab- ola which can be cut from a given right cone ? Put BC=a,AB-&, and CE=x, then BE=a-x,FE=v / a.r-a; a , Geom., Prop. XXII., Cor.,B. IV., and FG=2 . sin. xdx d. versed sin. x~d(R — cos. x)— - . R Ex. 1. Required the differential of the cosine of 65° 10'. Ans. -0.000264. Ex. 2. Required the differential of the cosine of 5° 31'. Ans. —0.000028. As the arc increases, its cosine diminishes; hence its differ ential is negative. Proposition VI. — Theorem. (221 .) The differential of the tangent of an arc is equal to the square of radius, into the differential of the arc, divided by Uic square of the cosine of the arc. Let tt=tanir. x. Q- R sin. x ^ nce tang. x= , we have, Art. 184, cos. x Differentiation of Circular Functions. 169 R cos. xd sin. x—K sin. xd cos. x d. tang. X- cos. 2 X (cos. 5 .r + sin. 2 x)dx cos. 2 X But cos. 2 x+sin. 2 &=R 2 . Hence R 2 e?x d. tang. £= 5 — . Ex. 1. Required the differential of the tangent of 45°. 4ns. 0.00058. Ex. 2. Required the differential of the tangent of 64° 14 Ans. 0.00154. Proposition VII. — Theorem. (222.) The differential of the cotangent of alt arc is negative, and is equal to the square of radius into the differential of the arc, divided by the square of the sine of the arc. Let u=coi. x. du=d cot. x=d tang. (90°-x). (1) But by the last Proposition, v R a d(90°-x) d^ g .(^-x) = co J {90O _^ y Also, d(90°-x) = — dx, and cos. 2 (90°— x) = sin. 2 x. Hence, by substitution, equation (1) becomes Wdx d. cot. x= sin. x Ex. 1. Required the differential of ihe cotangent of 35° 6' Ans. -0.00088. Ex. 2. Required the differential of the cotangent of 21° 35'. Ans. -0.00215. The preceding are the differentials of the natural sines, tan- gents, etc. The differentials of the logarithmic sines and tan- gents may be found by combining Proposition II. with the pre- ceding. 170 Differential Calculus. Proposition VIII. — Theorem. (223.) The differential of the logarithmic sine of an arc, is equal to the modulus of the system into the differential of the arc. divided by the tangent of the arc. By Proposition II., Md sin. x M cos. xdx d JOg. Sin. X — : = — pi — : . sin. x R sin. x R sin x But Trig., Art. 28, tang. x= '—. b ' & C0S- Xm Mdx Hence d log. sin. x= . tang, x Ex. 1. Required the differential of the logarithmic sine of 10' 30", the difference being taken for single seconds. The differential of x, which is 1", must be taken in parts of radius, which, on p. 150 of Tables, is found to be .00000485. Its logarithm is ... . 4.G85575. The modulus M log. . . 9.637784. Mdx 4.323359. tang. 10' 30" 7.484917. 0.000689=0.838442. Therefore 0.000689 is the difference between the logarithmic sine of 10' 30" and. 10' 31", which corresponds with p. 22 of the Tables. Ex. 2. Required the differential of the log. sine of 4° 28'. Ans 0.000027. Proposition IX. — Theorem. (224.) The differential of the logarithmic cosine of an arc is negative, and is equal to the modulus of the systeni into the tan- gent of the arc into the differential of the arc, divided by the square of radius. By Proposition II., Md cos. x M sin. xdx a log. cos. x= = p^ . cos. x R t_os. x ,„ „, . sin. x tan a result which is the same for all values of b. Differentiating equation (2), we obtain g=0. (3) This last equation is entirely independent of the values of a and b, arid is equally applicable to every straight line which can be drawn in the plane of the co-ordinate axes. It is called the differential equation of lines of the first order. (230.) If we take the equation of the circle x-+if = W, (1) and differentiate it, we obtain 2xdx+2i/dy=0, dy - X (9\ Tx—'y (2) Equation (2) is independent of the value of the radius R, and ncnce it belongs equally to every circle referred to the same co-ordinate axes. If we take the equation of the parabola y'=2px, (1) and differentiate it, we find 2ydy=2pdx ; 176 Differential Calculus. whence —-=—. (2) dx y * ' But from equation (1), p = ~- Hence equation (2) becomes dy y ■ dlr~¥x ( 3 ) This equation is independent of the value of the parameter 2p, and hence it belongs equally to every parabola referred to the same co-ordinate axes. (231.) If we take the general equation of lines of the second order, which is, Art. 132, y t =mx+nx !> , (1) and differentiate it, we obtain 2ydy=m dx + 2 nxdx. (2) Differentiating again, regarding dx as constant, we obtain 2dif + 2yd 2 y=2ndx\ or, dividing by 2. dy 2 + yd 2 y= ndx\ (3) Eliminating m and n from equations (1), (2), and (3), we obtain y\lx n '+x i dy- + yx^d'y — 2xydxdy = 0, which is the general differential equation of lines of the second- order. Hence we see that an equation may be freed of its constants by successive differentiations ; and for this purpose it is neces- sary to differentiate it as many times as there are constants to be eliminated. The differential equations thus obtained, to gether with the given equation, make one more than the num- ber of constants to be eliminated, and hence a new equation may be derived which will be freed from these constants. The differential equation which is obtained after the con- stants are eliminated, belongs to a species of lines, one of which is represented by the given equation. (232.) We have seen, Art. 201, that the tangent of the angle which a tangent line at any point of a curve makes with the axis of abscissas, is equal to the first differential coefficient of the ordinate of the curve. We are enabled from this princi- ple to deduce general expressions for the tangent and subtan- gent. normal and subnormal of any curve. DIFFERENTIAL CALCULUS OF CtTRVES. 177 Proposition I. — Theorem. (233.) The length of the subtangent to any point of a curve referred to rectangular co-ordinates, is equal to the ordinate multiplied by the differential coefficient of the abscissa. In the right-angled triangle PTR, we have, Trig., Art. 42, 1 : TR : : tang. T : PR ; dy that is, 1 :TR: dx ' V- Hence the subtangent TR=y dx dy R Proposition II. — •Theorem. (234.) The length of the tangent to any point of a curve re- ferred to rectangular co-ordinates, is equal to the square root of the sum of the squares of the ordinate and subtangent. In the right-angled triangle PTR, TF=PR 2 + TR 2 ; dx 2 that is, TF-=y-+y"- ' Hence the tangent dy- I , „dx* I dx* dy 9 *V ' dy* Proposition III. — Theorem. (235.) The length of the subnormal to any point of a curve, is equal to the ordinate multiplied by the differential coefficient of the ordinate. In the right-angled triangle PRN, we have the proportion 1 : PR : : tang. RPN : RN. But the angle RPN is equal to PTR ; hence 1 : PR : : tang. PTR : RN ; that is, 1 : y dx ay Hence the subnormal RN=y-f-. M 178 Differential Calculus. Proposition IV. — Theoreih. (236.) The length of the normal to any point of a curve, is equal to the square root of the siun of the squares of the ordinate and subnormal. In the right-angled triangle PRN, PN 2 = PR 2 +RN 2 : that is, PN 2 =t/ 2 +^ dx 1 Hence the normal PN =yy- =y\/i + dx* ' ay dx 2 ' (237.) To apply these formulas to a particular curve, we dx d'li must substitute in each of them the value of — or -f-, obtained dy dx • by differentiating the equation of the curve. The results ob- tained will be true for all points of the curve. If the values are required for a given point of the curve, we must substitute in these results for x and y the co-ordinates of the given point. Let it be required to apply these formulas to lines of the second order whose general equation is y*=mx-{-nx*. Differentiating, we have dy m+2nx ??i + 2nx dx~ 2y ~2Vmx+nx*' Substituting this value in the preceding formulas, we find dx 2{mx-{-nx i ) The subtan^ent ■y dy m-\-2nx The tangent = yV +^=\/ *f (?nx + nx mx + nx +41 — \ m-\-2nx The subnormal \'/ dy m+2nx dx~ 2 ' The normal -Vx»+y»='R\ which corresponds with Art. 40. Ex. 2. Find the equation of the tangent line to a parabola. 182 Differential Calculus. SUBTANGENT AND TANGENT OF POLAR CURVES. (244.) The subtangenl of a polar curve is a line drawn from the pole perpendicu- lar to a radius vector, and limited by a tangent drawn through the extremity of the radius vector. Thus, if MT is a tangent to a polar curve at the point M, P the pole, and PM the radius vector, then PT, drawn per- pendicular to PM, is the subtangent. Proposition V. — Theorem. (245.) The length of the subtangent to a polar curve is equal to the square of the radius vector, multiplied by the differential coefficient of the measuring arc. Represent the radius vector PM by r, and the measuring arc ba by t (the radius Pa of the measuring circle be- ing equal to unity). Suppose the arc t to receive a small increment aa', and through a' draw the radius vector PM'. With the radius PM describe the arc MN; draw the chord MN, and draw PT parallel to MN. Now aa' is the increment of t, and \ M'N is the increment of?-; and in or- '' "' der to find the differential coefficient of r (t being considered the independent variable), we must find the ratio of the incre- ments of t and r, Art. 174, and determine the limit of this ratio by making the increment of/ equal to zero. By Geom., Prop. XIII., Cor. 1, B. VI., we have 1 : aa' : : PM : arc MN. arc MN Hence PM (1) Also, the similar triangles M'NM, M'PT furnish the pro- portion M'N : chord MN : : MP : PT. SUBTANGENT AND TANGENT OF PoLAR CURVES. 183 Whence M'N= t^t ' ^ Consequently, from equations (1) and (2), aa> _ arc MN PT M 7 N~cIoTa r MN X PM X PM" which is the ratio of the increments of t and r; and we must now find the limit of this ratio when the increment of t is made equal to zero. It is evident that the ratio of the arc MN to the chord MN will be unity, Art. 218, Cor. 2 ; also, PT will be the subtangent, and PM' will become equal to PM, which is represented by r. dt PT Hence ^-r, PT r * dt which is the value of the subtangent. PT Cor. The tangent of the angle PMT is equal to p^|, which rdt therefore becomes -7-. which represents the tangent of the an- dr l gle which the tangent line makes with the radius vector. Pkoposition VI. — Theorem. (246.) The length of the tangent to a polar curve, is the square root of the sum of the squares of the subtangent and radius vector. For the tangent MT is equal to VMP 2 +PT 2 , which is equal / r*d¥ t0 r V 1+ "^T (247.) It is required to apply these formulas to the spirals. The equation of the spiral of Archimedes, Art. 148, is _ t dt Whence -j-—2tt. dr ■ Substituting the values of r and — in the general ejpressior, for the subtangent, Art. 245, we have subtan"ent=—. 2n 184 Differential Calculus. Tf t—2rr, that is, if the tangent be drawn at the extremity of the arc generated in one revolution, we have the subtangent=2n=the circumference of the measuring circle. If t=2?nT, that is, if the tangent be drawn at the extremity of the arc generated in m revolutions, we have subtangent— m . 2m-x ; that is, the subtangent after m revolutions, is equal to m times the circumference of the circle described with the radius vector of the point of contact. 248.) The equation of the hyperbolic spiral, Art. 151, is a Whence -7-= — . dr a Substituting this value in the general expression for the sub- tangent, we have r-e subtangent= — — a ; that is, in the hyperbolic spiral the subtangent is constant. (249.) The equation of the logarithmic spiral, Art. 155, is t=\og. r. Whence dt— , r and ~=M, dr which represents the tangent of the angle which the tangent line makes with the radius vector, Prop. V., Cor. ; that is, the tangent of the angle which the tangent line makes with the radius vector is constant, and is equal to the modulus of the system of logarithms employed. DIFFERENTIALS OF AN ARC, AREA, SURFACE, AND SOLID OF REVOLUTION. Proposition VII. — Theorem. (250.) The limit of the ratio of the chord and arc of any curve ts unity. Let ADB be an arc of any 7 curve, AB=c the chord, and let Differentials of an Arc, Area, etc. 185 the tangents AC, CB be drawn at the extremities of the arc. It is evident that the arc ADB is greater than the chord c, but less than the sum of the two tangents a and b. By Trigonometry, Art. 49, a c sin. C a+b sin. A + sin. B sin sin. A . b sin. B - and -=—. — ^. c sin. C Therefore •sin B sin. (A + B) c sin. C By Trigonometry, Art. 76, sin. A + sin. B_cos. A(A — B) sin. (A+B)~ - cos. £(A + B)* a+b cos. i(A-B) Hence — ^ o i , A ,r>\" c cos. £(A + r>) Conceive now the points A and B to approach each other, and the arc ADB to decrease continually, the angles A and B will manifestly both decrease, and they may become less than any assignable angle whatever ; therefore A— B and A+B both ap- proach continually to ; and cos. i(A-B) and cos. |(A+B) ap- proach to unity, which is their common limit. Hence the limit of the ratio of a+b to c is a ratio of equality ; and as the arc ADB can not be greater than a+b, nor less than c, much more is the limit of the ratio of the arc to the chord, a ratio of equality. Proposition VIII. — Theorem. (251.) The differential of the arc of a curve referred to rect- angular co-ordinates, is equal to the square root of the sum oj the squares of the differentials of the co-ordinates. We have found, Art. 250, that the limit of the ratio of the chord and arc of a curve is unity ; hence the differential of an arc is equal to the differential of its chord. Let x represent any abscissa of a curve, AR for example, and y the cor- responding ordinate PR. If now we give to x any arbitrary increment h, and make RR' = />, the value of?