A TREATISE ON THE DIFFERENTIAL AND INTEGRAL C A LC iL U S, AND ON THE CALCULUS OF VARITATIONS. BY EDWARD IH COURTENAY, LL. D. LATE PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF VIRGINIA. NEW YORK: A. S. BARNES & BURR, 51 & 53 JOHN STREET. SOLD BY BOOKSELLERS, GENERALLY, THROUGHOUT THE UNITED STATES. 1860. ENTEnRE according to Act of Congress, in the Year One Thousand Eight Hundred ard Fifty five, by A. S. BARNE, & COMPANY, in the Clerk's Office of the District Court of the United States, for the Southern District of New York. JON 1S, &. DENYSE, STEREOTYPERa, 183 WVilliam-street, New York. EDWARD H. COURTENAY. IN the publication of the following Treatise on the Differential and Integral Calculus by Edward H. Courtenay, two Institutions have an equal interest-the Military Academy where he was graduated in the year 1821, and the University of Virginia, where he died in the Fall of 1853. Mr. Courtenay was born in the City of Baltimore, on the 19th of November, 1803. He entered the Military Academy as a cadet in September, 1818, and was the youngest member of the Class of that year. The Course of Study embraced a term of four years. In three years Mr. Courtenay made himself highly proficient in all the branches, and was graduated at the head of his class, in July, 1821. In his initiatory examination he made a strong impression on the mind of the examiner, who remarked, when the examination was.concluded, that " a boy from Baltimore, of spare frame, light complexion and light hair, would certainly take the first place in his class." We transcribe the following record from the Register of the United States Military Academy. "EDWARD II. COURTENAY-Promoted Bvt. Second Lieut., Corps of Engineers, July 1, 1821.-Second Lieut. July 1, 1821.-Acting Asst. Professor of Natural and Experimental Philosophy, Military Academy, from July 23, 1821, to Sept. iv EDWARD H. COURTENAY. 1, 1822; and Asst. Professor of Engineering, from Sept I, 1822, to Aug. 31, 1824.-Acting Professor of Natural and Experimental Philosophy, Military Academy, from Sept. 1, 1828, to Feb. 16, 1829; and Professor, from Feb. 16, 1829, to Dec. 31, 1834.-Resigned Lieutenancy of Engineers, Feb. 16, 1829; and Professorship of Natural and Experimental Philosophy; Dec. 31, 1834.-Professor of Mathematics, University of Pennsylvania, from 1834 to 1836.-Division Engineer, New York and Erie Railroad, 1836-37.-Civil Engineer, in the service of United States, employed in the construction of Fort Independence, Boston Harbor, from 1837 to 1841.- Chief Engineer of Dry Dock, Navy Yard, Brooklyn, N. Y., 1841-42.-Professor of Matlhematics, University of Virginia, since 1842.-Anthor of Elementary Treatise on Mechanics, translated from the French of M. Boucharlat, with additions and emendations, designed to adapt it to the use of the Cadets of the U. S. Military Academy," 1833.-Degree of A. M., conferred by University of Pennsylvania, 1834; and of LL. D., by Hampden Sidney College, Va., 1846." * Mr. Courtenay, while employed as Engineer in the construction of the works in Boston Harbor, was associated with that distinguished officer, Colonel Sylvanus Thayer, of the Corps of Engineers. The year before Mr. Courtenay entered the Military Academy, as a Cadet, Colonel Thayer had been appointed Superintendent. He was then engaged in laying the foundation of the system of instruction and discipline which has imparted so much reputation to that institution. It was among the most agreeable and cherished remembrances of Mr. Courtenay's life that he enjoyed the entire confidence and friendship of so interesting and distinguished a.man. The relation of principal and pupil, in a public institution became the basis of a sincere and generous friendship; and when the news reached the north that Courtenay was dead, no eye was moistened by a tear of warmer sympathy than that of the Superintendent who had guided his youth and admired'his life. EDWARD H. COURTENAY. V The author of this notice examined Mr. Courtenay when he entered the Military Academy, was associated with him in the Academic Board, and knew him intimately in all the situations which he subsequently filled; and yet feels quite incompetent to do justice to the memory of so perfect a man and so dear a friend. The painter who has a faultless form to delineate or a perfect landscape to transfer to the canvas, is embarrassed by the very perfection of his subject. He has nothing to put in opposition to the beautiful-no shading that can give full effect to the living light. Characters which afford strong contrasts are easily drawn-it is the perfect character which it is difficult to sketch. The intellectual faculties of Professor Courtenay were blended in such just proportions, that each seemed to aid and strengthen all the others. He examined the elements of knowledge with a microscopic power, and no distinction was so minute as to elude the vigilance of his search. He compared the elements of knowledge with a logic so scrutinizing that error found no place in his conclusions; —and he possessed, in an eminent degree, that marked characteristic of a great mind, the power of a just and profound generalization. His mind was quick, clear, accurate and discriminating in its apprehensions-rapid, and certain, in its reasoning processes, and far-reaching and profound in its general views. It was admirably adapted both to acquire and use knowledge. The intellectual faculties, however, are but the pedestal Vi EDWARD H. COURTENAY. and shaft of the column-the moral and social faculties are its entablature or crowning glory. It is these faculties which shed over the whole character a soft and attractive radiance, exhibiting in a favorable light the majesty of intellect and the divine attributes of truth, justice and beneficence. It was the ardent desire and steady aim of Professor Courtenay, during his whole life, to be governed by these principles, and there are few cases in which the ideal and the actual have been brought more closely together. Modest and unassuming in his manners even to diffidence, he was bold, resolute and firm in asserting and maintaining the right. Liberal in his judgments of others, he was exacting in regard to himself. He could discriminate, reason, and decide justly. even when his own interests were involved in the issue. His love of truth and justice was stronger than his love of self or of friends. His intercourse with others was marked by the gentlest courtesies. He was an attentive and eloquent listener. Differences of opinion, appeared to excite regret rather than provoke argument, and his habitual respect for the opinions, wishes and feelings of others, imparted an indescribable charm to his manners. As a professor lie was a model. He was clear, concise, and luminous in his style and methods. Laborious in the preparation of his lectures, even to the min-ltest facts, he was at all times prepared to impart information. His manner, as a teacher, was highly attractive. He never by look, act, word, or emphasis disparaged the efforts or undervalued the acquirements of his pupils. His pleasant smile and kind EDWARD H. COURTENAY. Vii voice, when he would say, " Is that answer perfectly correct?" gave hope to many minds struggling with the difficulties of science and have left the impression of affectionate recollection on many hearts. At the Military Academy, on the banks of the Hudson, where Mr. Courtenay was educated, and where he first labored to advance the interest of instruction and science, his name is recorded on the list of distinguished graduates, and honorably enrolled among the most eminent Professors of that Institution. There his labors and memory will live long together. At the University of Virginia he has left a name equally dear to that distinguished Faculty of which he was an ornament and to the many pupils whom he there taught. When these, in later years, shall revisit their Alma Mater, to revive early and cherished