/ will become equal to P'R', which we will represent l8G Differential Calculus. by y'. If we draw PD parallel to the axis AR', we snail have the chord PF= VPD 2 +FD ,J = v7* 2 +P'D 2 . ^ ^. ^, dy, d\i If , ...... But r'\J=y'— pj^Hj^- + other terms involving highei powers of A, Art. 195. Substituting this value of P'D in the expression for the chord. we have dx* V dy' + ^ + ' etC - Therefore — -=v/ 1 +^+, etc., a v fife which expresses the ratio of the increment of the function to that of the variable, and we must find the limit of this ratio by making the increment equal to zero, Art. 174. In this case the chord becomes equal to the arc, which we will represent by z, and the terms omitted in the second mfem ber of the equation containing h disappear ; hence dz J d^_ dx * dx* and, multiplying by dx, dz = vdz'+dy*. (252.) To determine the differential of the arc of a circle, take the equation xdx whence xdx+ydy=0, or dy— , x*dx* dx and dz=\/dx*+ — ^=— Vz*+y°. v y y __ But Vx i +y 2 =R, and y= VR' 2 -x\ Hence dz= - "— . See Art. 226. VR'-x' Proposition IX. — Theorem. (253.) The differential of the area of a segment of any curve referred to rectangular co-ordinates, is equal to the ordinate into the differential of the abscissa. Let APR be a surface bounded by the straight lines AR, Differentials of an Arc, Area, etc. 1R7 PR, and the arc AP of a curve ; it is re- quired to find the differential of its area. Let x represent the abscissa AR, and ij the corresponding ordinate PR. If we give to x an increment h, and make RR'=//, the value of y will become P'R', which we will represent by y'. Since the limit of the ratio of the chord and arc of a curve is unity, the limit of the ratio of the area included by the ordi- nates PR, P'R' and the arc PP', to the trapezoid included by the same ordinates and the chord PP', must be a ratio of equality. Now the trapezoid PRRT'=RR'xi(PR-i-PR')=i*(y+y')- Hence -, =^.{y+y)- h But Hence that is, ax cpy If y >=y+-SLh+-^'^-+, etc., Art. 195. dz 2 d\i h PRR'P'_ dyh h ~ y+ dx 2 — =!/+-t:7t+» etc -» which expresses the ratio of the increment of the function to that of the variable, and we must find the limit of this ratio by making the increment equal to zero, Art. 174. But in this case, all the terms in the second member of the equation which contain h disappear, and representing the area of the segment by s, we have ds dx =y, or ds=ydx. (254.) Ex. To find the differential of the area of a circular segment, take the equation y-= W-x\ Whence y = VIV— x~. Hence ds=ydx=dx V IV — x~. The equation of the circle, when the origin of co-ordinates is placed on the circumference, is y= V2rx—x'\ and hence the differential of the area becomes dxV2rx—x i . 188 Differential Calculus. Proposition X, — -Theorem. (255.) The differential of a surface of revolution, is equal to the circumference of a circle perpendicular to the axis, multi- plied by the differential of the arr of the generating curve. Let the curve APP' be revolved about the axis of X, it will generate a surface of revolution; and it is required to find the differential of this surface. Put AR=a; and PR=?/. If we give to x an increment A=RR', the value of y will become P'R', which we will repre sent by y'. In the revolution of the curve APP', the points P and P' will describe the circumferences of two circles, and the chord PP' will describe the convex surface of a frustum of a cone. Also since the limit of the ratio of the chord and arc of a curve is unity, the limit of the ratio of the surface described by the chord to the surface described by the arc must be a ratio of equality. Now the surface described by the chord PP' is equal to PP' — X(circ. PR+«Vc. P'R'), Geom., Prop. IV., B. X pp which equals (2-ny-\-2-y'), or Hence PP'XTr(y+3/'). the surface of frustum PP =*(y+y'). But dy , dh/ /i 2 y yJr ~t + 7^9" + ' etc '' Art * 195 ' and Hence dx dl J-, 7j'+y=2y+~h+, etc the surface of frustum *y* pp' =7r v%+#+> etc -), which expresses the ratio of the increment of the function tc that of the variable, and we must find the limit of this ratio by making the increment equal to zero, Art. 174. But in this case, all the terms in the second member of the equation which contain h disappear, and representing the arc Differentials of an Arc, Area, etc. 18D AP by z, and the surface described by the arc AP by S, we have — =2ny, or dS=2~ydz ; dz and, by substituting for dz its value, Art. 251, we nave d$ = 2mj(dx 2 +dif)K where 2^y is the circumference of the circle described by the point P. Proposition XL — Theorem. (256.) The differential of a solid of revolution is equal to the area of a circle perpendicular to the axis, multiplied by the dif- ferential of the abscissa of the generating curve. Let the surface APR be revolved about the axis of X, it will generate a solid of revolution, and it is required to find its differential. Put AR=z and PR=y. If we give to x an increment A=RR', the value of y will become P'R', which we will represent by y'. In the revolution of the surface APR', the trapezoid PRR'P will describe the frustum of a cone, and the limit of its ratio to the solid described by the surface included by the ordinates PR, PR', and the arc PP', is a ratio of equality. Now the solidity of the frustum described by the trapezoid PRR'P', Geom., Prop. VI., B. X., is j7TxRR'(PR 2 +PR' 2 +PRxPR'), or 3' h(y*+y"+yy'). the solidity of the frustum -.,,,„. A ce _, J-^l — i =\My +y' +yy% . Hen But Hence Also, Therefore du , dry li 3 sf-y+#+^2 + ' ete -' ArL19B - yy'=y t +£ky+> etc - ody y>+y'*+yy'=3y*+^hy+, etc.; 190 Differential Calculus. that is, solidity of frustum , ody =M% a +-^Ay+, etc.), h which expresses the ratio of the increment of the function to that of the variable, and we must find the limit of this ratio by making the increment equal to zero, Art. 174. But in this case, all the terms in the second member of the equation which contain h disappear, and representing the vol- ume of the solid generated by V, we have or dV=ny*dx, where -ny" 1 is the area of the circle described by PR. DIFFERENTIAL OF THE ARC AND AREA OF A POLAR CURVE Proposition XII. — Theorem. (257.) The differential of an arc of a polar curve, is equal to the square root of the sum of the squares of the differential of the radius vector, and of the product of the radius vector by the differential of the measuring arc. Let PM, PM' be two radius vectors of j. polar curve, and let MC be drawn from iVI perpendicular to PM'. Then, in the fight-angled triangle M'CM, we have chord M'M: CM CM 7 Also, Therefore VM'C'+CM 2 tane. CM'M. chord M'M = Vl+tang. 3 CM'M. We must now find the limit o : this ratio, by making the in- crement of the radius vector equal to zero. The limit of the ratio of the chord MM' to the arc MM' is unity, Art. 250. Also, M'C approaches to M'N, which is the increment of the radius vector, and the limit of their ratio is unity ; and the an- rdt gle CM'M becomes PMT, which is equal to — , Prop. V., Cor ar Hence, representing the arc by z, we have Asymptotes of Curves. 191 dz dr r*dt % dr' or dz=Vdr z +r i dt\ which is the differential of the arc of a polar curve. Pr OPOSITION XIII.- -Theorem. (258.) The differential of the area of a segment of a polar curve, is equal to the differential of the measuring arc, multiplied by half the square of the radius, vector. Let PMD be any segment of a polar curve, and let the measuring arc receive M^ a small increment aa'; the increment of the area will be PMM'. The area of the sector PMN, Geom., Prop. XII., Cor. B. VI., is equal to MN X ^-. And since aa! : MN : : 1 : PM, MN PM" sector PMN PM 2 aa Therefore aa' 2 Now since the limit of the ratio of PM' to PM is a ratio of equality, the limit of the ratio of PMM' to the circular sector PMN is a ratio of equality. Taking the value of this ratio when the increment is equal to zero, representing the segment by s, and the measuring arc by t, we have ds r 2 Tt = ^ . r*dt or ~2~' which is the differential of the area of a segment of a polai curve. ASYMPTOTES OF CURVES. (259.) An asymptote of a curve is a line which continually approaches the curve, and becomes tangent to it at an infinite distance from the origin of co-ordinates. In some curves the distance between the origin of the co *92 Differential Calculus. ordinates and the point in which the tangent meets the axes increases continually with the abscissa, so that when the ab- scissa x becomes infinite, this distance is infinite. In other curves, even when the abscissa becomes infinite, the tangen 1 cuts the axes at a finite distance from the origin. It is then called an asymptote to the curve. Let A be the origin of co-ordinates, and let TP be a tangent to the curve at a point whose co-ordinates are AR=x* and YJi—y. If from the subtangent dx TR, which equals y -=-, the abscissa AR be subtracted, the remainder, AT=y- — x, dy T (1) R is the general expression for AT, the distance from the origin at which the tangent intersects the axis of X. Also, Hence PC=BC tang. PBC=%. dx AB: ,VK-VC=y-% x , (2) which is a general expression for AB, the distance from the origin at which the tangent intersects the axis of Y. (260.) If, when x and y become infinite, either of the ex- pressions (1) and (2) reduces to a finite quantity, we may con- clude that the curve has asymptotes ; but if both be infinite, then the curve has no asymptotes. If both the expressions are finite, the asymptote will be inclined to both the co-ordinate axes ; if one of the values becomes finite and the other infinite the asymptote will be parallel to one of the co-ordinate axes if both become zero, the asymptote will pass through the origin of co-ordinates. (261.) Ex. 1. It is required to determine whether the hy- perbola has asymptotes. The equation of the hyperbola, when the origin of co-ordi nates is at the center, is (Art. 98) T>2 Differentiating, we find Asymptotes of Curves. 19S dx _Ay_ x'-A 3 dy V ~ Wx~ x ' . m dx A a Therefore AT=V-j — x= v J ay x A a The expression represents the distance from the origin of co-ordinates at which the tangent intersects the axis of X. When x is supposed infinite, this expression becomes equai to zero. Hence the hyperbola has asymptotes which pass through the center. Ex. 2. It is required to determine whether the parabola has asymptotes. The equation of the parabola is y*=2px. Differentiating, we find dx if y—=?-=2x. dy p dx Hence AT=y- — x=x. When x is infinite, this quantity becomes infinite ; therefore the parabola has no asymptotes. Ex. 3. The equation of the logarithmic curve is x=\og. y. or 3/=« x - Tf x be taken infinite and negative, then 1 y=— =0; that is, the axis of abscissas is an asymptote to the curve. See fig., page 108. N SECTION VI. RADIUS OF CURVATURE— EVOLUTES OF CURVES. (2G2.) The curvature of a curve is its deviation from the tangent ; and of two curves that which departs most rapidly from its tangent, is said to have the greatest curvature. Thus, of the two curves AC, AD, having the common tangent AB, the latter de- ^^-' "^-^ B parts most rapidly from the tangent, and / x^ is said to have the greatest curvature. D (263.) The curvature of the circumference of a circle is evidently the same at all of its points, and also in all circum- ferences described with equal radii, since the deviation from the tangent is the same ; but of two different circumferences, that one curves the most which has the least radius. Thus, the circumference ADF departs more rapidly from the tangent line AB than the circumference ACE, and this devia- tion increases as the radius decreases, and conversely. In different circum- ferences the curvature is measured by the angle formed by two radii drawn through the extremities of an arc of given length. Proposition I. — Theorem. (264.) The curvature in two different circles varies inversely as their radii. Let R and R' represent the radii of two circles, A the length of* a given arc measured on the circumference of each ; a the angle formed by the two radii drawn through the extremities of the arc in the first circle, and a' the angle formed by the corresponding radii of the second. Then, by Geom., Prop XIV., B. III., we have 360A 2ttR : A : : 360° : a ; whence a— r> ; Radius of Curvature. 196 360° : a' ; whence a 360A 360A a : : 2ttR ' 2ttR" a' :: 1 R 1 : R' ; A B D M 360A and 2-R : A : : 360° : a' ; whence a' =7—57. 2ttK 3f>0 A SfiO A Therefore or that is, the curtature in two different circles varies inversely as their radii. (265.) Let DBE be any curve line, ABC a tangent at the point B, and B3I a normal at the same point, then will ABC be a tangent to the circum- ference of every circle passing through B, and having its center in the line BM. The curve DBE may, therefore, be touched by an infinite number of circles at the same point B. Some of these circles, having a greater curvature than the curve, fall wholly w T ithin it ; while others, having a less degree of curvature, fall between the curve and the tangent. Of this infinite number of circles, there is one which coincides most in- timately with the curve, and is hence called the oscillatory cir- cle, or circle of curvature, and its radius is called the radius of curvature of the curve. The osculatory circle may be found in the following manner. (266.) Let there be two curves w T hich meet at the point P, and let us designate the co-ordinates of one curve by x and y, and the co-ordinates of the second curve by x' and y'. If we suppose x to receive an increment and become x+h, we shall have _,_ clii d~y If ffy h 3 FR'=y +-fh +-^ - + -jh — +, etc., J ax dx 2 dx 2.3 P"R'=y'+^ 7 / i +33i-+33i^+ > etc. (1) (2) dif dry' If d 3 y' h 5 dx' dx" 2 dx' 3 2.3 But since the point P is common to the two curves, we musi have y—y'' Also, since the first differential coefficient represents the tan ffent of the an^le which a tangent line makes with the axis oi "96 Differential Calculus. abscissas, if we suppose the two curves to have a common tangent at P, we must have ax ax' Now if all the terms in the first of these developments are equal to the corresponding terms in the other, the curves will be identical ; and the greater the number of terms which are equal in the two developments, the more intimate will be the contact of the curves. Since the general equation of the circle contains but three constants, the equality of y and of the first and second differential coefficients in the equations of the curve and circle will give three equations by which the magnitude and position of the circle may be determined ; and therefore a circle will coincide most nearly with a given curve, when its first and second differential coefficients are equal to the first and sec- ond differential coefficients of the equation of the curve. (267.) Since the contact of the osculatory circle with a curve is so intimate, its curvature is regarded as measured by means of the osculatory circle. Thus, if we assume two points in the curve PP', and find the radii r and r' of the circles which are oscu- latory at these points, we shall have curvature at P : curvature P' : : - : — ; r r that is, the curvature at different points varies inversely as the radius of the osculatory circle Proposition II. — Theorem. (2G8.) The radius of curvature at any point of a given curve is equal to dz 3 dxd z y where x and y are the co-ordinates of the given point, and z the arc of the given curve. The general equation of the circle, Art. 38, is (x- a y+(y-by=n\ ivhere a and b are the co-ordinates of the center of the circle and R is the radius. Differentiating this equation, and dividing by 2, we have (x — a)dx+(y— b)dy=0. Radius of Curvature. 