recollections-to strengthen the bonds of early friendships and renew their resolves to be good and great, they will find that a wide space has been made vacant. They will realize in sorrow that a favorite professor has been transferred from the halls of instruction to the grove of pines which borders the town, and which contains the remains of the revered dead. Thither they will go, in the twilight of the evening, to visit the grave of a man of science-their able teacher and faithful friend. In reviewing his life and contemplating his character, they will exclaim" Mark the perfect man and behold the upright; for the end of that man is peace." FISHKILL LANDING, Mlarch 10th, 1855, 5 1 NOTICE. The following work was left by Professor. Courtenay, in manuscript, in a highly finished condition: and yet, it must be regretted that it could not receive the final corrections of the author. A premature death, at the meridian of life, placed the work in other hands, and any slight inaccuracies of language which may now appear, would doubtless have been corrected, if the sheets could have passed under the eye of the author. It is a cause of thankfulness, however, that the work was entirely completed by Professor Courtenay; and in its publication the plan, language and even the punctuation, have been followed with a fidelity due to the memory of a fiiend. The work will be found more full and extensive than any which has yet appeared in this country on the same subject and the part which relates to the Calculus of Variations will be especially acceptable to the Americar public. It is perhaps not improper to add, that the Publishers have generously offered to publish the work on very favorable terms, and that the profits, whatever they may be, will go to the family of the author. CONTENTS. THE DIFFERENTIAL CALCULUS. PART I. PAGE. CHAPTER I. FIRST PRINCIPLES...... 13 CHAPTER II. DIFFERENTIATION OF ALGEBRAIC FUNCTIONS... 23 Examples..... 28 CHAPTER 1II. TRANSCENDENTAL FUNCTIONS....31 Examples of Logarithmic and Exponential Functions 32 Trigonometrical Functions..... 35 Geometrical Illustration....38 Circular Functions.. 39 Examples........42 CHAPTER IV. SUCCESSIVE DIFFERENTIATION... 45 Examples......46 CHAPTER V. MACLAURIN'S THEOREM...... 49 Examples......50 Applications....... 53 CHAPTER VI. TAYLOR'S THEOREM....... 60 Examples........ 62 Applications........64 To Differentiate u=F(p,q) where p=fx and q=fix.. 68 To Differentiate u= F (p,q,r,s, &c.) where p,q,r,s, &c. are functions of the same variable.....70 To Differentiate u=F(p,x) where p=fx.... 71 Partial and Total Differential Co-efficients...71 Examples......72 Differentiation of Implicit Functions..... 73 X CONTENTS. PAGE. CHAPTER VII. ESTIMATION U0 TIE VALUES OF FUNCTIONS HAVING THE INDETERMINATE FORM.......77 0 The For...... 77 Exampie;.... 80 The Form........84 The Form oo X 0.. 85 The Form wc -......85.+00 The Forms 0,, -..... 85 Examples...... 86 CHAPTER VIII. MAXIMA AND MINIMA FUNCTIONS OF A SINGLE VARIABLE.. 90 Conditions necessary to render a Function of a Single Variable a Maximum or Minimum......91 Maximum and Minimum Values of an Implicit Function of a Single Variable......95 Examples........ 97 CHAPTER IX. FUNCTIONS OF Two INDEPENDENT VARIABLES... 111 To Differentiate a Function of Two Independent Variables. 114 To Differentiate a Function of Several Independent Variables.. 114 To Differentiate Successively a Function of Two Independent Variables 115 Implicit Functions of Two Independent Variables... 116 Given u = pz, and z = F(x,y) to Differentiate u without Eliminating z 117 Elimination by Differentiation......117 To Determine whether any Proposed Combination of x and y is a Function of some other Combination.... 120 Development of Functions of Two Independent Variables by Maclaurin's Theorem.......121 Lagrange's Theorem...... 122 Examples........ 125 CHAPTER X. MAXIMA AND MINIMA FUNCTIONS OF Two INDEPENDENT VARIABLES. 129 Conditions necessary to render a Function of Two Independent Variables a Maximum or Minimum.... 130 Examples........ 133 CHAPTER XI. CHANGE OF THE INDEPENDENT VARIABLE.... 137 Examples........ 138 CHAPTER XII. FAILURE OF TAYLOR'S THEOREM..... 141 Examples....... 143 CONTENTS. xi THE DIFFERENTIAL CALCULUS. PART II. APPLICATION OF THE DIFFERENTIAL CALCULUS TO THE THEORY OF PLANE CURVES. PAGE. CHAPTER I. TANGENTS TO PLANE CURVES.-NORMALS.-ASYMPTOTES... 147 Differential Equation of a Tangent to a Plane Curve.. 148 Differential Equation of a Normal to a Plane Curve... 149 Expressions for the Tangent and Subnormal.. 150 Applications to the Parabola, Ellipse, and Logarithmic Curve.. 150 Expressions for the Tangent, the Normal, and the Perpendicular on the Tangent of a Plane Curve.....151 Expressions for the Polar Subtangent, Subnormal, Tangent, Normal, and Perpendicular to the Tangent of a Plane Curve referred to Polar Co-ordinates....... 152 Examples. The Spiral of Archimedes. The Logarithmic Spiral. The Lemniscata of Bernouilli...... 154 Rectilinear Asymptotes...... 156 Conditions that Determine the Existence of Rectilinear Asymptotes. 150 Applications to the Hyperbola, the Logaritbmic Curve, the Cissoid, the Parabola, the Hyperbolic Spiral, the Spiral of Archimedes, the Logarithmic Spiral, and the Lituus... 157 Circular Asymptotes.......159 CHAPTER II. CURVATURE AND OSCULATION OF PLANE CURVES... 160 Differential of the Arc of a Plane Curve in Terms of the Differentials of the Co-Ordinates.....161 Conditions of the Osculation of Curves, in the Different Orders of Contact 162 To Determine the Radius of Curvature and the Co-Ordinates of the Centre of the Osculatory Circle... 164 Examples........ 167 At the Points of Greatest and Least Curvature of any Curve, the Osculatory Circle has Contact of the Third Order.. 169 Curves Intersect at the Point of Contact, when the Order of Contact is Even; but do not Intersect when it is Odd... 170 Formula for the Radius of Curvature when the Independent Variable is Changed........171 Radius of Curvature of Curves referred to Polar Co-Ordinates. 172 Examples........173 Radius of Curvature of Curves referred to the Radius Vector and the Perpendicular on the Tangent..... 174 xii CONTENTS. PAGE. CHAPTER III. EVOLUTES AND INVOLUTES.. 176 To Determine the Evolute of a Given Curve y = Fx. 176 Applications to the Parabola, Ellipse and Equilateral Hyperbola. 176 Normals to the Involute are Tangents to the Evolute.. 178 The Difference of any Two Radii of Curvature is Equal to the Intercepted Arc of the Evolute......179 Evolutes of Polar Curves Given by the Relation between the Radius Vector and the Perpendicular, on the Tangent... 180 CHAPTER IV. CONSECUTIVE LINES AND CURVES... 182 To Determine the Points of Intersection of Consecutive Lines or Curves 182 Consecutive Normals to any Plane Curve.. 183 The Curve which is the Locus of the Points of Intersection of a Series of Consecutive Curves, touches each Curve of the Series, and is called the Envelope........184 Examples of the Determination of Envelopes... 185 CHAPTER V. SINGULAR POINTS OF CURVES....190 Multiple Points...... 190 To Determine whether a Curve has Multiple Points of the First Species 190 Examples....... 191 To Determine whether a Curve has Multiple Points of the Second Species.... 194 Conditions for Determining Conjugate or Isolated Points.. 195 Examples... 196 Cusps........ 198 Examples........ 19 Points of Inflexion.....201 Examples........ 202 Stop Points....... 203 Shooting Points....... 203 Points of Contrary Flexure of Spirals.... 204 CHAPTER VI. CURVILINEAR ASYMPTOTES..... 205 Conditions necessary to Render Two Curves Asymptotes to each other 205 General Form for the Value of the Ordinate in Curves that admit of a Rectilinear Asymptote. 206 Examples.... 206 CHAPTER VII. TRACING OF CURVES...... 208 General Directions...... 208 Examples........ 209 CONTENTS. Xiii THE DIFFERENTIAL CALCULUS. PART III. THEORY OF CURVED SURFACES. PAGE. CHAPTER 1. TANGENT AND NORMAL PLANES AND LINES... 