197 Differentiating again, regarding dx as constant, we obtain dx'+dy'+^-^dry^O. dx'+df dy ax~-\-ay . . Whence y— o= ^ — , {}) dy(dx*+dy*\ and x - a= Tx\—dY~)' () Substituting these values in the equation of the circle, we have df(d_£+dyy ( dx* + dyy R = dx~\~lhr) + \TW~J ' or R 5 = and dx*+dy = ^-j . 198 Differential Calculus. 2nydx 1 —(m+2nx)dxdy Also, c?y=—2 ^_ > *. _[4nj/ 2 — (w+2»x) 2 ]^x 2 _— m 2 204 Whence Also, Whence Differential Calculus. y=-h. x - a =~T^y- h y \ f 2ry—y' X2y, = — 2V2ry—y* x=a—2V2ry—y*. Substituting these values of x and y in the transcendenla' equation of the cycloid, Art. 140, x=arc(ve? , sed si?ie=y)~ V2ry—y*, we obtain a— 2 V— 2rb— b'= arc (versed sine——b)— V — 2rb—b*, or a=arc(versed sine=—b) + V — 2rb— b'\ which is the transcendental equation of the evolute referred to the primitive origin and the primitive axes. This is also the equation of a cycloid whose generating cir- cle is equal to that of the given one, and whose vertex coincides witn the extremity of the base, lying, however, below the base, as appears by substituting — b for y in the equation of Art. 140. Thus, the evolute AA' of the cycloid is an equal cycloid ; the arc AA' is identical with AB, and the vertex B is trans- ferred to A. SECTION VII. ANALYSIS OF CURVE LINES. (279.) If it was possible to resolve an equation of any de- gree, we might fol.ow the course of a curve represented by any Algebraic equation, by methods explained in Analytical Geometry. By assigning to the independent variable different values, both positive and negative, we could determine any number of points of the curve at pleasure. The Differential Calculus enables us to abridge this investi- gation, and may be employed even when the equation of the curve is of so high a degree that we are unable to obtain a gen- eral expression for one of the variables in terms of the other. The first object aimed at in such an analysis, is to discover those points of a curve which present some peculiarity; such as the point at which the tangent is parallel or perpendicular to the axis of abscissas. Such points have been named singu- lar points.. A singular point of a curve is one which is dis- tinguished by some remarkable property not enjoyed by the other points of the curve immediately adjacent. Proposition I. — Theorem. (280.) For a point at which the tangent to a curve is paral- lel to the axis of abscissas, the first differential coefficient is equal to zero. For the first differential coefficient expresses the value of the tangent of the angle which the tangent line forms with the axis of abscissas, Art. 201 ; and when this line is parallel with the axis, the angle which it forms with the axis is zero, and its tangent is zero. Proposition II. — Theorem. (281.) For a point at which the tangent to a curve is perpen- di'-ular to the axis of abscissas, the first differential coefficient is equal to infinity. For the first differential coefficient expresses the vaiue of 206 Differential Calculus. the tangent of the angle which the tangent line forms with the axis of abscissas ; and when this angle is 90 degrees, its tan- gent is infinite. Ex. 1. Itts required to determine at what point the tangent to the circumference of a circle is parallel to the axis, and where it is perpendicular. Take the equation x*+y*=R\ By differentiating, we obtain dy x dx y' and placing this equal to zero, we find x=0. But when x=0, we have y=±R; hence the tangent is parallel to the axis of abscissas at the two points where the circumference intersects the axis of ordinates. If we make dy x y „ ■/== — =oo, or --=0, dx y x we find y=0. But when y—0, we have x=±R; that is, the tangent is perpendicular to the axis of abscissas at the two points where the circumference intersects the axis of abscissas. Ex. 2. It is required to determine at what point the tangent to a cycloid is parallel to the base, and when it is perpendicu- lar to the base. Proposition III. — Theorem. (282.) If a curve is convex toward the axis of abscissas, the ordinate and second differential coefficient will have the same sign. Let PP'P" be a curve convex toward the axis of abscissas ; and let x and y be the co-ordinates of the point P. Let x be increased by any arbitrary increment RR', which we will rep- Analysis of Curve Lines. 207 resent by k, and take R'R" also equal to h. Draw the ordinates P'R', P"R" ; draw the line PP\ and produce it to B; join P' and P", and draw PD, P'D' parallel to AR. We shall then have PR=*/, p, R , =y+ £+£-+, etc. Also, dy , cT-yAh 2 Hence dx dx* P'D=P'R'-PR= (1) (2) R n' R" dv , dry h* ,„ N Subti acting equation (1) from equation (2), we have P'D'=P"R"-P , R'=^A+^^-+, etc. (4) dx dx 2 Subtracting equation (3) from equation (4), remembering that BD' is equal to P'D, we have P'B=P"D'-P'D=^/r+, etc. (5) Now when we suppose h to be taken indefinitely small, the sign of the second member of equation (5) will depend upon that of the 'first term; and since the first member of the equa- tion is positive, the second must also be positive ; that is, the second differential coefficient is positive ; and the ordinate, be- ing situated above the axis of abscissas, is also positive. If the curve is below the axis of abscissas, we shall have dry -p"b=p"d'-p'd=-7+h' i +, etc. ; and since the first member of this equation is negative, the sec- ond will also be negative ; that is, the second differential coef- ficient is negative. Whence we conclude that if the curve is convex toward the axis of abscissas, the second differential co- efficient will be positive when the ordinate is positive, and neg- ative when the ordinate is negative. 208 Differential Calculus. Proposition IV. — Theorem. (283.) If a curve is concave toward the axis of abscissas, the ordinate and second differential coefficient will have contrary signs. Let PP'P" be a curve concave to- ward the axis of abscissas ; and let x and y be the co-ordinates of the point P. Let x be increased by any arbitrary increment RR', which we will represent by h, and take R'R" A also equal to h. Draw the ordinates P'R\ P"R" ; draw the line PP', and produce it to B. Join P' and P", and draw PD, P'D' parallel to AR. We shall then have ?R=y, B P>^ D' *?y D / R R' R" \ d ? \ V d' V ^ X b „ „ dii d 2 y A a Also, P^ =y+ | 2 A + ^ + ,etc. d*y4W 2 0) (2) (3) Hence P'D=P'R'-PR=^A+^ |+, etc. Subtracting equation (1) from equation (2), we have P»D'=P"R"-P'R'=A+^ ^+, etc. (4) dx dx 2 Subtracting equation (3) from equation (4), remembering that BD' is equal to P'D, we have d 2 v P"B=P"D'-P , D=-p^+> etc. (5) Now since the first member of this equation is negative, the second member must also be negative ; that is, the second dif- ferential coefficient will be negative, while the ordinate is posi- tive. If the curve is below the axis of abscissas, we shall have d*V +j t ,''Z,=p"^_ J0 ^=^/i 2 +, etc., where the second differential coefficient is positive, while the ordinate is negative. Analysis of Curve Lines. 209 Ex. 1. It is required to determine whether the circumference of a circle is convex or concave toward the axis of abscissas The equation of the circle is x 2 +y 3 =R a ; dy x whence -7-— — ax y $y_ x->+if_ R» A1S0 ' dx>~ y* ~ y" ■ which is negative when y is positive, and positive when y is negative. Hence the circumference is concave toward the axis of abscissas. Ex. 2. It is required to determine whether the circumference of an ellipse is convex or concave toward the axis of abscissas. (284.) Definition. A point of inflection is a point at which a curve from being convex toward the axis of abscissas, be- comes concave, or the reverse. Proposition V. — Theorem. For a point of inflection, the second differential coefficient must be equal to zero or infinity. When the curve is convex toward the axis of abscissas, the ordinate and second differential coefficient have the same sign, but when the curve is concave, they have contrary signs. Hence, at a point of inflection, the second differential coeffi- cient must change its sign. Therefore, between the positive and negative values there rousl be one value equal to zero or infinity ; and the roots of the equation dry d 2 y d?^ 0r dx-^' will give the abscissas of the points of inflection. Having discovered that the second differential coefficient for a certain point of a curve is equal to zero or infinity, we increase and diminish successively by a small quantity h, the abscissa of this point; and if the second differential coefficient has con- trary signs for these new values of x, we conclude that here is a point of inflection. Ex. 1. Determine whether the curve whose equation is y=a + (x— b) 3 , has a point of inflection. 210 Differential Calculus. By differentiating, we find and £■*-* g-*-* A When x=b, the first differential coefficient is zero, and the tangent is parallel to the axis of abscissas at the point whose co-ordinates are x=b, y=a. When xb, the second differential coefficient is positive; that is, the second differential coefficient changes its sign at the point of the curve of which the ab- scissa is x=b; consequently there is an in- flection of the curve when x=b. On the left of P the curve falls below the tangent line TT', while on the right of P it runs above TT'. Ex. 2. Determine whether the curve whose equation is y=a— (x— b) 3 , has a point of inflection. Ans. The curve is first convex and then concave toward the axis of ab- scissas, and there is an inflection at the point x=b. Ex. 3. Determine whether the curve whose equation is y=3x+18x*-2x% has a point of inflection. By differentiating, we find -l=3+3Gx-6x\ ax — T' and ^=36-12z. ax 2 Putting the second differential coefficient equal to zero, we obtain x=3. Take, therefore, AB=3, and draw the ordinate BC : C is the point of inflection. If x be be- tween and 3, 36— 12a; is posi- Analysis of Curve Lines. 211 tive ; therefore the part AC of the curve is convex to AB ; but when x is greater than 3, 36— 12x is negative, and there- fore the curve is concave toward the axis. (285.) Definition. A multiple point is a point at which two or more branches of a curve intersect each other. Proposition VI. — Theorem. For a multiple point, the first differential coefficient must have several values. It is obvious that where two branches of a curve intersect, there must be two tangents which have different values ; and since the first differential coefficient expresses the tangent of the angle which the tangent makes with the axis of abscissas, this coefficient must have as many values as there are inter- secting branches. For a multiple point, the first differential coefficient generally reduces to the form of -, which represents an indeterminate quantity, Algebra, Art. 130. Ex. 1. It is required to determine whether the curve repre- sented by the equation y*=a*x 9 —x*, has a multiple point. Extracting the root of each member, we have y=±x{a*-xy. (1) By differentiating, we obtain dy , a*-2x* • dx =± 1' < 2 > ax (a'-xY We see from equation (1) that every value of x gives two values of y with contrary signs ; hence the curve has two branches, which are symmetrical with respect to the axis of X. Also, when x=±a, y=Q ; that is, the curve cuts the axis of x at the points B and C, at the distances +a and —a from the origin. When x=0, y=0 ; hence the twf> branches intersect at the origin A, which is therefore a multiple point. At this point there are two tangents given by equation (2), which, when x=0, reduces to 212 DlFFERENTIAS CALCULUS. dx =±a. Hence one tangent line makes an angle with the axis of abscissas whose tangent is +a, the other an angle whose tangent is —a. Ex. 2. It is required to determine whether the curve repre« cented by the equation y-^(x-a)\x-b), has a multiple point. Ans. The point whose co-ordinates are x=a, y=0, is a mul- tiple point. (286.) Definition'. A cusp is a point at which two or more branches of a curve terminate and have a common tangent. If the branches lie on different sides of the tangent, it is called a cusp of the first order ; if both branches lie on the same side of the tangent, it is called a cusp of the second order. Since the axes of reference may be chosen at pleasure, we shall, for convenience, suppose the tangent at a cusp to be per- pendicular to the axis of abscissas. If the tangent is parallel to the axis of abscissas, we have but to transpose the terms abscissa and ordinate in the two following theorems. Proposition VII. — Theorem. (287.) A point of a curve at which the tangent is perpendicu- lar to the axis of abscissas, and the contiguous ordinates on each side of that point are real, and both greater or both less than the ordinate of the given point, is a cusp of the first order If P be a point of a curve at which the tangent is perpendicular to AX, and if the ordinates P'R', P'R", however near they may be taken to PR, are both greater than PR, it is evident that P will be the point of meeting of two branches which have PR for their common tangent, as repre- sented in the annexed figure. If P'R', P'R" are both less than PR, P will be the point of meeting of two branches which have PR for their common A R'llll" ■X Analysis of Curve Lines. 