214 General Differential Equation of a Tangent Plane... 214 Equations of a Line Normal to a Curved Surface...216 Equations of a Line Tangent to a Curve of Double Curvature. 217 Equation of a Plane Normal to a Curve of Double Curvature.. 217 Examples of Tangent Planes to Surfaces...218 CHAPTER II. CYLINDRICAL SURFACES, CONICAL SURFACES, AND SURFACES OF REVOLUTION 219 General Differential Equation of all Cylindrical Surfaces. 219 Equation of a Cylindrical Surface which Envelopes a Given Surface, and whose Axis is Parallel to a Given Line.... 220 WVhen a Cylinder Envelopes a Surface of the Second Order,.the Curve of Contact is an Ellipse, Parabola. or Hyperbola.. 222 General Differential Equation of Conical Surfaces... 222 Equation of a Conical Surface which Envelopes a Given Surface, and whose Vertex is at a Given Point... 224 When a Cone Envelopes a Surface of the Second Order, the Curve of Contact is an Ellipse, Parabola, or Hyperbola.. 225 General Differential Equation of Surfaces of Revolution.. 226 A Curved Surface Revolving about a Fixed Axis; to Determine the Surface which Touches and Envelopes it in Every Position. 227 Examples.. 227 CHAPTER 1II. CONSECUTIVE SURFACES AND ENVELOPES.... 230 Equations of the Intersection of Consecutive Surfaces.. 231 The Locus of the Intersections of a Series of Consecutive Surfaces Touches each Surface in the Series....231 Examples.. 233 CHAPTER IV. CURVATURE OF SURFACES....... 235 Conditions Necessary for Contact of the Different Orders.. 235 The Ellipsoid, Hyperboloid and Paraboloid can have Contact of the Second Order... 236 To Determine the Radius of Curvature of a Normal Section of a Surface 236 The Sum of the Curvatures of Two Normal Sections through the same Point, and Perpendicular to Each Other, is Constant.. 238 Principal Radii of Curvature at a Given Point.. 239 Properties of the Principal and Normal Sections... 240 XiV CONTENTS. PAGE. At Every Point of a Curved Suiface a Paraboloid tnay be Applied, having Contact of the Second Order.. 242 Meusnier's Theorem.... 244 Lines of Curvature...... 244 THE INTEGRAL CALCULUS. PART I. CHAPTER I. FIRST PRINCIPLES........ 248 Integration of Simple Algebraic Forms.... 249 Examples........ 250 CHAPTER II. ELEMENTARY TRANSCENDENTAL FoMS.... 253 Logarithmic Forms....... 253 Examples..... 253 Circular Forms...... 254 Examples....... 257 Trigonomnetrical Forms...... 259 Examples....... 259 Exponential Forms....... 260 Examples.......260 CHAPTER III. RATIONAL FRACTIONS...... 261 CASE I. When the Denominator can be resolved into Real and Unequal Factors of the First Degree..... 261 CASE II. When the Denominator contains Equal Factors of the First Degree........265 Examples......266 CASE III. When the Simple Factors of the Denominator are Imaginary......... 268 General Examples. 271 CHAPTER IV. IRRATIONAL FRACTIONS....... 274 CASE I. When the Fraction contains only Monomial Terms. 274 CASE II. When the Surds in the Expression have no Quantity under the Radical Sign, but of the Form (a + b)... 275 CASE III. When there are no Surds except of the Form (a+bx-'+c2x2) 277 Examples....... 277 CONTENTS. XV PAGE. CHAPTER V. BINOMIAL DIFFERENTIALS.. 280 Conditions of Integrability... 281 Examples. 281 CHAPTER VI. FORMULAS OF REDUCTION.... 283 Formulae (A), (B), (C) and (D).. 284 Applications of Formulae (A), (B), (C) and (D). 286 CHAPTER VII. LOGARITHMIC AND EXPONENTIAL FUNCTIONS... 291 Logarithmic Functions...... 291 Examples...... 291 Exponential Functions...... 95 Examples........ 296 CHAPTER VIII. TRIGONOMETRICAL AND CIRCULAR FUNCTIONS.... 298 Trigonometrical Functions...... 298 Formule (E), (F), (G), (H), (I) and (K)... 299 Applications of Formule (E), (F), (G), (H), (I) and (K)..305 Formulae (L) and (M)..... 309 Circular Functions....310 CHAPTER IX. APPROXIMATE INTEGRATION.... 312 Method by Expansion....... 312 Examples...... 312 Bernouilli's Series....315 CHAPTER X. INTEGRATION BETWEEN LIMITS AND SUCCESSIVE INTEGRATION. 316 Integration between Limits..... 316 Precise Signification ofbXdx.... 317 Successive Integration. 319 To Develop the ntz Integral of /'XdxT in a Series. 319 To Deduce the Development off2.Xdxs from that of X. 320 Examples...... 320 Xvi CONTENTS. PAGE. THE INTEGRAL CALCULUS. PART II. RECTIFICATION OF CURVES. QUADRATURE OF AREAS. CUBATURE OF VOLUMES. CHAPTER I. RECTIFICATION OF CURVES...... 322 Formula for the Length of the Arc of a Plane Curve referred to Rectangular Co-ordinates..... 322 Examples of its Application.... 323 To Determine what Curves of the Parabolic Class are Rectifiable. 326 Formula for the Rectification of Polar Curves... 327 Examples of its Application...... 327 Formula when the Curve is referred to the Radius Vector and the Perpendicular on the Tangent. 328 CHAPTER II. QUADRATURE OF PLANE AREAS...... 330 Fornmulae....... 29 Examples...... 330 Formula for Polar Curves.. 333 Examples.. 334 Formula for Curves given by Relation between the Radius Vector and the Perpendicular on the Tangent.... 335 Examples.... 335 CHAPTER III. QUADRATURE OF CURVED SURFACES... -. 337 Formula for Surfaces of Revolution.... 338 Examples...... 338 Formula for any Curved Surface referred to Rectangular Co-ordinates 340 The Tri-rectangular Triangle and the Groin... 342 CHAPTER IV. CUVtATURE OF VOLUMES. 344 Volume Generated by the Revolution of a Plane Figure about an Axis 344 Examples.... 345 Volume of all Solids Symmetrical with respect to an Axis. 347 Examples.... 348 Volume of a Solid Bounded by any Curved Surface referred to Rectangular Co-ordinates. 350 Examples........ 351 Volume of a Solid Bounded by a Surface whose Equation is referred to Polar Co-ordinates..... 353 Examples.... 354 CONTENTS. Xvii THE INTEGRAL CALCULUS. PART III. INTEGRATION OF FUNCTIONS OF TWO OR MORE VARIABLES. CHAPTER I. PAGE. INTEGRATION OF EXPRESSIONS CONTAINING SEVERAL INDEPENDENT VARIABLES.... 356 Conditions of Integrability of Exact Differentials... 356 Formula for the Integration of the Form du = Pdx + Qdy. 358 Examples.... 361 Homogeneous Exact Differentials...363 Examples...... 365 CHAPTER II. DIFFERENTIAL EQUATIONS..... 367 Differential Equations of the 1st Order and Degree.. 368 1ST CASE in which the Variables may be Separated, Ydx + Xdy = 0. 369 2D CASE, the Form XYdx -- X Yldy = 0. 369 3D CASE, the Homogeneous Form (xanym + axnl+]ya-I.... + pxln+ ym-c)dx........ 370 4TH CASE, the Form (a + bx + cy)dx + (al + bilx + cly)dy = 0. 371 5TH CASE, the Linear Equation, dy + Xydx = Xdx.. 372 6TH CASE, Riccati's Equation, dy + by2dx = axndx... 375 Factors Necessary to Render Differential Equations Exact. 382 Geometrical Applications of Differential Equations of the 1st Order and Degree.... 389 CHAPTER III. DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND OF THE HIGHER DEGREES.... 394 d d When the Equation can be Resolved with Respect to -^.. 394 When it Cannot be Solved with Respect to d- but Contains only One of the Variables, and may be solved with respect to it. 397 2D CASE, when the Equation is Homogeneous with Respect to x and y 399 3D CASE, the Form y = x d + ( which 0 (dy) does not contain x or y.. 400 4TH CASE. the Form y = Px + Q, where P and Q are functions of p. 401 CHAPTER IV. SINGULAR SOLUTIONS OF DIFFERENTIAL EQUATIONS... 403 Conditions that Render a Singular Solution Possible...404 Singular Solutions Illustrated Geometrically.. 406 Conditions for Finding Singular Solutions without First Determining the Complete Primitives...... 407 XViii CONTENTS. PAG., CHAPTER V. INTEGRATION OF DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 411 The Form F(x, d......411 The Form F (y, ).... 412 The Form F(......413 dx. The Form F (x, dy d).... 413 The Form F(y, d,' dz2)=. 415 dx dx,)'415 The Form F(v, z, p)0...... 416 CHAPTER VI. Dp-tyERENTIAL EQUATIONS OF THE HIGHER ORDERS... 418 The Form F(d, d-) 0.. 418 \dxn- dx-~The Form F(d d-)..... 418 CHAPTER VII. SI? LTANEOUS DIFFERENTIAL EQUATIONS.... 420 Examples of the Integration of Several Systems of Equations. 420 CALCULUS OF VARIATIONS. CHAPTER I. 41 CT, AND FIRST PRINCIPLES..... 425 General Principles.... 425 kppli, aons....... 427 CHAPTER II. Al, TICt. CAL OF GENERAL FORMULE TO FUNCTIONS OF ONE. VARIABLE. 434 CHAPTER III. SUccESS Tm VARIATION...... 448 CHAPTER IV. MAXIM -. ND MINIMA....... 451 Ma;xma and Minima of One Variable.... 451 Restive Maxima and Minima of One Variable.. 473 Applications..... 460 DIFFERENTIAL CALCULUS. CHAPTER I. FIRST PRINCIPLES.? in all mathematical calculations, the quantities which are p,^-ented for our consideration belong to one of two remarkable cla sees: namely, constant quantities, which are such as preserve the same values throughout the limits of one investigation; or var, able quan!tities, which may assume successively different values, the number of such values being unlimited. The first letters of the alphabet, as a, b, c, &c., are usually employed to denote constant quantities, and the last letters z, y, x, &c. are used to represent such quantities as are variable. 2. When two quantities x and y are mutually dcependent upon each other, so that a knowledge of the value of one will lead to that of the other, they are said to be functions of each other. Thus, in the equations y ax, y =- bx2 +- cz - e, y = ax3 + bx2 - x + e, the value of y is determined as soon as that of x is known; and accordingly y is said to be a function of x. In like manner, an assumed value of y will fix the corresponding values of x, and therefore x is a function of y. There is this difference, however, between the two cases: when the value 14 DIFFERENTIAL CALCULUS. of x is assumed, that of y is obtained by a sinple substitution; whereas the determination of the value of x from that of y requires the solution of an equation. Hence, y is called an explicit function of x, but x is said to be an imp2licit function of y. The general fact that y is an explicit function of x is written thus: y = Fx, or y = gx, when the character F or qg stands as the representative of certain operations to be performed on the quantity x, the result of which operations will be a quantity equal in value to y. And wh-en we wish to imply that the values of x and y are connected by an unresolved equation, or that y is an implicit function of x, we write (.x, y) = O, or (x, y) = 0. For the purpose of illustration, let there be taken the three equations y=ax + b (1), y = ax2 + bx + c (2), y =- ax3 + bx2 + C + e (3), and suppose x to receive an increment h in each equation, converting it into x + h, and causing y to assume a new value yl Then if the form of each function, or value of y, be supposed to remain unchanged, the three equations (1), (2), and (3), will become respectively vt = a(x x+h)-t b (1)+ y, = a(x + h)2 + b(x h) + c (5), and 1 = a(x + h)3 + b(-x + h)2 + (x + h) e (6). Subtracting (1) from (4) we obtain y1-y=ch (7). FIRST PRINCIPLES. 15 from (2) and (5) we get y, - y = a(2xh + h) + bh (8). And from (3) and (6) y,- y a (3x2h + 3xh2 - h3) + b(2xh + h2) + ch (9). From (7) we deduce, by division, Y =a (10); from (8) Y = a(2x + A) + b (11); =-2ax + ah + b; and from (9) Y - Y - (322 + 3xh + h2) + b(2x + h) + c (12). The results, (10), (11), and (12), express tne ratio between the increment h assigned to x, and the corresponding increment y - y imparted to y. The values of this ratio, in the three examples selected, present remarkable differences. In the first example, this ratio retains the same value a, whatever may be the value assigned to the increment h. In the second example it consists of two parts, one = 2ax + b, entirely independent of h, and the other = ah, which varies with h. If the value of h be supposed to diminish, the ratio 2ax - b + ah (11), will become more and more nearly equal to 2ax + b; and, finally, when h becomes indefinitely small, the ratio is reduced to this latter value. The corresponding increments h and. Y - y, when indefinitely 16 DIFFERENTIAL CALCULUS. small, are called the differentials of the quantities x and y, and the limiting value -of the ratio Y1- Y is called the differential coefficient, because it is the multiplier of the differential of x necessary to produce the differential of y. The differentials of x and y are written dx and dy, the character d being the symbol of an operation to be performed on x or y, not a factor: and the differential coefficient is written ldx Moreover, one of the variables (usually x) is called the independent variable, its increment dx (although small) being arbi trarv; while the other y, whose increment dy depends on that of x, is called the dependent variable or simply the function. In the third example, the ratio Y1-Y h reduces, at the limit when h = 0, to dy dx = 3ax2 + 2bx + c. These examples illustrate the fact that two indefinitely small quantities may yet have a finite ratio; and they suffice to show that the form of the differential coefficient, which is usually a function of x, will depend very materially on the form of the original function y. (3.) The considerations just presented analytically admit of geometrical illustration. For, whatever may be the relation between x and y, the former may be regarded as the abscissa, and the latter as the ordinate of a plane curve; and the determination of the relation between the corresponding increments of x and y, is reduced to finding the change in the length of the ordinate produced by an arbitrary change in the length of the abscissa. FIRST PRINCIPLES. I1 It is the chief object of the Differential Calculus to investigate the laws of increase of functions having various'formls, when such changes are produced by an arbitrary change in the value of the independent variable upon which the values of the functions depend. Geometrical considerations will also point out very clearly how it happens that a given augmentation of the variable x will, in different stages of its magnitude, produce widely different increments of the function y. Referring to the an- X nexed diagram, it will be apparent that near - the vertex C of the \ curve CPE, a slight O X C De D increase in the value of the abscissa x will produce a comparatively large increase in the value of the ordinate y; but when the tangent to the curve forms a smaller angle with the axis OT, as at P, the sca.me increment in x will produce a much smaller increase of y; and if the tangent be nearly parallel to OX, the increment received by / will be very small in comparison with that given to x. Finally, by continuing to increase x, the ordinate y may first cease to increase, and may afterwards actually decrease, or the increment of y may become negative; and these different results will occur without any change in the form of the function y. 4. One of' the first inquiries presented for consideration is the determination of the general form of the function F(x -.- i); for, since we desire to compare y = Fx with Y1 = F(x + h), it is important to know what form F(x + h) will assumle when expanded into a series of terms involving x and h. Ilence the following 2 18 DIFFERENTIAL CALCULUS. Proposition. To determine the general form of the development of any function of the algebraic sum of two quantities, such as F(x + h), arranged according to the powers of the second 1. 1st. There must be one term in the development of the form Fx, and the other terms must contain h. For, since the development is supposed to be general, and therefore true for all values h, it ought to be applicable when h = 0, in which case the undeveloped function F(x + h) reduces to Fx. This condition will be satisfied by supposing the first term in the development to be ix, and all the succeeding terms to contain powers of h, since the supposition h = 0 will then give rise to an equation, Fx -- Fx, which is identically true. And no other conceivable form of development would& lead to this result. We may therefore write F(x + h) - Fx + Aha + Bhb + Ch" + &c. (1), in which the coefficients A, B, C, &c., will usually be functions of x; and the exponents a, b, c, &c., undetermined constants. 