213 tangent, but the branches will be situated as in the figure annexed. Ex. 1. It is required to determine wheth- er the curve represented by the equation y=a+2(x-by, has a cusp of the first order. By differentiating, we obtain dy 4 R'RR" dx 3(x-by When x=b, this coefficient becomes infinite, and the tangent will be perpendicular to the axis of abscissas at the point whose co-ordinates are x=b, y=a. Let us now substitute for x, in the equation of the curve, b+h and b—h successively; we shall obtain in each case a y=a+2h 3 ; and hence y is less when x=b, than for the adjacent values of x either greater or less than b. Hence there is a cusp at the point whose co-ordinates are £=&, y=a. Ex. 2. It is required to determine whether the curve repre- sented by the equation y=a-2(x-bf, has a cusp of the first order. If we substitute for x, in the equation of the curve, b+h and b—h successively, we shall obtain in each case 2 y=a— 2h 3 , and hence y is greater when x=b, than for the adjacent values of x either greater or less than b. Hence there is a cusp at the point whose co-ordinates are x=b, y=a. Ex. 3. It is required to determine whether the curve repre- sented by the equation x*=y\ has a cusp of the first order. 214 Differential Calculus. Proposition VIII. — Theorem. (288.) A point of a curve at which the tangent is perpendicu- la?' to the axis of abscissas, and the contiguous abscissa upon one side of the given point, has two values both greater or both less than the abscissa of the given point, is a cusp of the second order. If P be a point of a curve at which the tangent is perpendicular to AX, and & if corresponding to the ordinate AR', there are two abscissas P'R', P"R', both greater than PR, however near they may be taken to PR, it is evident that P is the. point of meeting of two branches which have PT for their common tan- gent, as represented in the annexed figure. If P'R', P"R' are both less than PR, P will be the point of meeting of two E.' branches which have PT for their com- mon tangent, but the branches will be situated as in the figure annexed. Ex. It is required to determine wheth- er the curve represented by the equation i has a cusp of the second order. By differentiating, we obtain dy_ 1 iy±W~ We see from the equation of the curve that the curve has two branches, both of which pass through the origin of co-or- dinates. When y=0, x—0, and the first differential coefficient reduces to infinity ; and hence the axis of Y ordinates is tangent to both branches of the curve at the origin of co-ordinates. If y is supposed to be negative, x is imaginary; hence the curve does not extend below the axis of abscissas. If we suppose y—+h, we shall have R Analysis of Curve Lines. 215 x=h*±h\ 5 When k is less than unity, h 2 is less than h\ and x will have two positive values, PR and P'R ; hence the point A is a cusp of the second order. By a similar course of investigation, the cusps may be de- termined when the tangent is inclined to both the co-ordinate axes. (289.) Definition. An isolated point is a point whose co-or- dinates satisfy the equation of a curve, while the point itself is entirely detached from every other in the curve. Proposition IX. — Theorem. For an isolated point, the first differential coefficient is equal to an imaginary constant. For since, by supposition, the proposed point is entirely de- tached from every other point of the curve, there can be no tangent line corresponding to that point, and consequently the value of the first differential coefficient must be imaginary. Ex. It is required to determine whether the curve represent- ed by the equation y' i =x(a+x) 2 , has an isolated point. Extracting the square root, we find yz=±(a+x) y/X. Hence, when x is negative, y will be imaginary. If x=0, y=0, which shows that the curve passes through the origin A. For every positive value of x, y will have two real values, which shows that the curve has two branches extending indefinitely toward the right. The equation is also satisfied by the values £= — a, y—0. Hence the point P, whose abscissa is —a, is detached from all others in the curve, and is called an isolated point. The form of the curve is *uch as exhibited in the annexed figure. (290.) From the preceding propositions it will be seen that, in order to trace out a curve from its equation, we first dis- 216 Differential Calculus. cover the most remarkable points by putting x and y success- ively equal to zero or infinity, and also the first and second differential coefficients equal to zero or infinity. Then, to trace the curve in the neighborhood of the points thus determined, when they appear to present any peculiarity, we increase one of the co-ordinates by a very small quantity, and observe the effect upon the other co-ordinate. Having determined the singular points, and examined the course of the curve in their 'mmediate vicinity, we can easily trace the remainder of the curve, by assigning to x and y arbitrary values at pleasure. INTEGRAL CALCULUS. SECTION I. INTEGRATION OF MONOMIAL DIFFERENTIALS — OF BINOMIAL DIFFERENTIALS — OF THE DIFFERENTIALS OF CIRCULAR ARCS. Article (291.) The Integral Calculus is the reverse of the.* Differential Calculus, its object being to determine the expres- sion or function from which a given differential has been de- rived. Thus we have found that the differential of x* is 2xdx therefore, if we have given 2xdx, we know that it must have been derived from x\ or x* plus a constant term. (292.) The function from which the given differential has been derived, is called its integral Hence, as we are not cer- tain whether the integral has a constant quantity or not added to it, we annex a constant quantity represented by C, the value of which is to be determined from the nature of the problem. (293.) Leibnitz considered the differentials of functions as indefinitely small differences, and the sum of these indefinitely small differences he regarded as making up the function ; hence the letter S was placed before the differential to show that the sum was to be taken. As it was frequently required to place S before a compound expression, it was elongated into the sign /, which, being placed before a differential, denotes that its in- tegral is to be taken. Thus, f2xdx=x 2 +C. This sign / is still retained evei i by those who reject the philosophy of Leibnitz. (294.) We have seen that the differential coefficient ex- presses the ratio of the rate of variation of the function to that of the independent variable. Hence, when we have given a certain differential to find its integral, it is to be understood 218 Integral Caiculus. that we have given a certain quantity which varies uniformly, and the ratio of its rate of variation to another quantity de- pending on it and given quantities, to find the value of that quantity. Thus, if we have given du=3x 2 dx, to find its integral, we have given a quantity x which varies uniformly, and the ratio of its rate of variation to that of u, to find the value of u. And since f3x*dx=x*, we know that u equals x 3 , or x 3 +C Ex. There is a quantity x which increases uniformly, and the rate of its variation, compared with another quantity de- pending on it, is as 1 to ax" ; required the value of this quan- tity when a=9 and £=10. Let u—the quantity required. Then dx : du : : 1 : ax*. Hence du=ax' 2 dx, and fdu=fax*dx, ax 9 u=—. Hence the number required is fXl0 3 =3000. (295.) We have seen (Art. 176) that the differential of the product of a variable multiplied by a constant, is equal to the constant multiplied by the differential of the variable. Hence we conclude that the integral of any differential multiplied by a constant quantity, is equal to the constant multiplied by the integral of the differential. Thus, since the differential of ax is adx, it follows that fadx = ax= afdx. Hence, Proposition I. — Theorem. If the expression to be integrated have a constant factor, this factor may be placed without the sign of integration. Thus, fabx' 2 dx=abfx i dx. (296.) We have seen (Art. 179) that the differential of a function composed of several terms is equal to the sum or dif Integration of Monomial Differentials. 219 ferenoe of the differentials taken separately. Hence the in- tegral of a differential expression composed of several terms is equal to the sum or difference of the integrals taken separate- ly. Thus, since the differential of a*x*— 2ax 3 — x is 2a 2 xdx—6ax 2 dx—dx, we conclude that f(2a 2 xdx — 6ax'dx—dx) is a i x i —2ax 3 —x. Hence we derive Proposition II. — Theorem. The integral of the sum or difference of any number of diffe? entials, is equal to the sum or difference of their respective in- tegrals. (297.) We have seen (Art. 177) that every constant quan- tity connected with the variable by the sign plus or minus will disappear in differentiation ; that is, the differential of u -J-C is the same as that of u. Consequently, the same differential may answer to several integral functions, differing from each other only in the value of the constant term. Hence Proposition III. — Theorem. In integrating, a constant term must always be added to the integral. Thus, fdu=u+C. (298.) We have found (Art. 186) that the differential of x m+1 is (m+l)x m dx. dx m+1 ( z m+1 \ Hence x m dx= — rr = d[ — rr ) • ?n+l \m-\-\J x m+1 Therefore is the function whose differential is x m dx, or m-\-\ x m+1 fx m dx = — — - + C. J m+\ Hence Proposition IV. — Theorem. To find the integral of a monomial differential of the form x m dx, increase the exponent of the variable by unity, and then divide by the new exponent and by the differential of the variable. Ex. 1. The rate of variation of the independent variable .r, 220 Integral Calculus. is to the rate of variation of a certain algebraic expression aa I to j-x 9 ; it is required to find that expression. x 9 dx Ex. 2. What is the integral of — — ? Ex. 3. What is the integral ofx 2 dx? A aX r» Ans. — +C. x Ans. — -fC. y Ans. §:c 3 -|-C. dx i Ex. 4. What is the integral of — or x 2 dx ? ° ^x Ans. 2x 2 +C. dx Ex. 5. What is the integral of — or x~ 3 dx ? x~ 2 1 Ans. — — or — r-^+C. 2 2x dv Ex. 6. What is the integral of ax*dx+—— % ? ax 3 ' . , Ans. — — {-x'+L. O Ex. 7. If the side of a square increases uniformly at the rate ot T \ of an inch per second, what is the area of the square when it is increasing at the rate of a square inch per second ? Ans. (299.) There is one case in which the preceding rule fails. It is that in which the exponent m is equal to —1. For in this case we have, according to the rule, fx l dx- x - 1+1 x° 1 =— =r=ao. -1+1 which shows that the rule is inapplicable. dx But x 1 dx is the same as — , and we know (Art. 215) that x v this expression was obtained by differentiating the logarithm of the denominator. Therefore fdx fx~ l dx or / — =log. .r-t-C. Integration of Monomial Differentials. 221 . . /• adx Also, / —a log. x+C Hence we have Proposition V. — Theorem. If the numerator of a fraction is the product of a constant * quantity by the differential of the denominator, its integral is the product of the constant by the Naperian logarithm of the de- nominator. dx Ex. 1. What is the integral of — — ? ° a+x Ans. log. (a+x)-\-G. %bxdx Ex, 2. What is the integral of — -r-= ? & a+bx 2 Ans. log. (a-\-bx 2 )-\-C. adx Ex. 3. What is the integral of -j— ? Ans. ax^dx Ex. 4. What is the integral of — =—•? ° x 3 Ans. Proposition VI. — Theorem. (300.) Every polynomial of the form (a-\-bx-\- cx 2 +, etc.) n dx, \n which n is a positive whole number, may be integrated by raising the quantity within the parenthesis to the nth power, multiplying each term by dx, and then integrating each term separately. This is an obvious consequence of Proposition II. Ex. 1. What is the integral of (a+bx)~dx ? Expanding the quantity within the parenthesis, and multi- ■ plying each term by dx, we have a'dx + 2abxdx + b 2 x 2 dx. Integrating each term separately, we obtain &V a'x+abx 2 +-^-+C, o which is the integral sought. 222 Integral Calculus. Ex. 2. What is the integral of (5+7a?)*dx? Ans. Ex. 3. What is the integral of (a+3xydx? Ans. (301.) We have seen (Art. 188) that any power of a poly- nomial may be differentiated by diminishing the exponent of the power by unity, and then multiplying by the primitive ex- ponent and by the differential of the polynomial. Thus the differential of (ax+x*) 3 is 3(ax+x' i y(adx+2xdx). Hence we deduce Proposition VII. — Theorem. In order to integrate a compound expression consisting of any power of a polynomial multiplied by its differential, increase the exponent of the polynomial by unity, and then divide by the new exponent, and by the differential of the polynomial. Ex. 1. What is the integral of (a+3xyGxdx ? Ans . <2+5£T +c . o Ex. 2. What is the integral of (2x 3 -l)6x'dx ? Ans. (302.) The preceding rule is equally applicable when the exponent of the polynomial is fractional. Ex. 3. What is the integral of (x+ax) 2 (dx+adx) ? 