2d. None of the exponents, a, b, c, &c., can be negative. For if there could be a.term of the form Bh-b or b it would become infinite when h- 0, thus rendering the developed expression infinite, while the undeveloped expression would become simply Fx, and this latter would probably be finite. 3d. None of the exponents can be fractional. For if there could be a term of the form Eh' or E. /, such term would have as many different values as there are units in s; that is, it would have s values; and each of these values FIRST PRINCIPLES. 19 could be combined in succession with the aggregate of the other terms of the series. Now if each of these other terms, except the first term Fx, be supposed to have but one value, the sum of all the terms containing h will have s different values. And if Fx be susceptible of n different values, the entire development will admit of n X s values, since each value of Fx may be combined, in succession, with each value of the remaining terms. But F(x + h) being of the same form with Fx, must have the same number n of values. Thus, for example, if I I F(x + h) = (x + h)3, then Fix = X3 and both will have three values. If F(x J+ h) = a(x + ah)2 + b(x + h)5, then Fx = ax2 + bx5, and both will have five values, &c. Thus, in the case supposed above, where there was one fractional exponent, F(x + h) would have n values when undeveloped, but n X s values when developed-a manifest absurdity. We conclude therefore that the exponents a, b, c, &c., in the general development, must be positive integers; and in order to make the development include every possible case, we write F(x + h) = Fx 4- Ah + Bh2 + Ch3 - Dh4, &c., including every power of h. If in any particular case some of these terms should be unnecessary, it will suffice to suppose the corresponding coefficients A, B, C, &c., to reduce to zero. We have a familiar example of the expansion of F(x -- h) in the well known binomial theorem. Thus, if F(x + h) =,(X + h,) = xt + Znx-lh + (- ) 2 n(n - )(n - 2) xn-3,i3 -+ &c., 1. 2 1.2.3 20 DIFFERENTIAL CALCULUS. we shall have Fx=xn, A=-2Xn-, B= n(n - 1) x-2, 1.2 1 = n(n -- ) ( -2) xn-3, &C. -- 1.2.3 where A, B, C, &c., are functions of x. The following are likewise examples of the development as ap. plied to particular cases. 2. Let Fx = ( ) + bxn: then F(x + h) = (a + +x h + ) (x + +h)n, which expressions, when expanded by the binomial theorem, give F(x - h) =(a + x)+ (a x) i- - (a + x) - + &c., 2 8 + bxn + nbx n-1, - (n - ) bXn-2%2 + &c. 1.2 = Fx + [2bxn-1 + (a + x) + b(- 1)n -- ( + ) h2 +&c. which corresponds with the general form. 3. Let Fx = log x: then i'(x ) = log(x+) =logx( I - =)lorxlog( x + log ( + h7 /2 h3 h4 ) - log(- x + + &. 2x2 3x3 4.$,4 where i/ denotes the modulus of the system of logarithms. MM if.F(x.+4-) A=Fx h — h-2 h2+ h3 - 4 h +&c. xhih 2$X 3ox 4X4 which also corresponds to the general form. FIRST PRINCIPLES. 21 It may be well to observe, that although the form of the development of F(x + I) is always such as has been indicated while x retains its general value, yet it is possible (in some cases) to assign certain particular values to x which shall cause the development in this form to become impossible. Thus, if in the second of the above examples, we put x a, the true development of F(x + h) will become simply F(x+ h) 1 + + (- a) bn(- a)-1 h, + &c., in which one fiactional exponent appears. The same supposition causes all the coefficients involving negative powers of a + x to become infinite in the general expansion. It will be shown hereafter that the particular cases in which the general development is inapplicable, are always indicated by some of the terms of the development becoming infinite. At present it is sufficient to remark that the number of such cases is compara tively small, and that they will receive a special examination. 5. From the development of F(x + A), we derive a direct and general method of finding the differential of any proposed functioD y = Fx. For, if we give to x an increment h, we shall have Y = F(x A- h) = Fx + Ah + Bh2 + Ch13 + &c...y-y =- F( (x + h) - Fx =AhBh+ C32 - + &c.. -~ Y = A + Bh + C/2 + &c. And by passing to the limit, when h - 0, we get dy dy A, whence dy = Adx. dx 22 DIFFERENTIAL CALCULUS. Thus it appears that the coefficient A of the 1st power of h in the development of F(x - h) is the differential coefficient of the proposed function, and this multiplied by dx gives the required differential of y. It will be found convenient, however, to form rules for differentiating functions of the various forms likely to arise, and to this investigation we proceed next. CHAPTER II. DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 6. Prop. To differentiate the product of two functions of a sin. gle variable. Let -= yz, Ahere y and z are given functions of the same independent variable x, and let x take an increment h, converting u, y, and z, into ul, Y1, and z1. Then, since y1 and z1 will each be a function of x + h, we shall have l = + Ah + Bh2 + Ch3 + &c., and z z + Alh + Bih2 + C1h3 + &c..*. ~ = ylz = yz + (Az + A1y)h + (Bz + Bly + AA1)h2 + (Cz + Cly +AB1 + AB)h/3 + &c. " 1-. _Y - i- = Az + Aly + (Bz + Bly + AA)h + (Cz + Cly + AB1 + -41B)h2 + &. and when h - O, this becomes du dy dz = Az + Aly. = + y, i dx c dz dy dz since A =- and A1= dx dx And by multiplying by dx, we get du = zdy + ydz. 24 DIFFERENTIAL CALCULUS. Thus the diferential of thle product yz of two functions is found by multiplying each function by the differential of the other fIlctionl, and adding the results. 7. Prop. To differentiate the product of several functions of a single variable. 1st. Let q --- vyZ, where v, y, and z, are functions of the independent variable x. Put yz = s; then u =vs, and by the last proposition, du v= Vs + sdv, and also ds = ydz + zdy. Substituting the values of s and ds in that of du, there results du v (ydz - zdyy) + yzdv = vydC +- vzdy +- y+ dv. 2d. Let U = svyz. Put yz = wu; then t = svow,.. du svdw + swdv + vwds = s'(yd.z - szdy) 4 syzdv + v-zlds, or, d =u svydlz - svy - y - syzdv +-'vytd; and the same method could be a )pplied to the product of a greater number of functions. Hence we have the followying rule for the differential of the product of several functions:.AIultiply the di{/ferential of each factor by the continued prod'uct of all the otter fcctors, and addl the resullts. 8. Prop. To differentiate a fraction whose numerator and denaom. inator are functions of a single' variable. Lett u =,- where y and z are functions of X. s ALGEBRAIC FUNCTIONS. 25 Then y uz, and this differentiated by the rule for products, gives dy udz + zdu =- - dz +- zdu z. zdy = yclz + z2du, zdY - ydZ and by reduction du Thus the rule is as follows: Mlultiply the di ferential of the numerator by the denominator, and the dtfferentlil of the denominator by the numerator; subtract the second product from the Jirst, and divide the remainder by the square of the denomiznator. 9. Pr7op. To differentiate a power of a single variable. 1st. Let - = x?, where n is a positive integer. Regarding x" as the product x. x. x. x, &c., of n equal factors each x, and applying the rule for differentiating a product, we get du = x1'-1dx + x"-ldx + xn-ldx + &c., to n terms... du -- nxcZ-1dx, and the rule in this case is the following: Multitply the given power (x") by the exponent (giving nxn); then diminish the exponent by unity (giving nx~n-); and finally, multiply by t/e differential cf the root (producing nxn-dxlx). 2d. Now suppose the exponent n to be a positive fraction - a Tlen u = xC.'.: =- xa, where the exponents a and c are both positive integers. H-ence, by the application of the rule just established for such cases. we have cuc-ldu == axa-ldx fixa —1 a xa —1 a-l —a+ a a-. 1, d. du _. d = -dx -= - s cdx CUC.1 (" c a a — {x) 26 DIFFERENTIAL CALCULUS. and the rule for differentiating the power is the same as when the exponent is a positive integer. 3d. Let the exponent be a negative integer, or u = x-" I x Then u = — xn x"n+l and this differentiated by the rule for fractions, gives X l+l - (n, + 1)Xn+1 ndx du _ - -~' — dx _-nx"-ldx. x2n-t2 X,'+" And the rule is still the same. 4th. Let the exponent be a negative fraction, or let iu = x Then uc = xa, and by the first and third cases, a X-a —1 C tlc-ldu = - al-a-ldx, or, du =- -. dx. a x-a-ld.r a - -1.*. du= x ~ d., c c and the formula is still the same. We might have deduced the rule for cliffelrentiating a power, as alike applicable to all cases, by employing the binomial theorem; for, since the second term in the developmlent of (x + )", is nx"-lh, for all values of n, cdx we must have ()= nx-, or, d (x4)-nx8-1dx. It is intended, however, to demonstrate the truth of the binomial theorem by the aid of the differential calculus, and hence the necessity of establishing the rules for differentiation, without reference to that theorem. Remark. If the function which it is proposed to differentiate contain a constant factor, such factor will appear in the differential. ALGEBRAIC FUNCTIONS. 27 Thus d (ax) = adx, for when x takes the increment h, the function ax becomes U3U -a adiu du ul = a (x - h) and.. an a.'h dX Similarly if u = a. Fx, where F denotes,any function, then ul = aF (x + h) and du = ad (Fx). 10. Prop. To differentiate the algebraic sum of several functions of a single variable. Let u = As + Bv- Cy + Dz, where s, v, y, and z, are functions of x. Then when x takes the increment h, As becomes As- = A (s + A1,h + B1h2 + C1I3 &c.). Bv becomes Bvy - B (v + A2h -- B22 + C2h3 &c.). Cy becomes.Cy = C (y + A3h + B3h2 4 C33h &c.). Dz becomes Drz = D (z -- A4h + B4h2 + C+4h &c.)... u becomes uz= As - Bv - Cy + Dz + (AA1 + BA2 - CA3 +DA4) h + &c.. = (A. -+ B2 - C(AA- +3 + A4) dx. But A4dx = cls, A2d = dv, Adx = dy, A4dx = dz...du = Ads + Bdv- Cdy + Ddz. And the rule is as follows: Differentiate the terms successively, and take the alyebr(cic sum of the result. Remark. If a constant be connected with a variable quantity by the sign +- or -, such constant will disappear by differentiation. Thus, when we have u = a + Fx, then,t, = a + F(x + h) = a + Fx + Ah + Bh2, &c., = u -- Ah +- Bh2, &c..~. du = Adx, the constant a having disappeared. 28 DIFFERENTIAL CALCULUS. EXAMPLES. 11. 1. To differentiate y - 43 + 7x2 - 8x- 5. Applying the rule for powers to each term we obtain dy = 4 X 3 x'dx + 7 X 2xdx - Sdx = (12x2 + 14 - S)dx. dy 12x2 + 14x -8. dx 2. y' = ax2(bx + c) =: abx3 + acx. Differentiating this as a product, we get dy = 2ax(bx + c)dx + ax2bdx- = (3abx2 + 2acx)d7x. Or by first performing the multiplication indicated, and then di~ ferentiating as a sum, the same result is obtained... abx2 + 2acx. dX ~~3 y- ^-~4x3 Y - ( + x2)3 Differentiating by the rules for fractions and powers, we obtain dy 12x2(b + X2)3dx -(b + x2)2 X 4x3 X 2xdx dy _ (b + x2)6 122(b - x2) - 24 4 12x2(b -.2) - dx dx. (b + X2)4 (b + x2)4 dy 1 x2(5 - x2)' dx (b + 2)4 4. y = /a+- bx= 2 (a + bx2)2 1 -~ ~ ~ ~ dy 2 ha+ x dy = I + -bx2) X 22bxdx.. - _ ~ f dx ^a-bx2 ALGEBRAIC FUNCTIONS. 29 5. U = x(1 + 2)(1l + 3). d- = (1+ X2)(1 + 3)+ x(1 + 3) 2 + x(1 + 2) X 3x2 = 1 + 2 +- x3 + x + 2x2 - 2x5 + 3x3 + 3x5 = 1 + 3x2 - 4x +- 6z5. 6. u = 1 V^+ 4 = [+ 2 +(14+-2) 22 du 1 [ ( X2) X + (1+ 2) ] 2(X + + +X2) + ( — X+ (1+x-)~ x [l+(l+ x 2 /x + / + x V1 + 2-/ — 2 7. u =bxx du bx. dx c dut 6~c 8. _u -- -b c- -- b.= -C 6C~-7 8. C6 5 6 dx x* v - 9. )x.. - 3,:. 2 dX + xI X( + 1 -x du x (2 + 1 )dx +x3(x2 +1) 2 X 2 dx. 32 2 1 _ 1 +. d* 3x3 4x6(X + 1) 12: V + I o. = + x + (l4 + /-y + 2. i0.+ - - 1 + /1 -- _ - _ ___ du - x2(1 - x2) - (1 + /~ - ) l + x/i -Z.11, u - 1 + + - - A (I + x2) x + 1+ 2 -(1+ A2) 30 DIFFERENTIAL CALCULUS. di1 +- 22x - 2 +/1 =- x2' 2 "'- v'-/^s+ + (,c 23-. 12. a -= 4 Ca - - )6 - [a-* bi + (c2 - 3 4 1d3. ia- x + (2 2) a 4X 2+ b, 2+' &o., _ contXnue e] 3T) 4.r 2Vor, - a x.. u 2+ -, b d 13. - bV/4a q- 4x + a 1Te / fc a o z a+ x + i/a + x &., continued inceft nitely. because they require only the performance of the common algebraic opere atixn ofU, andditio,, ultipli 2tio, divisin, iig exponent, or in connection with logarithms, sines, cosines, ta1 gents, circular arcs, &C., of which the following are examples: aX, x, y dX -k4na+i4x+ The functions considered hit, (log. These are called lgebraic fuctions because they require only the performance of the common algebraic operansce of adition, suraction, and they will biplicon, divisiere in the next of powers, and etracion f roo Thee is a seco nd very extensive class of functions, in which the variable enters as a-In exponent, or in connection with logarithms, sines, cosines, tangents, ciircular arcs, &c., of which the: following are examples: axic, x log x, sin X, (COS X)sinx, sin-1x, (log X)tanx, &c. These are called transcendental functions, and they will be considered in the next chapter, CHtAPTER 111. TRANSCENDENTAL FUNCTIONS. 12. Prop. To differentiate u = log x. Let x take the increment hl, converting u into l - = log (x + A). Then ul =log ( x + ) log x ( + log x -x log ( ). Ih h0 /3 / or U1= U + - + - - &c. ) X\ 2x2 3x3 44 / where M is the modulus of the system. dt d(log x) M M dx d x x x Hence the rule is as follows: Multiply the cifferential of the variable by the modtulus of the system iln which the logarithm is taken, and divide the product by the variable. If the logarithms belong to the Naperian system whose modulus is equal to unity, we shall have dx d(log x) -. As the essential properties of logarithms are the same in all systems, while the form of the differential is simplest in the Naperian system, the logarithms employed throughout the Calculus will 32 DIFFERENTIAL CALCULUS. always be the Naperian, unless the contrary is distinctly specified, and the rule for differentiating a logarithm will be simply this: Divide the differential of the quantity by the quantity itself. 13. Prop. To differentiate an exponential function as u -- a, the base a being constant. Passing to logarithms we have log u = — log a. ds. d(log u) - d(x log a) or = log a. dx; uu. du =l log a. u. dx = log a. a. ldx and - l(g a. a. And the rule for differentiating an exponential is this: M:ltidply the erxponen.tial (ax) by the di./ 1rential cf the exponrent (dx), and that p'rodtuct by te J eia lo it the hei an lo i of e base (log a). Cor. If ac- e, the Naperian base, we shall have log e = 1;.. d(ex) = exx, and d(e) e. dx Remarkc. The rule for difierentiating logarithmic functions will often be found useful, evenl when the original finction is algebraic, since by passing to logarithms we may give the function a simpler form. Excanmpes cf Logarithmic and Exponential Function8. 14. 1. Let u = log (X +/1 +- ). d{(x ci-~ +2) + (1 + ) -du' - dx + + 1 + X i + 2T 1 s _ + + ___ dx da. 1 ( -.VI- T —-- - d =+.. _ (X + _VI + 2) Vj' + X2 V11 -~- X dx V-2/ F97 TRANSCENDENTAL FUNCTIONS. 33 2. u= x(a + 22)/a2 — x. Passing to logarithms we have log u = log x + log (a2 + x2) + log (2 - x2). d dx d (a2 + 2) I d(a2 _- 2) ~ -- - - x a2 +x +2 a2 2 - dx 2xdx xdx x a2 + 2 a2 - 2 a + a^ d- (a2 + X2) -2 2 2 + 2 x2(a2 +X2) a4 + a2x2 - 4x4 /a2 a- X2 x2a1 ~- x 3. u - log 2 - x V2A- 1 + x Multiplying numerator and denominator by the numerator we have u = log 2x 1-2x log (22 + 1 - 2Iz=l- + i2 = log212 2- 1X -- - du 4x -2-2 2-(2 T 2 -21) - 2 dx 2x2 + 1 -2x - @ + 4. X = a/V. 1 Then logu = a -- log x. 2b X dud du x x'X 1, and dut = a.. aJIi = dxx dx. Thus the rule for differentiating a power is still the same, when the exponent is imaginary 3 34 DIFFERENTIAL CALCULUS. 5. u = x. Then log u = log x. du dx. = log x. dx +x.-= (log x.+ 1)dx x du,. = xz(log x + 1). dx 6. U = Zxx This signifies that x is raised to a power whose exponent is x, and it must not be confounded withl (X)x, which latter implies that xx is raised to the xth power. du du dz Then log u = x log x.. - log x(log x D- 1)x dx + x _1 7. a = eX where e is the Naperian base. du x log u = x log e = xx. *. = ex xx (log + 1). 