3 Ans. %(x+ax) 2 +C. dx Ex. 4. What is the integral of jl {1+xf Ans. Ex. 5. What is the integral of (ax*+bx s y(2ax+3bx 2 )dx? Ans. i Ex. 6. What is the integral of {ax+bx' 1 f{a-\-2bx)dx? Ans. (303.) Any binomial differential of the form du = ( a + bx n ) m x n ~ l dx, in which the exponent of the variable without the parenthesis Integration of Binomial Differentials. 223 is one less than the exponent of the variable within, may be integrated in the same manner. Let us put y=a+bx n . Then dy=bnx*~ l dx, 1 d ^ n-lJ -and t~ = x dx. on _, . , dy Therefore du=y m 1 -, * bn and u=-. — — rr-. — |-C, (a+bxT +1 , Hence we deduce (m+l)bn {a+bx*) m ' t {in + \)bn Proposition VIII.—Theorem. To integrate a binomial differential when the exponent of the variable without the parenthesis is one less than that within, in- crease the exponent of the binomial by unity, and divide by the product of the new exponent, the coefficient, and the exponent of the variable within the parenthesis. Ex. 1. What is the integral of du=(a+3x*yxdx ? Let us put y=a+3x* ; dy whence dy=6xdx, or xdx=—. b y 3 dy Therefore du=^—^-, D y* (a+Sx-y and tt =_=___ +C . i Ex. 2. What is the integral of (a-\-bx f ) 2 mxdx ? m i Ans. -Aa+bxY+C. xdx Ex. 3. What is the integral of vV+* a Ans. Va'+x'+C. 3 Ex. 4 Whit is the integral of (a+bxy~exdx? 56 224 Integral Calculus. Ex. 5. What is the integral of (cf+xyanx^dx ? Ans. Ex. 6. If x increase uniformly at the rate of one inch pei second, what is the form and value of the expression which is \-\-x ncreasing at the rate of -== inches per second, when ° V2x+x* x= 10 inches? Ans. (304.) To complete each integral as determined by the pre- ceding rules, we have added a constant quantity C. While the value of this constant is unknown, the expression is called an indefinite integral. But in the application of the calculus to the solution of real problems, the complete value of the integral is determined by the conditions of the problem. We may de- termine the value of the constant, or make it disappear entire- ly from the integral, in the following manner. If we suppose the independent variable and the integral to begin to exist at the same instant, then when x=0, the integral =0, and conse- quently C=0. Again, if we suppose the integral to begin to exist, or to nave its origin when x becomes equal to a given quantity a, the value of C may then be determined. When the value of the constant has been determined, and a particular value assigned to the independent variable, the value of the integral is then known, and is called a definite in- tegral. Ex. 1. Represent the base of the tri- r angle ABC by x, and the perpendicular „. by nx, then the area of the triangle is E/^i \nx*, whose differential is >*f \ nxdx. /^ \ \ If we take the integral of nxdx accord- A D F B ing to Prop. IV., we obtain fnxdx = hix* + C, which represents the area of the triangle ABC. The constant is determined by observing that the base x, and the area of the triangle begin to exist at the same time ; hence when x=0, the integral =0 ; that is, \nx 2 +G = 0, and consequently C=0. Integration of Binomial Differentials. 22a Again, suppose we wish to obtain an expression for the area of the trapezoid EDFG, contained between the two perpendic- ulars DE, FG. We must first obtain the area of the triangle ADE. Suppose the variable x to be equal to AD, which we ^ill represent by a, then the area of ADE will be expressed by Next suppose the variable x to become equal to AF, which we will represent by b, then the area of AFG will be expressed by ±nb*+C. Subtracting the former expression from the latter, we obtain the area of the trapezoid EDFG, \nb^— \na*. (305.) Hence we find that the constant C may be made to dis- appear by giving two successive values to the independent varia- ble, and taking the difference between the two mtegrals corre sponding to these values. When we take the excess of the value of an integral when the independent variable has become equal to b, above its value when it was only equal to a, we are said to integrate between the limits of x—a and x=b. This is indicated bv the sign / . «/ a l^,^ ! „_,*(*-«') Thus, / nxdx=\nb 2 —\na' i = Ex. 2. Integrate / 2xdx, and illustrate the case by a geo> metrical example. Ans. Ex. 3. Integrate / 3x-dx ; illustrate the case by a geomet rical example, and determine its numerical value when a=4 and 6=6. Ans. Ex 4. Integrate / -xdx ; illustrate the case by a geomet- rical example, and determine the value of the definite integral between the limits a=2 and 6=3. -4.71 s. Ex. 5. Integrate / -x'dx ; illustrate the case by a geomet- 22Q Integral Calculus. rical example, and determine the value of the definite integral between the limits a=4 and 6=6. Ans. Ex. 6. What is the value of / 2(e+x)dx, when a=W, b=20. and e=4 ? and illustrate the exercise by a geometrical figure. Ans. Ex. 7. What is the value of / ^{e+nx^^nxdx, when a=4, b=6, e=4, and n=2l Ans. pb dx Ex. 8. What is the value of / — - — , when a— 2, b=3, and J a e+x e=4l Ans. INTEGRATION BY SERIES. (306.) If it is required to integrate an expression of the form in which X is a function of a:, it is often best to develop X into a series, and then, after multiplying by dx, to integrate each term separately. This is called integrating by series, since we thus obtain a series equal to the integral of the given expres- sion, from which we may deduce the approximate value of the integral when the series is a converging one. dx Ex. 1. It is required to integrate the expression t^t - . By the binomial theorem, we find or (l+x)~ 1 = l—x+x i —x 3 -\-x*—, etc. l+x Multiplying by dx, we have =dx—xdx-\-x' i dx—x s dx+x i dx—, etc. ; l+x and integrating each term separately, we obtain A if^-l+l-7+ir-' etc - +c - dx Ex. 2. It is required to integrate the expression „ Differentials of Circular Arcs 227 By the binomial theorem, we find or (l+x*)- 1 =l-x*+x*-x' i +, etc. 1+x- dx Whence =dx— x"dx+x*dx — x a dx+, etc., l+a; 2 /' dx x 3 x b x 1 , _ 1 -+p-*-8 + 677 + » etC '' +a dx Ex. 3. What is the integral of ? ° a—x Ans. dx Ex. 4. What is the integral of ^ ? ° . (a— a;) JLws. dx Ex. 5. What is the integral of Vl-x* Ans. INTEGRATION OF THE DIFFERENTIALS- OF CIRCULAR ARCS. (307.) We have found in Art. 227, that if z designates an arc, and y its sine, the radius of the circle being unity, dz=-^=. Vl-if Hence f r V =z+C. If the arc be estimated from the beginning of the first quad- rant, the sine will be when the arc is 0, and consequently C equals zero. Therefore the entire integral is / I =the arc of which y is the sine. Vl-y* If it were required to integrate an expression of the form Va -y it may be done by the aid of an auxiliary variable. y Assume v=- or y=av. Then dy=adv^aud Vd 2 —y 2 —aVl — v^ S28 Integral Calculus. Substituting these values in equation (1), we have , dv dz=- Hence or Ex. Integrate the expression dz Vl-v- z=the arc whose sine is v, y 2= the arc whose sine is -. a dy V4-f Ans. z=the arc whose sine is \y. (308.) We have found in Art. 227, that if z designates an arc. and y' its cosine, the radius of the circle being unity, -dy' dz- Hence Vl-y" J Vl-v' 2 To determine the constant C, we see that if the arc be estimated from B, the beginning of the first quadrant, the cosine becomes when the arc becomes a quadrant, which is represented by \it ; hence the first member of the equation becomes equal to \tt when y' = 0. But under this supposition the arc whose cosine is becomes \tt\ hence C = 0, and the entire in tegral is -dy' f- :the arc whose cosine is y'. Vl-y" it it were required to integrate an expression of the form Vcf-y' 2 it may be done by the aid of an auxiliary variable, as in Art. 307, and we shall find • • y' z=the arc whose cosine is — . a (309.) We have found in Art. 227, that if z designates an arc, and t its tangent, dt dz ~TTf' Differentials of Circular Arcs. 229 Hence /lT? =Z+a If the arc is estimated from the beginning of the first quad- rant, we shall have %=0 when I rXT^ '■> hence C=0, and the entire integral is — — =the arc of which t is the tangent. If it were required to integrate an expression of the form a +t it may be done by the aid of an auxiliary variable. t Assume v=- or t=av; then dt=adv. Substituting in equation (1), we have adv 1/ dv \ ^ = a 2 + aV~a\l+uV Hence z=-X arc whose tangent is v, a 1 . t or z=-Xarc whose tangent is -. a a (310.) We have found, Art. 227, that if % designates an arc, and x its versed sine, dx ax— , — ==■ V2x—x* /' dx n V2x^ =Z+C - If the arc is estimated from the beginning of the first quad« rant, we shall have C = 0, and the entire integral is /' dx =the arc of which x is tte versed sine. y/2x— x 2 If it were required to integrate an expression of the form 230 Integral Calculus. *=-7==. CO V2ax— x it may be done by the aid of an auxiliary variable. Assume v=— or x=av ; a then dx=adv. Substituting in equation (1), we have adv dv dz=- V2a*v-a*v* V2v-v* Hence z=the arc whose versed sine is v, x or z=the arc whose versed sine is -. a INTEGRATION OF BINOMIAL DIFFERENTIALS. Proposition IX. — Theorem. (311.) Every binomial differential can be reduced to the form p x m - l {a-\-bx n ydx, in which the exponents m and n are whole numbers, and n is positive. 1st. For if m and n were fractional, and the binomial were of the form i \ p x 3 (a l -\-bx 2 )' 1 dx i we may substitute for x another variable, with an exponent equal to the least common multiple of the denominators of the given exponents, by which means the proposed binomial will be transformed into one in which the exponents of the variable are whole numbers. Thus, if we make x=z*, we find x^(a+bxfydx=6z\a+bzydz, in which the exponents of z are whole numbers. 2d. If n were negative, or the expression were of the form p x m ~\a-itbx-ydx we may put x=-, in which case we shall obtain Integration of Binomial Differentials. 231 -z- m -\a+bzydz, in which the exponent of the variable within the parenthesis is positive. 3d. If the variable x were found in both terms of the bino- mial, and the expression were of the form p x m - l (ax T +bx*ydz, we may divide the binomial within the parenthesis by x\ and EI multiply the factor without the parenthesis by x*, and we shall obtain x m+ T-\a+bx n -ydx, in which but one of the terms within the parenthesis contains the variable x. Proposition X. — Theorem. (312.) Every binomial differential, in which the exponent of the parenthesis is a whole number and positive, can be integrated by raising the quantity within the parenthesis to the proposed power, multiplying each term by the factor without the paren thesis, and then integrating each term separately. This results directly from Proposition II. Ex. 1. Integrate the expression du=x*(a J rbx*ydx. Expanding the binomial, we obtain du=d i x' i dx + 2abx f 'dx + b q x*dx. A.nd integrating each term separately, we find aV abx° 6V _ u= 1 1 r-C. 3 3 9 Ex. 2. Integrate the expression du=x*(a+bx*) 3 dx. aV 3a*bx* SaVx* b'x" Ans. ^=^+-^-+-g-+lo' +C Ex. 3. Integrate the expression du=x*(a+bxydx. Ans. 232 Integral Calculus. Ex. 4. Integrate the expression du=x b (a+b 1 x*) s dx. Ans. Proposition XL — Theorem. (313.) Every binomial differential can be integrated, when the exponent of the variable without the parenthesis, increased by unity, is exactly divisible by the exponent of the variable within. For this purpose, we substitute for the binomial within the parenthesis, a new variable having an exponent equal to the denominator of the exponent of the parenthesis. Let us assume a+bx n =z\ Then (a+bz n )*=z*. (1) 2 q — a Also, x n = — - — and and, by differentiating, -PrO* mx ^-a^c^r* (2) Multiplying together equations (1) and (2), and dividing by m, we obtain m x m - 1 (a+bxydx=-jz^- 1 {—r-j dz, which, according to Prop. X., can be integrated»when — is a whole number and positive. If — is negative, we may, by Formula D, Art. 323, increase the exponent until it becomes positive. Ex. 1. Integrate the expression 3 du=x 3 (a + bx 1 ) 2 dx. Assume a+bx*=z' i . Then (a+bxy=z\ (1) Also, x% =—jT (2 ) Integration of Binomial Differentials. 233 7 zdz /o\ and xdx=-j-. \<*) Multiplying together equations (1), (2), and (3), we obtain du=x\a+bz*) s dx=z* .—^-dz. z T az b Hence u= W~5b 2 Replacing the value of z, we find (a+bxrf a(a+bx*y , n lb 56 Ex. 2. Integrate the expression du=x\a+bx*) 2 dx. Assume a+bx 2 =z\ Then (a+bx*)*=z. (1) Also, X ^ Z ~T' • (2) 7 z d z /o\ and xdx—-r-- {#) /z*-a\'z*dz Hence du— 1 — -, — J -r-, z (Zz — 2az 4 dz +a'z'^g or ^w = fi "* z 7 2«z 5 aV and M= W-5F + W +U Restoring the value of z, we find (a+bz* 7 (a+ZQ 3 2a(a+&:c a ) 2 a*(a+bx*) 2 W 56 3 36 3 Ex. 3. Integrate the expression du=x*(a+bx v fdx. _s\5 3(a+bx>) * 3a(a+bx*) 3 3a\a+bx')' s , n Ans. u=——y g^ + ^ . U Ex. 4. Integrate the expression ..I du=x*(a—x' 2 ) 2 dx. If we put a— x*=z\ we find 234 Integral Calculus. du= — (a — z*)dz. z* Whence u=— az+— +C, o or u——a{a—x)-\ hC o Ex. 5. Integrate the expression du=x\a?+x*)-'dx. If we put z=a a +;c Q , we find zdz a 4 dz and m=- — «"z+— log. 2+C-, 4 2 or / 9 I S\ 9 4 U =(12±) -a'la'+x^+^log.W+x^+C. Proposition XII. — Theorem. (314.) Every' binomial differential can be integrated, when the exponent of the variable without the parenthesis, augmented by unity, and divided by the exponent of the variable within the parenthesis, plus the exponent of the parenthesis, is a whole number. p The binomial x m ~ 1 (a+bx n )' ] dx, may be written x^-'U^+bJx^^dx, p np or x m -\ax- n + bfx*dx, m[ n p x p which equals x q (ax~" +b) q dx, which, according to the preceding Proposition, can be inte- grated when np m+- L — is a whole number, or ( — r-— J is a whole number. \n qj Ex. 1 Integrate the expression du=a(\-\-x*) 2 dx. Integration of Binomial Differentials. 235 Put vV=l+z'. Then (l+x 2 )- 1 ^- 9 ^" 3 - <*> 1 Also, Whence xtf — lY' and 1=*>'-1)'. . (3) Multiplying together equations (1), (2), and (3), we have Jw=a(l+^ 2 ) 2 dx = — jr. Whence a =-=-==+C Ex. 2. Integrate the expression _j = — +u, r+2x 2 - — j, r or «= £T" . Ex. 3. Integrate the expression _i dx p ut »=a;+Va ! +x ! . Then i^dx+-^=^ . ■Vrf+tf Therefore V = 7^+?' 2S6 Integral Calculus. Consequently we have X = / — = / — =log. v=\og. [x+ VoM^ 1 ]. J Va'+x 2 J v Ex. 4. Integrate the expression *dx c?X 2 =- Va 2 +x 2 Put D=(aV+^) 2 . __, _ a 2 xdx+2x s dx VV hence dv= — , (aV+.r 4 ) 2 a?dx 2x*dx or dv= f+- (a'+z 2 )* (a 2 +:£*)* or e?u=a'= V2ax—x 2 Make ?n, in Formula c, equal to 3, and the formula reduces to 5a, X 3 =— X-^-V2ax- x> 3 o wnere X, has the same value as in Ex. 3. Q 242 Integral Calculus. Ex. 5. Integrate the expression V2ax—x i Make m, in Formula c, equal to 4, and the formula reduces to la. X 4 =-— X 3 — -V2ax—x\ 4 4 where X 3 has the same value as in Ex. 4. (320.) Formula c reduces the differential binomial x m dx I y/2ax—x t f x m ~ 1 dx to that of / -; J V2ax — x 2 and, in a similar manner, this will be found to depend upon x m ~*dx S: I a' ' v 2ax— x and so on ; so that after in operations, when m is a whole num- ber, the integral is found to depend upon dx f y/2ax—x i . x which represents the arc whose versed sine is - Art. .110. Ex. 6. It is required to find the integral of x 2 dx V2ax—x' i Substituting, in Formula c, § for m, we obtain /x 2 dx 4a f x-dx 2x 2 ==— / — = — —-V2ax—; V2ax—x~ 3./ v2ax—x i 3 i x' 2 dx dx But y/2ax—x i V2a—x Also V2a— a /dx , z = -2V2a— x. ■\l 9. n — f rr C x fa 8<3 /~ % X /TT- Hence / — = V2a—x — —V2a—x. J V2qx-x* 3 3 (321.) Formula A will only diminish the exponent m when Integration of Binomial Differentials. 243 m is positive ; but we may easily deduce from this formula an- other which will diminish the exponent when it is negative. For this purpose, multiply Formula A by the denominator b(np+m+l), and transposing the term which does not contain the sign of integration, we obtain Formula B. x . m -n +1 ^ + hx ny +i _ b ( np+m+l)fx m (a+bx a ydx fx m "(a+bxydx=- a(m-n+l)~ Ex. 1. Find the integral of dx - x ,orx-\l+x*) 3 dx. ;r(l+0' Substituting, in Formula B, — 2 for m—n, 1 for a, 1 for 6, 3 for n, an d —3- for p, we obtain fx-\\ +x s )~*dx^ -x~ l (l +x*f+fx(l +x<)~ h dx. Ex. 2. Find the integral of dx -3 3, or a: -2 (2— x a ) 2 dx. x*(2-xy Substituting, in Formula B. —2 for m—n, 2 for a, -1 for 6, 2 for n, and — I for 77, we obtain fx-\2-x*) *dx= y —^- L — +f(2-x*) 2 dx. Proposition XIV. — Theorem. (322.) The integral of any differential of the form x m (a+bx n ydx, may be made to depend upon the integral of another differential of the same form, but in which the exponent of the parenthesis is diminished by unity. Let us put v=x% 244 Integral Calculus where s is an exponent to which any value may be assigned as may be found most convenient. Differentiating, we find dv=sx*~ 1 dx. If then we assume udv=x m (a + bx a ) p dx, Art. 315, we must have u= (a+bx n ) p ; s and, by differentiating, du = ( m ~ S + 1) ar~(a + bxydz+^z^*(a+bar)*- l dx. s x s But (a+bxy=(a+bx") (a+bx n ) p -\ Hence a(m—s + l)+b(m—s+l+np)x n m _ s . n .__ x , du=— — — £J —x m s (a+bx n ) p l dx. Let the value of s be taken such that m— s+1 +w/?=0 ; that is, s=??i + l+np, , —anpx m -*(a+bxy- l dx we shall have a«= ; — . 72^> + ?7Z+l Substituting the values of w, v, du, and rfy here given u. formula (1), Art. 315, we obtain Formula C. , v , x m+ \a+bxy+anpfx m {a+bx n ) p - i dx Jx m (a + bx n ) p dx — j — — , J v np+m+l by which the value of the required integral is made to depehd upon another having the exponent of the binomial less by unity. The value of this new integral may, by the same formula, be made to depend upon that of an integral in which the exponent of the binomial is still further diminished ; and so on until the exponent of the binomial is reduced to a fraction less than unity. Ex. 1. Find the integral of the expression dx vV+x 2 . • We may diminish the exponent of the binomial by unity bv substituting, in Formula C, for m, a 3 for a, 1 for 6, 2 for n, £ for p, and we obtain Integration of Binomial Differ entials. 245 xVcf+x" 1 a~ C dx fdxVa a +x*= +-J - But by Ex. 3, Art. 314, dx f Va'+x ==log. [x+ Vd'+X*]. Hence fdxVa*+x-= +— log. [x+ Va 3 +x\l. Ex. 2. Find the integral of the expression dxVx^—a? . xVx 2 — a 2 a 2 , _ . t Ans. — log. [x+ Vx 2 — a 2 L (323.) Formula C will only diminish the exponent of the parenthesis when the exponent is positive ; but we may easily deduce from this formula another which will diminish the ex ponent when it is negative. For this purpose, multiply Form- ula C by the denominator vp-\-m + \, and transposing the term which does not contain the sign of integration, we obtain Formula D. fx™(a+bxy->dx= - x ™ +i ( a + bx r+( n P+ m + 1 )fz m ("+te a Yd* Ex. 1. Find the integral of (2—x 2 ) 'dx. Substituting, in Formula D, for m, 2 for a, -1 for b, 2 for n, and — | for jo — 1, we obtain f{2-xr*dx=l{2-xr\ or 2 V 2V2-X*' Ex. 2. Find the integral of xdx _i ■ — i, or x(l+x s ) 3 dx. (l+O 3 Substituting, in Formula D, 1 for m, 1 for a, 246 Integral Calculus. 1 for 6, 3 for n, and — Tf for p—l, we obtain I X* 2 2 /r(l+:c 3 ) 3 dk=--(l+a: , ) 3 +2/a:(l+a:') 3 2r I _. ■ =(2r)\2r-y) 2 cfy. Integrating by Art. 303, we obtain f(2r-y) A dy=-2(2r-yy+C. Hence z= - {2r)*2 V2r-y + C, = — 2V2r(2r—y)+C. If we estimate the arc from the point B where y=2r, we shall have, when z— 0, y=2r; hence 0=0+C, or C = 0, which shows that there is no constant to add, and consequent- ly the entire integral will be z = — 2 v , 2r(2r— y), which represents the length of the arc of the cycloid from B to any point D whose co-ordinates are x and y. But we see from the figure that BE = 2?'— y. Also, BG 2 =BCxBE, Geom., Prop. XXII., B. IV. Hence BG= v'BCxBE= V2v(2r-y), _ and consequently the arc BD = 2BG, or the arc of a cycloid, estimated from the vertex of the axis, is equal to twice the corresponding chord of the generating circle; hence the entire arc BDA is equal to twice the diameter BC, and the entire curve ADBH is equal to four times the diameter of the generating circle. Ex. 4. It is required to find the length of an arc of the com- mon parabola. 250 Integral Calculus. The equation of the parabola is y*=2px. Differentiating, and dividing by 2, we have • ydy=pdx : y~ whence dx*=—dy. V • Substituting this value in the differential of the arc, we obtain dz=\/dy*+^dy\ dy J V P*+f- Integrating according to Ex. 1, Art. 322, we obtain 2= .v^ + £ Iog . (y+ ./7TF)+c. If we estimate the arc from the vertex of the parabola, we shall have ?/=0 when z=0 ; hence 0=| log. p+C, or C=-| log. p ; and consequently yy/p'+y* P locr (y±Vf±f\ 2p 2 °;\ p / Ex. 5. It is required to find the length of an arc of the log- arithmic spiral. The differential of an arc of a polar curve, referred to polar co-ordinates, Art. 257, is dz= Vdr+i-dt\ The equation df the logarithmic spiral, Art. 155, is t=\og. r. Udr Consequently dt—- r Hence, by substitution, we find dz= Vdr'+bVdr*, =drVl+M\ For the Naperian system M=l, and we find dz=drV2; whence z~rV2+C. Rectification of Curves. 251 U we estimate the arc from the pole where r=0, we have z = ?V2; that is, in the Naperian logarithmic spiral, the length of an art estimated from the pole to ant/ point of the curve, is equal to the diagonal of a square described on the radius vector. Ex. 6. It is required to find the length of an arc of an ellipse. The equation of an ellipse, Art. 69, Cor. 6, is y=(l-0(A 2 -:r). Differentiating, we obtain dy = (l-e> dx y x Vl—e" ~ VA 2 -x 2 ' Substituting this value in the differential of the arc, we obtain dz=dx\/l + \A' A 2 -or dx VA'-eV VA 2 -.r 2 , , / 7x* Mx\f \ — r T VA 2 -.r 2 Developing y 1 — -ry in a series, we obu _.ain A 2 kdx eV eV 3eV The several terms of this series may be integrated as in Art. 317, and we obtain * =x ~h^. -2^ x -d^ 6 -' etc -' (1) where X c represents the arc of a circle whose radius is A and sine is x, A.X„ x „ 3A 2 .X 2 x> .— a 5A'.X, x» X 6 =— -— --VA'-^etc. b 0*0 Integra;. Calculus. In order to obtain one fourth of the circumference of the el ripse, we must integrate between the limits £=0 and x=A But when x=A, VA 2 —x 2 =0; hence the values of the quani : ties, X 2 , X„ etc., become A.X„ X 2 = X 4 = 2 ' 3A a .X, X fi = 4 5A 2 X 3A 3 .X 2.4 ' 3.5A\X, 6 2.4.6 and consequently equation (1) becomes e 3e 4 3.3.5e ', etc. Z=X (1-; ; — , etc.), 2.2 2.2.4.4 2.2.4.4.6.6 for one fourth of the circumference of the ellipse, where X is one fourth of the circumference of the circle whose radius is A. Hence the entire circumference of the ellipse is equal to 3.3.5e 6 ; — , etc.) e- 3e 4 2.2.4.4.6.6 QUADRATURE OF CURVES (325.) The quadrature of a curve is the measuring of its area, or the finding a rectilinear space equal to a proposed curvilinear one. When the area of a curve can be expressed in a finite number of algebraic terms, the curve is said to be quadrable, and may be represented by an equivalent square. We have found, Art. 253, that the differential of the area of a segment of any curve, referred to rectangular co-ordinates, is ds=ydx, where s represents the area ABPR, and x and y are the co-ordinates of the point P. To apply this formula to any particular curve, we must find from the equation of B the curve the value of y in terms of x, or the value of dx in terms of y and dy, and sub- stitute in the formula ds=ydx. The in- ^~ tegral of this expression will give the area of the curve. Ex. 1. It is required to find the area of the common parabola The equation of the parabola is y*=2px* -c II X Quadrature of Curves. 253 whence, by differentiating, , V dl J dx = . P y'dy T he re fore ydx — , p if and, by integrating, s=— — (-C. If we estimate the area from the vertex of the parabola, the constant C will be equal to zero, because when y is made equal to 0, the surface is equal to ; hence the entire integral is ■¥ which equals ^Xy 2 = — X2px=%xy ; that is, the area of a segment of a parabola is equal to two thirds of the area of the rectangle described on the abscissa and ordinate Ex. 2. It is required to find the area of any parabola. The general equation of the parabolas, Art. 136, is y" = ax ; whence, by differentiating, we obtain ny"~*dy=adx, ny n ~ i dy and dx — Therefore ydx = a ny*dy a And, by integrating, Art. 298, (n+l)a n V" ^ or s=— — -X — X*/+C, n + 1 a ji = — r- r.r?/+C, by substituting z for its n+1 value — . a If we estimate the area from the vertex of the parabola, the constant C will be equal to zero ; hence n n+1 * Hence the area of any portion of a parabola is equal to the i54 Integral Calculus. rectangle described on the abscissa and ordinate multiplied by 7 • n the ratio n + 1 If ?z = 2, the equation represents the common parabola, and the area equals t x l/' If 7i=l, the figure becomes a triangle, and the area equals \*y ; that is, the a ea of a triangle is equal to half the product of its base and perpendicular. Ex. 3. It is required to find the area of a circle. The equi-aon of the circle, when the radius equals unity, is i y=(i- x y. The se-sc ,d member of this equation being developed by the binomial theorem, we have _ x 1 x* x° 5x° y ~ 1 ~'2~~8~TG~128~ > etC ' . x 2 dx x"dx x B dx 5x e dx Henc* y*z=dz-—~ ___ etc., and integrating each term separately, we have . , x 3 x b x 1 5x° s =fydx=x-----—-—-, etc., +C. D If we estimate the arc from the point D, when x=0, the area CDEH is 0, and consequently C = 0. The preced- ing series, therefore, expresses the area of the segment CDEH. A If the arc DE be taken equal to 30°, the ^ine of 30°, or its equal CH, which is x, becomes =h, and we lave 1 1 ; — , etc., CDEH=~ 2 48 1280 1433G = .4783055. ]Rut as x—\, y= v/f ; therefore the area of the triangle CEH=lx n/^.2165063. H^nce the area of the sector CDE = .2G17992, wMch, multiplied by 12, gives 3.14159, etc., for the area of the whole circle Quadrature of Curves. 25b Ex 4. It is required to find the area of an ellipse. The equation of the elljpse, referred to its center and axes, is B and consequently the area of the semi-ellipse will be equal to fydx=-jrfdx VA'"-r. But dxVA'—x 2 is the differential of the area of a circle whose radius is A. Art. 254 ; hence the area of the ellipse =-r-Xthe area of the circumscribing circle. A ° But the area of the circumscribing circle is equal to ~A 2 ; hence the area of the ellipse is equal to B or ttAB. Ex. 5. It is required to find the area of a segment of an hyperbola. The equation of the hyperbola, referred to its center and axes, is Ay-BV=-A 2 B 2 : whence y= — Vx' — A 3 A Bdx ds = ydx = — r— Vx' 2 — A' 2 Consequently Integrating according to Ex. 2, Art. 322, we obtain _, vV— A" A.B, _ ,— — — -> „ s==Bx — 2A 2" l0S * [X+ Vx '~ A ]+ ' fn order to determine the constant C, make x=A, in which case 5=0, and we have A.B = log. A+C; A.B that is, C=-£- log. A. Hence B.rV^-A 2 A.B 2A 2~ 256 Integral Caiculus. which represents the segment APR; hence the entire segment APP' is hxVx 2 -a: 2 -A.B log. $ x+Vx*—A* ) which equals A.B log. or . „ . $Ay+Bz\ sy-A»Blog.{ J AB \. Ex. 6. It is required to find the area of a cycloid. The area of the space ABC is most conveniently obtained by first finding the area of the space ABD, contained between the lines AD, DB, and the convex side of the curve. LetBC = 2r,AG=.r,FG=y; whence FE=2/-— y=v. We shall then have d(ADEF)=ds=vdx=(2r-y)dx. But the differential equation of the cycloid, Art. 275, is ydy dx= — = y/2rv— y-y ds=dyV2ry—y-, Hence and s =fdy V 2 ry — y" + C. . But this is evidently the area of a segment of a circle whose radius is r and abscissa y (Art. 254) ; that is, the area of the segment CHI. If wc estimate the area of the first segment ADEF from AD, and the area of the segment CHI from the point C, they will both be when y=0 ; the constant C, to be added in each case, will then be 0, and we shall have ADEF=CHI; and when y = 2r, ADB=the semicircle CHB= — . 