8. u = X x. Then log = ex log x c x/ 1\ " dx = xe logx t - ex. 9. = log (nx). Then du = d( ) _. nx X This result is the same as when u = log x, as might have been anticipated, since log (nx) = log n + log x, and log n is constant. 10, = log (logx). Then du= d (log) dx logx ggx.logx du 1 dx x log TRANSCENDENTAL FUNCTIONS. 35 11. u = (log x)n = log nx. Then du = z log n-~ x. d(log x) du _. log n-1 x dx x 12. u =e lg a +x. Then log u = log a + x2 U = /a - - (a2 + x2) and dx.V/2 x2 log' z 13. u = e du -- e. d (logZx) du n log. ~ -_ n. e logn-lx. dx x 14. u = x4 Iog2x - -4 log x - - x 3d = log2x + x3 log x -3 log x - x3 -+ x-3 3 - log2x. dx 2 2i 8 8 15. u - e7(x4 - 43 + 12x - 24x + 24) du d — e (x - 4x3 12x2 x- 24x - 24) + e2(4z3 - 122 + 24x - 24) = ex. x. Trigonnometrical Functions. 15. The trigonometrical functions sin x, cos x, tan x, &c. will next be considered, but the determination of the forms of their differen.. tials will be facilitated by the following arc arc arc Prop. The limit to the ratios 7 -- and ~, when the sin chord tang arc is diminished indefinitely, is unity. sin cos rad-versin versin Proof Since -ta-. radi = _ - _,ad "t3an razdliuJs rad rad 36 DIFFERENTIAL CALCULUS. and since the last term in this equality can be rendered smaller than any assignable quantity by taking the arc sufficiently small, it follows that the limit to the ratio n is unity. tan But both the chord AB and the arc AB c i A are intermediate in value between the sine BD and the tangent AT. Hence at the limit, when the arc is indefinitely small, arc are are sin sin chord tan -tal 16. Prop. To differentiate y = sin x. In the well known trigonometrical formula, sin a -sin sin - (a -2 ) cos (a + b), make a - x + h and b - x. Then (a - b), and (a + b) x +.. *. sin ( ) Sill x =Sill 2. cos ( h) sin (x + h) — sin x 2 sin h. cos (x + ).. h h sin - h 2 1 2= ~ cos ( + ^). 12' But at the limit when h = 0, sin - h 1I, and cos (x + h) =cosx. -h 2 TRANSCENDENTAL FUNCTIONS. 37 dy d(sin x) -Y d - =x cos, and d(sin x) = cos x. dx. dx dx 17. Prop. To differentiate y = cos x. Here y = cosx = sin (~ - ^) where 7r -= semi-circumference of the circle whose radius 1... dy = dsirQ ~ - ) cos(2 - x).( ( - - -sinxdx dy d cos x dy dC =- sin x; dx dx the negative sign prefixed to the value of this ratio signifies that the cosine.decreases as the arc increases. 18. Prop. To differentiate u = tan x. ix osx. dsin s x -sin x. d cos x duz = d(tan x) = d = cos X Cos2X cos2x + sin2x dx = 2 ~ ~ =dx -= - sec2x. dxo CXS2X COS2X du d tan x e.dx = dx _ =sec2x. dx dx 19. Prop. To differentiate u = cotx. dU = d(cot ) =dtan(j-x) = sec2 1 x. d (- r- x) - cosec2x. dx du d cot x'' = ~- dx — d- cosec2x. d~x dx 20. Prop. To differentiate u = sec x. Tere sc 1 1 - cos x sin x, d.e Hlere u -- secx = -- *.. du = d = - 2 - - cos x cos x cosx cos2Z du d s(c x or, du = tan x. sec x. dx and... =. = tan x. secx. dx dx 38 DIFFERENTIAL CALCULUS. 21. Prop. To differentiate u = cosecx. du = d (cosec x) = d sec 7 ~ -x \2 / = tan( - - x) sec( - - x)d(2- x) - cot x. cosec xdx. du d cosec x x *. = dx - cot x. cosec x. 0 dx aIx 22. Prop. To differentiate u = versin x. du = d(versin x) = d( - cos x) = sin xdx. du d versin x * x = -.sin X. dx dx 23. Prop. To differentiate u = coversin x. du -= d(coversinx) = d. versin( 2 r- Sx)= silln - x11(d -- du d coversin x = -cos.. X. = - - - cos. dx dx 24. In each of these expressions, x represents the length of an are described with a radius equal to unity, and the radius does not appear in the formulae: but it is necessary to remember that, in each case% R- 1 must be understood to enter into the formula as often as may be required to make the two members of the equation homogeneous. Geometrical Illustration. 25. The results just obtained may be illustrated geometrically in such a manner as to convey a more precise view of the comparative small changes imparted to the several trigonometrical functions, by an arbitrary small change in the arc upon which they depend. TRANSCENDENTAL FUNCTIONS. 39 Thus let ab represent an arc x described with rad = 1, and bb1 = dx a small in- r crement given to x. Then eb - sin x, ce = cos x, at = tan x, ct = sec x, sbl = d. sin x, sb = d. cosx, tt, = d. tan x, rt = d. sec x. Also when bb6 is diminished continually, the small figures bsWb and trt1 will continu- c'e a ally approach to the forms of right angled triangles, becoming in definitely near to such forms at the limit. Moreover, the two small triangles will then be similar to cbe. Hence we shall have the proportions cb: ce:: bb,: brs or 1: cosx:: dx: dsinx=cosxdx. cb: eb:: bb: bs or I: sinx:: dx: dcosx= sin xdx. The latter result should be written d cos x = sil x. dx, because the cosine diminishes as the arc increases. Again we have the proportions ca: ct::'t: tt ).' ca X cb: (ct)2:: bb: tt1 and cb: ct:: bb: rt ) or 12: sec2:: dx: dtan x.. d tan x - sec2xdx. Also ca: at::rt: rt1'. ca X cb: at X ct:: bb: rt. cb: ct: bb rt or 12: tanx.secx:: dx: secx.. dsec x = tan x. sec x. dx. In the same mannelr, expressions for d cotx, dcosecx, &c., could be obtained. Circular Functions. 26. We will now consider the circular functions, sin-lx, tan-lx, &c., which expressions are read, the arc whose sine is x, the arc whose tangent is x, &c. 40 DIFFERENTIAL CALCULUS. In these cases, it is the arc which is the function, or dependent variable, the independent variable being the sine, or the tangent, &c. 27. Prop. To differentiate y = sin-Ix. Since this notation is intended to imply that y is the arc whose sine is equal to x, we must have as an equivalent relation x = sin y. dx = cos y. dy and = = dx cos y /1 -- sin2y d sin-lx 1 dx / 2 28. Prop. To differentiate y cos-lx. Here x = cos y,.. dx =- sin y. cly dy _ 1 dx sin y / -Cosy 1_ 2 d cos-1x dx x2 29. Prop. To differentiate u- tan-x. = tan u,.. dx - see2u. du du _ 1 I dx sec2u 1 + tan2u 1 + x2 d tan-x 1 dx 1 + x2 30. Prop. To differentiate u = cot-lx. x = cot u,.'. dx --- csec2. du du 1 I 1 dx cosec2u 1 +cot2u I -+ x2 d cot-r 1 dx 1 + 2 TRANSCENDENTAL FUNCTIONS. 41 31. Prop. To differentiate u = sec-lx. x = sec u,.. dx = tan u. sec u. du du 1 I *dx tan u. sec sec s 2e2T ~ I- x-2x d sec-lx *' dx x - dzx x2 —1 32. Prop. To differentiate u = cosec-lx. x = cosec,.. dx =- cot u. cosec u. du du 1 dx cot u. cosec u cosec Vcoseu - 1 1 x x — 1 d cosec-x 1 d - dx x233. Prop. To differentiate u = versin-lx. z = versin.. dx = sin u. du = -/ 2 versin u - velsin2u du du 1 1 dx - 2 versin u - vers inl2 2x x d versin-x 1 or, dz - 34. Prop. To differentiate u = coversin-lx. x - coversin u dx = - cos u. du = - /2 coversin u - coversin2u d. du 1 1 dx coverlsin ~-/ coversin l2 - -x2 d coversin-lx 1 or, dx ^- v/: x2 42 DIFFERENTIAL CALCULUS. 35. The differentiation of trigonometrical and circular functions will now be illustrated by examples. EXAMPLES. 1. u = 3 sin4x. du = 3 X 4 sin3x. d sin x = 12 sin3. cos x. dx du dx = 12 sin3. cos x. 2. u = COS nx. du = - sin nx. d(nx) = - n sin nx. dx du /x. — n n sin nx. dx 3. u = ta=-n1nx. du _= n tanlll1 nx. d tan nx = n2 tan11-1,X. see2nx. dx dx. =?2 ~ -,.n —' x'. sec2nX. 4. u = sin 3x.'. cos 2x. du = (3 cos 3.. cos 2x -" 2 sin 3x. sin 2x)dv. -= d 3 cos 3x. cos 2x - 2 sin 3x. sin 2, -- cos 3x cos 2x -- 2 cos 5. dx 5. u = (sin z)". Then log u = x.l(-lo (sin x) du du =[log(sinx)+ x cotx]dx. u- = (sin x)x. [log (sin x)+ xcotx]. 6. u = (cos x)sin. Then log u = sin log (cos x) ~ d (cos x)sin'x [cos x log (cos x) -- Sil x tan. x]. 7. u = sin (cos x), du = cos (cos x)d cos x..'d ~= - sin x.cos (cos x). dx TRANSCENDENTAL FUNCTIONS. 43 xd 8. U = sill-, _ I + X2 9. U = log tan x. d ) (1 4- - +2)du sc2x 1- x dx tan x - sin x. cos x sin Qx 1 - i+ x2,1 -) — ++ X) d1 - * * lx I + 2 ~~9. 6^ ~ == log taln x. du sec^ 1 2 dx tan x sin. a. cos x sin. 2x 10. u l = lo \ = (1 + sin x) -- og (1 - sin X c/d I 1 cos x cos x 1 cos X dz 2 1 + sin x + - sin x I- sin2x cos x 11. uz = sin-l (3x - 4x3). du 3 - 12 __2 3 d.