2 But the area of the rectangle ADBC is equal to ACxAD = 7rrX2/~27T;-\ Hence the area AFBC=ADBC — ADB = |7rr"= three times the semicircle CHB ; and doubling this, we find the area included between one branch of the cycloid and its base, is equal to three times the area of the generating circle. Area of Spirals. 257 ds—- AREA OF SPIRALS. (32G.) The differential of the area of a segment of a polar curve. Art. 258, is 2 ' Ex 1. It is required to find the area of the spiral of Archi- medes. The equation of the spiral of Archimedes, Art 148, is t 2r7 ; dt '2r? ds=TTr"~dj- ; Trr 3 f If we make t=2n, we have r=; whence hence dr 247T 2 ' s= 3' which is the area PMA described by one revolution of the radius vector. Hence 3VL the area included by the first spire is equal to one third the area of the circle, whose radius is equal to the radius vector after the first revolution. If we make £=2(2~), we have 8tt s= 3' which is the whole area described by the radius vector during two revolutions. But in the second rev- olution, the radius vector describes the part PMA a second time ; hence, to ob- tain the area PNB, we must subtract that described during the first revolution ; hence the area PNB= = — , 3 3 3 Ex. 2. It is required to find the area of the hyperbolic spiral. The equation of the hyperbolic spiral, Art. 151, is R 258 Integral Calculus. a r= _. , a*dt whence "2?"' a 2 and s=--. Ex. 3. It is required to find the area of the logarithmic spiral. The equation of the logarithmic spiral, Art. 155, is t—\og. r. J Mdr blence dt= . r rdr When M=l, ds=—, <& T and s=— +C. 4 If we estimate the area from the pole where r=0 and C=0 we have r a S= 4" ; that is, the area of the Naperian logarithmic spiral is equal to one fourth the square described upon the radius vector. AREA OF SURFACES OF REVOLUTION. (327.) We have found (Art. 255) that the differential of the area of a surface of revolution is dS = 2nyVdx ,i + dy 2 ; whence S=f2ny Vdz"+dy*, (1) which is a general expression for the area of an indefinite por- tion of a surface of revolution ; the axis of X being the axis of revolution, and Vdx^+dy" 2 the differential of the arc of the generating curve. In order to obtain the area of any particular surface, we differentiate the equation of the generating curve, and deduce from it the values of y and dy in terms of £ and dx ; or of dx in terms of y and dy, which we substitute in expression (1). The integral of this expression will be the area required. Ex. 1. It is required to determine the convex surface of a cone. Area of Surfaces of Revolution. 259 B If the right-angled triangle ABC be re- solved about AB, the hypothenuse AC will describe the convex surface of a cone. Let AB=A, BC=b, and let x and y be the co-ordinates of any point of the line AC, referred to the point A as an origin ; we shall then have x : y : : h : b ; bx whence # = T" By differentiation, we obtain dy=jdx, and dy 2 =-j~idx' 2 . Substituting these values of y and dy" 2 in the general formula, we have bx f2ny V dx 2 + dy' =/2n-^dx V 1? + b% bx' = n^rVk*+b*+C. If we estimate the area from the vertex where x=0, we have C = 0, and bx' S=TT—Vh 2 +b\ If Making x=AB=h, we have the surface of the cone whose altitude is h, and the radius of its base b, 7vbVfi i +b 2 =2Trbx—; that is, the convex surface of a cone is equal to the circumference of its base into half its side. Ex. 2. It is required to determine the convex surface of a cylinder. If the rectangle ABCD be revolved about the side AB, the side CD will describe the convex surface of a cylinder. Let AB=h, and CA = &; the equation of the straight line CD will be y=b ; whence dy=0. Substituting these values in the general formula, we obtain D 260 Integral Calculus. f2ny Vdz*+dy*=f2nbdz, = 2nbx+C. It we estimate the area from the point A where x=0, C be- comes equal to 0; and if we make x—AR — h, we have the convex surface of the cylinder 2nbh ; that is, the convex surface of a cylinder is equal to the circum- ference of its base into its altitude. Ex. 3. It is required to determine the surface of a sphere. The equation of the generating circle, referred to the center as an origin, is x*+y*=R\ By differentiating, we obtain xdx-\-ydy=0 ; . xdx dy= — whence and dxf y x'dx* Substituting this value in the general formula, we obtain f27ry\/(^+l)dx°=f2TTdx vV+7", =f2nRdx, = 2ttRx+C. 0) To determine the constant, we will sup- pose the integral to commence at the center of the sphere ; and since the origin of co-or- dinates is at the center, the integral will be zero when x=0, and therefore the constant is equal to zero. Making .r=R, we have for the surface of a hemisphere 2;rR 2 , and theiefore the surface of the sphere is 4:rR 2 ; that is, the surface of a sphere is equal to four of its great circles. Ex. 4. It is required to determine the surface of a parabo* bid. Area of Surfaces of Revolution 201 A paraboloid is a solid described by the revolution of an arc AC of a parabola about its axis AB. The equation of the parabola is y*=2px, which, being differentiated, gives , ydy , , a yW dx= -, and ax —■ V P Substituting this value in the general formula, it reduces k o dS=2ny\/(fS^ di/Jjydy Vf+f. Integrating according to Art. 303, we obtain 2tt 2 To determine the constant C, let us suppose that y becomes zero, in which case S also reduces to zero, and the preceding equation becomes 0=^-+C; 2rrp* whence C= — 3 ' and supposing the integral to be taken between the limits y=0 and y—b, the entire integral will be Ex. 5. It is required to determine the surface of an ellipsoid. described by revolving an ellipse about its major axis. According to Art. 255, we have dS = 2rryVdx' + dy% or dS=2Trydz. But in Ex. 0, Art. 324, we have found kdx eV e*x* 3eV dZ " V^r^x~ i{1 ~2A~2AA~2AlA l ' etC ° ; 2-Aydx eV e*x* Se e x 6 hence ^=-^==(i_— -^^-— x .-, etc.,. But iL.=B. VA'-x 3 e*x* e*x* 3e 6 x 6 Hence dS ^mx{\-^-^^-—^--, etc.), 262 Integral Calculus. and integrating each term separately, we obtain Integrating between the limits x=0 and x— A, we shall ob- tain half the surface of the ellipsoid e 2 e 4 3e 6 = 2^(1---—-^^-, etc.), or the entire surface of the ellipsoid equals e 2 e 4 3e 6 4ffAB( i___ i — _ s _. lrt0 .). Ex. 6. It is required to determine the surface described by the revolution of a cycloid about its base. The general formula for the differential of the surface is dS=2nydz. But we have found in Ex. 3, Art. 324. dz J v 2ry—y Hence dS—2nydy 2TrV2iy 2 dy 2ry-y z V2ry-tf ' which, being integrated, will give the value of the surface re- quired. But, according to Ex. 6, Art. 320, / Hence irdy 8r .—— V2ry—y 2 3 2y y~~t^ 2r ~y- f 2nV2ry*dy n .-— r 8r 2y -, — 2n V2r[— — V2r— y — — V2r— y]+G. V2ry— y 2 If we estimate the surface from the plane passing through B, we shall have S = when y=2r, and consequently C = 0. If we then integrate between the limits y=0 and y=2r, we have half the surface= 3 3 2 7rr 2 ; hence the entire surface = 6 g- 4 7rr !l ; that is, the surface described by the cycloid revolved about its base, is equal to G4 thuds of the generating circle. CUBATUSE OF SoLIDS OF REVOLUTION. 263 CUBATURE OF SOLIDS OF REVOLUTION. (328.) The cubature of a solid is the finding its solid con- tents, or finding a cube to which it is equal. We have found, Art. 256, that the differential of a solid oi revolution is dV=7ry*dx ; whence Y=fny 2 dx, (1 ) where x and y represent the co-ordinates of the curve which generates the bounding surface, the axis of X being the axis of revolution. For the cubature of any particular solid, we differentiate the equation of the generating curve, and deduce from it the value of dx in terms of y and dy, or the value of y 2, in terms of x which we substitute in expression (1). The integral of this expression will be the solid required. Ex. 1. It is required to determine the solidity of a cylinder. Let b represent AC, the radius of the base, c p and h the altitude AB. Then V =JTx\fdx —Jidfdx, = nb\x + C. A ~~B Taking the integral between the limits x=0 and x=AB=h. we have V=nb*h; that is, the solidity of a cylinder is equal to the product of its base by its altitude. Ex. 2. It is required to determine the solidity of a cone. Let h represent the altitude of the cone, and r the radius of its base. We shall then have, by Ex. 1, Art. 327, ?/--x, and y'=pc\ Substituting this value of y 1 in the general formula, it be- comes r 8 dY~—nx 2 dx; h' r'lrx 3 whence V= „.,. +C. 3/r And taking the integral between the limits .r=0 and x=h, we obtain 264 Integral Calculus. h V=i7rr 2 A=7rr a X-; o that is, the solidity of a cone is equal to the area of its base into one third of its altitudi. Ex. 3. It is required to find the solidity of a prolate spheroid, or the solid described by the revolution of an ellipse about its major axis. The equation of an ellipse is Substituting this value of y 2 in the general formula, it be- comes B 2 dV=7r— (A*-x*)dz, and by integrating, we find B 2 x*\ V=.-(A'*--) + C. If we estimate the solidity from the plane passing through the center perpendicular to the major axis, we shall have when x—0, V=0, and consequently C— 0. Therefore Making x=A, we obtain for one half of the spheroid frrB'A; and consequently the entire spheroid equals |ttB 2 A, or |7tB 2 X2A. But ttB 2 represents the area of a circle described upon the minor axis, and 2A is the major axis ; hence the solidity of a prolate spheroid is equal to two thirds of the circu??iscribing cylinder. Cor. If we make A=B, we obtain the solidity of the sphere f7rR 3 =i7rD\ Ex. 4. It is required to find the solidity of the common pa raboloid. The equation of the parabola is y*=2px. CuJature of Solids of Revolution. 2G5 Substitut/.ng this value of y 1 in the general formula, it be- comes dV=2npxdx. Hence V==7rpa: a +C. To determine the constant, we suppose x to become equal to zero, in which case the solidity is zero, and C = 0. Taking the integral between the limits x—0 and z—h, and designating by b, the ordinate corresponding to the abscissa x = h, we have /St But 7r& a represents the area of a circle of which BC is the radius ; hence the solidity of the paraboloid is one half that of the circumscribed cylinder. Ex. 5. It is required to find the solidify of the solid generated by the revolution of any parabola about its axis. The general equation of the parabolas is y n =ax ; whence dz=— -, a nny n+1 dy and dV-- a ,n+2 Hence V= rSjr+ C ' (n+2)a 71 o" V « Xtti/ 2 X-+C, w+2 J a n -Try-x + C. 71 + 2 J But when .r=0, V=0, and therefore C=0 ; hence V= — ; — "ny x. n+2 J If n—2, the solid becomes the common paraboloid, and its solidity equals \^ifx. If n=l, the curve becomes a straight line, and the solid be- comes a cone, and its solidity equals \ny*x. Ex. 6. It is required to find the solidity of the solid gener- ated by the revolution of the cycloid about its base. 266 Integral Calculus. The general formula for the differential of a solid of revoiu tion is dV=Try*dx. But we have found for the cycloid, Art. 275, ydy dx—- Hence dV= y/2ry—y a rry'dy V2ry—y* which, being integrated, will give the value of the solid re- quired. The integral of this expression has already been found in Art. 319. Hence Y=Tr(-^-X 2 -j-V2ry-y*), 3 ?v y where yi i =— X t — - y/2?-y— y 1 , X!=X — V2ry—y% and X = arc of which r is the radius and y the versed sine. We must now integrate between the limits y=0 and y=2r. When 3/=0, all the above terms become 0. When y=2r, these values become X =77r, X 1 =X =7rr, _3r _37rr a e* 2 3 and Y=^f, which is one half of the solidity ; hence 5ttV s is the solid re- quired. But 7r(2r) 2 represents the base of the circumscribing cylinder, 2nr represents its altitude, and 87rV 3 represents its solidity. Hence the solid required is equal to Jive eighths of the cir- cumscribing cylinder. EXAMPLES FOR PRACTICE. ANALYTICAL GEOMETRY. Application of Algebra to Geometrij. (329.) Ex. 1. In a right-angled triangle, having given the hy- pothenuse (a), and the difference between the base and perpen- dicular (2d), to determine th e two sid es A M .s/£^+a, a *\/?^~d. Ex. 2. Having given the area (c) of a rectangle inscribed in a triangle whose base is (6) and altitude (a), to determine the height of the rectangle. a . fa 1 ac 2~ v 4 Ex. 3. Having given the ratio of the two sides of a triangle, as m to n, together with the segments of the base, a and b, made by a perpendicular from the vertical angle, to determine the sides of the triangle. t /a 2 -6 a , ./or Ex. 4. Having given the base of a triangle (2a), the sum of the other two sides (2s), and the line (c) drawn from the ver- tical angle to the middle of the base, to find the sides of the triangle. Ans . s± Va" + and b V ^W" Ex. 6. Determine the sides of a right-angled triangle, having given its perimeter (2p), and the radius (r) of the inscribed circle. Ans. The hypoth enuse equals p — r, and the other sides are p + r ± V{p — r f — 4pr 2 ' 268 Examples for Practice. Ex. 7. The area of an isosceles triangle is equal to a, and each of the equal sides is equal to c. What is the length of the base ? Ans. v / 2c 2 ±2\/c i — 4a 2 . Ex. 8. The area of an isosceles triangle is 100 square inches, and each of the equal sides is 20 inches. What is the length of the base ? Ans. 38.637 or 10.356. Ex. 9. The sum of the two legs of a right-angled triangle is s, and the perpendicular let fall from the right angle upon the hypothenuse is a. What is the hypothenuse of the triangle ? Ans. Vs 2 + a 2 —a. Ex. 10. The area of a right-angled triangle is equal to a, and the hypothenuse is equal to h. What are the two legs ? Ans. lVh 2 + 4a±±V]r — 4a. Ex. 1 1. From two points, A and B, both situated on the same side of a straight line, the perpendiculars AC = b and BD — a are let fall upon this line, and the distance, CD, between their points of intersection is equal to c. At what distance from C in the given straight line must the point F be taken, so that the straight lines AF and BF may be equal to each other ? ,, r.-n a 2 + c 2 — b 2 Ans. CF = — ■ . Ex. 12. The same construction remaining as in the preceding problem, where must the point F be taken in the given straight line, so that the angle AFC may be equal to the angle BFD ? Ans. CF=-^-. a+b Ex. 13. At what distance from C in the given straight line must the point F be taken, so that the two triangles ACF and BDF may contain equal areas ? Arts. CF=-^ 7 . a + b Ex. 14. At what distance from C must the point F be taken, so that the area of the triangle ABF may be equal to d ? * /^,Ti *CCt — DC Ans. CF= j-. a—b Examples for Practice. 269 Ex. 15. At what distance from C must the point F be sit- uated, so that the angle AFB may be a right angle ? Ans. CF = !±£Vc a -4a6. Ex. lG. At what distance from C must the point F be sit- uated, so that the circle passing through the points A, F, and B may touch the given straight line in the point F ? . - be ± Vab[c- + (a~bf] Ans. Lr = —. -. a — b Ex. 17. At what distance from C must the point F be sit- uated, so that AF may have to BF the ratio of n to m ? _, _, — n 2 c ± Vm 2 n 2 ( cr + b~ + c 2 ) — n*a 2 — m % U 2 Ans. CF=r — 5 5 — , nr — n* Ex. 18. One of the angular points of an equilateral triangle falls on the angle of a square whose side is a, and the other an- gular points lie on the opposite sides of the square. What is the length of a side of the triangle, and what is its area ? Ans. The side of the triangle is a( -\/6— -v/2), and its area is a\2\/3 — 3). Ex. 19. The area of a ring contained between two concentric circles is a, and its breadth is b. What are the radii of the circles ? a b . a b Ans. —. and -r — \--. 2bn 2 2bn 2 Ex. 20. Determine the radii of three equal circles, described in a given circle, which touch each other, and also the circum- ference of the given circle whose radius is R. Ans. R(2l/3-3). Ex. 21. Having given the three lines a, b, and c, drawn from the three angles of a triangle to the middle of the opposite sides, to determine the sides. Ans. %y/ 2a 2 + 2b*-c 2 , |V 2a 2 + 2c 2 ~-/r , |V26 2 + 2c 2 — cr. Ex. 22. Having given the hypothenuse (a) of a right-angled triangle, and the radius (r) of the inscribed circle, to determine the other sides. Ans. l{a + 2r± Vcr-4.ar— 4r 2 ). 270 Examples for Practice. The Straight Line. Ex. 23. Construct the line whose equation is Ex. 24. Construct the line whose equation is y 2 — 5x 2 . Ex. 25. Find the lengths of the sides of a triangle, the co- ordinates of whose vertices are a/=2, y'=3; x" = 4, y" = — 5 ; oc"'=— 3, y /// =— 6, the axes being rectangular. Ans. V68, ^50, V 10 ^. Ex. 26. Find the co-ordinates of the middle points of the sides of the triangle, the co-ordinates of whose vertices are (4, 6), (8, -10), (-6, -12). Ans.{Q, -2), (1, -11), (-1, _3). Ex. 27. The line joining the points whose co-ordinates are (6, 9), (12. —15) is trisected. Find the co-ordinates of the point of trisection nearest the former point. Ans. x=8, y=l. Ex. 28. The co-ordinates of a line satisfy the equation x 2 +y 2 — 4x—6y=l8. What will this equation become if the origin be removed to the point whose co-ordinates are (2, 3) ? Ans. X 2 + Y 2 = 31. Ex. 29. The co-ordinates of a line, when referred to one set of rectangular axes, satisfy the equation y 2 —x 2 = 6. What will this equation become if referred to axes bisecting the angles between the given axes ? Ans. XY = 3. Ex. 30. What points are represented by the two equations x 2 -\-y 2 — 2b, and x— y—\ ? Ans. (4, 3), (-3, -4). Ex. 31. The equation of a line referred to rectangular axes is 3x+4y + 20 = 0. Find the length of the perpendicular let fall upon it from the origin. Ans. 4. Examples for Practice. 271 Ex. 32. The equation of a line referred to rectangular axes is 2x+y — 4 = 0.' Find the length of the perpendicular let fall upon it from the point whose co-ordinates are (2, 3). a 3 Ans. —-. V 5 Ex. 33. Form the equations of the sides of a triangle, the co- ordinates of whose vertices are (2, 1), (3, —2), ( — 4, —1). Ans. 3a?+y=7; a?+7z/+ll = 0; 3y-a:=l. Ex. 34. Form the equations of the sides of the triangle, the co-ordinates of whose vertices are (2, 3), (4, —5), ( — 3, —6). Ans. ^ Ans. du = - Va 2 —x 2 Ex. 6. Wha^s the differential of the function _ a+x ? Va— x Ans. au = 2{a—xf ns. au= -.ax. Ex. 7. What is the differential of the function x U-- z+Vl-x 2 dx Ans. du-. u = a + — — g? Vl—x 2 + 2x(l—x 2 ) Ex. 8. What is the differential of the function 4-Vx 3+x 2 . 7 6(1— x 2 )dx Ans. du=y- 1 ' , . Ex. 9. What is the differential of the function a 2 —x 2 „ u=— =-= r : ar+crar+ar . 7 — 2x( 2a 4 + 2a 2 x 2 — x*)dx Ans. du = ; y . (a 4 + aV'+a? 4 ) 2 Ex. 10. What is the differential of the function x n 7 W = A?2S. C?W: (l+a?) n+1 Ex. 11. What is the differential of the function u = (a— x)y/ a 2 -{-x 2 ? {a 2 — ax-\-2x 2 )dx Ans. dn = Va 2 + x 2 Ex. 12. What is the differential of the function u = (a 2 — x 2 ) Va + x ? Ans. du=\(a — 5x)Va+x.dx. 276 Examples for Practice. Ex. 13. What is the differential of the function Vx 2 -\-y 2 7 axydx—ax 2 du Ans. du= ^— ^-. %{x 2 + y 2 ) 2 Ex. 14. What is the differential of the function u = (2a 2 + 3x 2 )(a 2 -x 2 fl Ans. dn= — 15x 3 V a 2 — x 2 .dx. Ex. 15. What is the differential of the function u~ Va + x+Va—x „ Va + x—Va—x , a 2 -\-aVd z — x 2 7 Ans. du = ... dx. x 2 vd z —x 2 Ex. 16. What is the differential of the function a-\-2bx ? U ~(a-\-bxf ' . 7 — 2b 2 xdx Ans. du=- — T . (a + bxy Ex. 17. What is the differential of the function u=x log - , xl Ans. dn = (l -{-log. x)dx. Ex. 18. What is the differential of the function log. x . x . 7 (1— log. x)dx Ans. du—- % — - — . x* Ex. 19. What is the differential of the function x log. X ' 7 (log. x— l)dx Ans. du — - — 7; ~ — . (log. xy Ex. 20. What is the differential of the function u = (\og.x) n ? 7 7?(log. x) n ~ l dx Ajis. du — -±—£ — - Examples for Practice. 277 Ex. 21. What is the differential of the function u=\og. [a?+ Vx 2 + a 2 ] ? Ans. du=- Vcc 2 + ( Ex. 22. What is the differential of the function * a+ V« +# ' . , adx Ans. au=- iVci 2 -\-l Ex. 23. What is the differential of the function 1 c VI -fa? 2 — a? ' AftS. o'tt: Ex. 24. What is the differential of the function . < y / «+a?4- Va— x } ;=lo °- t "7^= — 7= 5 ■ v Va-fa?— va— a? Ans. du = wVa 2 —x 2 Ex. 25. What is the differential of the function ° ( Va- V* > , -y/a.dx A?is. au = -. — ;-. (a—x)yx Development into Series. Ex. 26. Develop into a series the function u=Va 2 -{-x 2 . Ans. u — a-\ -A -— , etc. 2a 2.4a 3 ^ 2.4.6a 5 ' Ex. 27. Develop into a series the function u=V2x— 1. Aws. u=y — HI— a; . etc.|. ' 2 2 5 278 Examples for Practice. Ex. 28. Develop into a series the function 1 Vb 2 — x 2 Ans. u = b~ l + -b~ 3 x 2 + -^-rb~ 5 x i + ' ' -6~ 7 a? 6 + , etc. 2 2.4 2.4.6 Ex. 29. Develop into a series the function u=(a 2 + x 2 ) 3 '. lo 54 5.2 2 5.2.1 _s Arcs. ^/; = a J + o a ^ +77^ ~ o~«a a ■£+, etc. o 0.0 0.0. y Ex. 30. Develop into a series the function 1 U — -Vtf+i 1 «*_ 5a- 8 5.9a? 12 5.9.13a? 16 M_ a _ 4a 5+ 4^~4.8.12a 13+ 4.8.12.16a"~' Ex. 31. Develop into a series the function i Z£ = (a 5 -f-a 4 a?— a 5 ) 5 . k 4 f 4.9 a? 3 4.9.14 a 4 A, 5 . ^^-ng^+gyas- w • l.2.3.4 +>etc - Maxima and Minima. Ex. 32. Fmd the values of a? which will render ^^ a maximum or a minimum in the equation u = x 4 — 8x 3 + 22a 2 — 24a: + 1 2. Ans. This function has a maximum value when x~2, and a minimum value when a=l or 3. Ex. 33. Determine the maxima and minima values of the function a 2 x \a—xy Ans. u has a maximum when ac=-\-a } and a minimum when as=—a. Ex. 34. Find the values of x which will render the function u = 3a 2 x 3 — b*x+c 5 a maximum or a minimum. b 2 Ans. There is a maximum corresponding to x=i— — , and b 2 a minimum corresponding to x~ +^-. Examples for Practice. 279 Ex. 35. Find the values of x which will render u a maximum or a minimum in the equation u = 3a: 4 — 1 6a: 3 +■ 6ar + 72a: - 1 . Ans. This function has a maximum value when x—-\-2, and a minimum value when x~ — 1 or +3. Ex. 36. It is required to find the fraction that exceeds its cube by the greatest possible quantity. Ans. +VJ. Ex. 37. It is required to inscribe the greatest rectangle in an ellipse whose axes are 2 A and 2B. Ans. The sides of the rectangle are A-\/2 and B-y/2. Ex. 38. The equation of a certain curve is a 2 y = ax 2 — x 3 . Required its greatest and least ordinates. Ans. When o?=fa, y is a maximum; when x—Q, y is a minimum. Ex. 39. It is required to circumscribe about a given parabola an isosceles triangle whose area shall be a minimum. , Ans. The altitude of the triangle is four thirds of the axis of the parabola. Ex. 40. Required the least parabola which shall circumscribe a circle whose radius is R. Ans. The axis of the parabola is fR, and its base is 3R. Ex. 41. What is the altitude of the maximum cylinder which can be inscribed in a given paraboloid ? Note. — A paraboloid is a solid formed by the revolution of a parabola about its axis. Ans. Half the axis of the paraboloid. Ex. 42. What is the diameter of a ball which, being let fall into a conical glass full of water, shall expel the most water pos- sible from the glass, the depth of the glass being 6 inches, and its diameter at top 5 inches ? Ans. 4-^g- inches. Subtangents and Subnormals. Ex. 43. Find the value of the subnormal of the curve whose equation is y 2 = 2a 2 log. a?. A ^ Ans. — . . 280 Examples for Practice. Ex. 44. Find the value of the subnormal of the curve whose equation is 3flj/ 2 + 3 = 2# 3 . Ans. — . a Ex. 45. Required the subtangent of the curve whose equa- tion is 9 x 3 y — — • a— x 2x(a—x) Ans. — — — — . 3a— 2x Ex. 46. Required the subtangent of the curve whose equa- tion is xy 2 = a\a— x). 2(ax—x 2 ) Ans. * -. a Ex. 47. Determine when the subtangent of the preceding curve is a minimum. Ans. When x—\a. Ex. 48. Find the value of the subtangent of the curve whose equation is x 2 y 2 — (a + x) 2 (b 2 — x 2 ). x(a-\-x)(b 2 —x 2 ) Ans. i rr~u — -- x 6 + ao* Curvature and Curve Lines. Ex. 49. Determine the radius of curvature at any point of the cubical parabola whose equation is y z = ax. Ans. R= (V+< 6a 2 y Ex. 50. Determine when the curvature of the preceding curve is greatest. 4/^2" Ans. When v = \/ — • J v 45 Ex. 51. Determine the radius of curvature at any point of the logarithmic curve whose equation is y = a x . 3 J (M 2 +« 2 P Ans. R = y ; , M being the modulus, and a the base. Examples for Practice. 281 Ex. 52. Determine the point ot greatest curvature of the logarithmic curve. m. • • M Ans. The point whose ordinate is equal to —7-. Ex. 53. Determine whether the curve whose equation is y 3 =x 5 has a point of inflection. Ans. This curve has a point of inflection at the origin. Ex. 54. Determine the point of inflection in the curve whose equation is ax 2 = a 2 y-\-x 2 y. Ans. There is an inflection at the point where y = \a. Ex. 55. Determine the point of inflection in the curve whose equation is x 2 y 2 = a 2 (ax — x 2 ). Ans. There is a point of inflection corresponding to each r . : * 3a , a 01 the points x=—, and y= ——/-• A Ex. 56. Determine whether the curve whose equation is (y — b) 3 = (x—a) 2 has a cusp at the point where the tangent is parallel to the axis ofY. INTEGRAL CALCULUS. Integration of Differentials. e di dx Ex. 1. Find the integral of the differential du= (a—xf Ans. u=: 1 "4(a— a?) 4 " Ex. 2. Find the integral of the differential Axdx dll: \\-x 2 f 2 Ans. u = - ^ + C. 1 —x* Ex. 3. Find the integral of the differential , 2adx du = — - — xV 2ax— x 2 . 2V2ax~x 2 „ Ans. u= f-C. 282 Examples for Practice. Ex. 4. Find the integral of the differential 7 xdx du = 3 • (2ax— x 2 ) 2 Ans. u=-\/ — t-C. a v 2a— x Ex. 5. Find the integral of the differential 7 x 8 dx du=- Va? + 6x 9 A Ex. 6. Find the integral of the differential 7 dx au= Va 9 + Gx 9 c , Ans. u— — |-U 27 VT+ X~ x 3 , 3X 5 3. bx 1 r Ans. w= *__+— -^-^+,etc., +C. Ex. 7. Find trie integral of the differential a; 3 + cr Ans. w=log. (# 3 + a 2 ). Ex. 8. Find the integral of the differential 5x 3 dx du — 3* 4 + 7' Arcs. w = fUog.(3^ 4 + 7). Ex. 9. Find the integral of the differential Ex. 16. Find the integral of the differential 7 x 5 dx du = - V2ax- ■x" . 9a C x A dx x^ . Ex. 17. Find the integral of the differential , x 5 dx x*-\-a' J ' . x* a 2 x 2 a*. . . Ans. u=- — -f- log. {a?+a z ). Ex. 18. Find the integral of the differential , x 3 dx du = — - Vl-x 2 . 2 {* xdx r 2 Ans. u=- I — _ i/i V Vl-x 2 3 V1 ' 284 Examples for Practice. Ex. 19. Find the integral of the differential (III: Va + bx 2 Ex. 20. Find the integral of the differential du = x\a + bx 3 )' J dx. 3 2x 2 (a + bx 3 y- i 4a . . , a Ex. 21. Find the integral of the differential cZtt: Ans. u—-—— log. (a + bx). Ex. 22. Find the integral of the differential xdx dlt: (a+bxf \b^2b 2 ){a + bxf Ex. 23. Find the integral of the differential x~^dx du-- a -f bx' 1 x Ans. u — - W. a to ' a-\-bx' Ex. 24. Find the integral of the differential dx du — x(a-\-bx 3 )' 1 r 3 Ans. u= — losr. 3a °'a-\-bx 3 ' Ex. 25. Find the integral of the differential dx du — ^ + 6^+8' Ans. M=-loof. . 2 ° #+4 Examples for Practice. 285 Rectification, Quadrature, etc. Ex. 26. Determine the length of the curve of a parabola cut off by a double ordinate to the axis whose length is 12, the ab- scissa being 2. Arts. 12.8374. Ex. 27. Determine the circumference of an ellipse whose two axes are 24 and 18 inches. Ans. 66.31056. Ex. 28. Required the equation of the curve whose area is equal to twice the rectangle of its co-ordinates. Ans. The equation is xif — a. Ex. 29. Determine the area of the logarithmic curve. Ans. s=M(y'—l). Ex. 30. Determine the area of an hyperbola whose base is 24 and altitude 10, the transverse axis being 30. Ans. 151.68734. Ex. 31. Determine the area of the curve whose equation is 9 9. 9 ° i