- ~ (3x 4x3)2 V - 1 12. u = log (cos x + v/-. sin x). du __ --. cos x - sin x dx cos x + V/- I. sin x 13.. IC = 1 b +a. cos x' 13. 1= - ~ *===: COS I ~1' ^/a2_ b2 a + b. COS X (b + a.cos \ 1= a + b. cos x) du =- _*2- /~/ - a. cos x\2 V \a- + b cos xJ a sinx (a + b cos x) - b sil (b + a cos x) d (a2 - b2) (a + b cos x) [((t + b cos x)2 - (b +a cos X)2] 2 44 DIFFERENTIAL CALCULUS. di (a2 - b2) sin x (2 - 2)2(a + b cos x)[(a2 - 2)(1 - os2x)o)] 1 a + b cos x 14. u = ex cos x. du d= e cos x - e sinx = ex (cos x -sin x). dx 15. u = tan-l ( l + X- x). du (1 +x2) ix- dx 1 + (V1 + - X)2 2 (1 + 2) 16. = log /sin x + log cos du 1 /cos sin x\ 1 dx 2 \ sin x cos x- tan 2x 17. u= log -- - + tan- x. 1 I x 2 = log (1 + x)- o (1 - x) + -tan- z. du 1' 1 1 dx 4(1 + x) 4(1-x) 2(1 + x2) 1-x4 ~18. _~ ~eax (a sin x - cos x) 18. u 2 ~ 9 ~ aCd- 1 du 1 dx= a. +' [aeax (a sin x - cos x) + aeaC cos x + ea sin x] dx ea sin X= eax sin a. CHAPTER IV. SUCCESSIVE DIFFERENTIATION. 36. When we differentiate a function u = Fx, the differential coefficient will usually be itself a function of x, and will therefore admit of being differentiated. This will simply be equivalent to examiinilg the comparative rates of increase of the independent du variable x and the variable ratio'* This differentiation will give dx rise to a second differential coefficient, which may also be a function of x, and this, in its turn, being differentiated will give a third differential coefficient, &c. 37. To illustrate this subject, let u = x3 be the proposed function. The first differential coefficient, duz = - 3x2, dx d du\ - d -6x dx third differential coefficient, ddu dx d dx dx As the third differential coefficient in this example proves con 46 DIFFERENTIAL CALCULUS. stant, the fourth and all succeeding differential coefficients will be equal to zero. 38. The preceding notation of successive differential coefficients beiln inconvenient, it is replaced by the following: ddu dx d2t For -' we write d2; dx dx2' dd for dx Cfor dd. we write d., ( " dx the symbols d2, d3, &c., indicating the repetition of the process of differentiation twice, thrice, &c., and not the formation of a power. On the contrary, the expressions dx2, dx3, &c., represent powers of dx. The second differential coefficient d2 may be obtained dx2 da immediately from the first differential coefficient d by differentiating this iatter as though dx was constant, (thus producing d ) and then dividing the result by dx. Now since the law according to which the independent variable x changes, in different stages of its magnitude, is entirely arbitrary, we adopt, as most simple, that law by which the successive increments of x are supposed equal; that is, we make dx constant. The same supposition will enable us to derive each successive differential coefficient front the preceding coefficient by a similar process of differentiation and division. EXAMPLES. 39. 1. u == x". = nx 1 n2 - 1)xn-2 dx dx2 n(~2d3it 1 d4u d, n(n - 1)(n - 0)X-3, dx4 n(" -- )( - 2)(, -- 3).-n &e. SUCCESSIVE DIFFERENTIATION. 47 This operation will terminate when,b is a positive integer; but if n be a negative integer or a fraction, the number of variable differential coefficients will be unlimited. cdy I d2y 1 d3y 1.2 2. Y-logx dx=X dx2 - x2 dx3 3' d4y 1.2.3 day 1.2.'....(1 - 1) _~ - ~and by analogy -~. -1) dx4 X24 dXn x' the upper sign will apply when n is odd, and the lower when n is even. 3,. u = sin x. du d2u d3d.t d4u d- = cOS x, - sin x, ~~ - ~ cos x, d = sin x, dx dx - dx3 dx4 and the succeeding differential coefficients will rectir in the same order. 4. y = cos x. dy d2y d3y d4y d =-sin x, d =- cos x, d3 -= sil x, ~d4 = cos x, dx dx2 dx3 dx4 and the coefficients will now recur in the same order. 5. ut = tan x. du d2u d3u d =sec2x', -- 2 sec2x. tan x, d3 = 4 sec2x tan2x - 2 sec4., &c. Here the law of formation of the successive coefficients is not obvious. 6. u =- a. du d2u1 d3u - = ax. log a, d- a. log2a, -- a-"x log3a, &c., t dx of te c b vry ed the law of the coefficients being very evident. 48 DIFFERENTIAL CALCULUS. 7. = ex. du d2u d3u dx = ex,~ d-= e d3 the coefficients being all equal. 8. u = sin (nx). du d2u, = n cos(nx), ~ — (~nsnx), &c. dx' - ~ ~O) dx in(n), &. The formation of successive differential coefficients will be found extremely useful in the expansion of functions by the methods which will be explained in the chapters immediately succeeding. CHAPTER V, MACLAURIN'S THEOREM. 40. The theory of Maclaurin is a very general and useful formula for the development or expansion of a function of a single variable, in a series involving the positive ascending powers of that variable, when such development is possible. 41. Prop. If y = Fx, where Fx denotes such a function of x as can be expanded in a series containing the positive ascending powers of x, then will the form of the development be the following: (Y) + \I)x + \d\ x + \ ( (.-2 + &2" in which the parentheses are used to denote the particular values of the quantities v, -dy- dy &c., enclosed therein, when x is taken equal to zero. Proof. By hypothesis, y can be expressed in the form y = A + Bx + Cx2 + Dx3 + Ex4 + &c., (1). in which A, B, C, &c., are unknown constants. d.. = B +2Cx - 3Dx2 + 4Eza3 + &c. d: 2 C - 2. 3Dx + 3. 4Ex2 + &c. vd3 2.3D 2.3.4Ex &c. dx3 2.3.4E + &c. &c. &c. 4 50 DIFFERENTIAL CALCULUS. Now making x = 0 in each of these expressions, we obtain (d 4)-2 (1)4 (&)% (Y) = A y B, (d )C - 2c,!) 2. 3D d^ -) 2.3. 4-, &c., &c..A = (y), _B -, c - \ dX / I. \ d1"0 I1 (d3Y 1 2 4 (y\ &c.,&c. - =.'2. a \ E x'4 C These values, being substituted in (1), reduce it to the form Y (Y ) 1 (dd +. 2 d31.\. 2dx. 23 ~d\-z) 1. 23. 4 + &c., (2). which agrees with the enunciation. This formula, called Maclaurin's Theorem, may be written thus dFx\ x (d2Fx\ x2 (d3Fx\ x3 Fx -= (F)x) + d- dx2 + ~ -. ) _! dx{1 4x.F2. 2l 1.2.4 + -?,xg-\ 1.2.3.4 - &c., (3); /x4 1 2.3\ 4 or again, if we represent the 1st, 2d, 3d, &c., differential coefficients, which are functions of x, by FPx, Fx, F3x, &c., the formula may be written X X2 X3 Fx = FO + F10 + F20 + F3 1.2 1.2.3 x4 + F41 2 - &c. (4). EXAMPLES. 42. 1. To expand y = (a - x)n. MACLAURIN'S THEOREM. ol Here n(a + -, dX2 = n(n-1) - (a + ) n-2, dx3 -- dY _n(n- )(n -2)(a + x)n-3, dY = n( - ) (-)( - 3) (a + X)n-4, &C. &C. Hence, when x = 0. (Y) a,n (d) nan-l, = n(n -) 2 (d3 _ n3 ) __ ( -I n 1)(_n n - 2)a -3 (d4y) - ( - 1) (n - 2) (- 3)a-, &c., &c. And, therefore, by substitution in Maclaurin's formula, y = (a + x)" = a + na"-x + n(n 1) an-2x2 1. 2 + n(n - ) (n - 2) n3x3 t.2.3 + (- 1)(- 2) (- 3) a -x4 + &C. 1.2.3.4 Thus we have a simple proof of the binomial theorem, applicable to all values of the exponent, whether positive or negative, integral or fractional, real or imaginary. 2. To develop y = log (1 + x), the modulus of the system being AM, dy M d d2y Mf d3y 1.2J1 dx I+x dX2 (1 + x)' dx3 -(1I - )3 d4y. 2. 83M dx4 =- (1 + X)4- &e. 52 DIFFERENTIAL CALCULUS..when x = O (y) = log1 o 0, (dy\ M!d21/\ M {(d3\ 1.2M (d4y\ 1. 2.311f {d/ 1 Td / -'. I.x1 -~1 ~' dx And by substituting these values in Maclaurin's frmula, we have 1 2 1 3 +- &c.) Y = log (1 ~ x) M(r - -X2 ~ -x3 X4 ~ &e.) 2 3 4 which is the fundamental theorem used in the computation of loga rithnms, and is, indeed, that which was employed in deducing the rule for differentiating logarithms. 3. To expand y = sin x. Here Fx = sin x.. F1x = cos x, F2x =-sin, F3 =- cos x, 4x = sin x, and the succeeding coefficients recur in the same order. O. F = sin 0 = 0, f10 cos 0 = 1, F20=-0, F30=-1, 40 = 0, 50 = I, &c.. by substitution in (4) the third form of M/aclaurin's theorem, we have *3 X5 X7 sin x = x - -- - ~- &c. 1.2. 3 1.2.3.4.5 1.2.3.4.5.6.7 This series converges very rapidly when x is small. 4. To expand y cos x. Fx -- cos, Fx — - in x, F - cos x, _Fx =- sin x, FF4x = cos x, and the succeeding coefficients recur in the same order, Fo = 1, FO = 0, F20 -1, F30 =, 40 =,, F F0 = 0, &c. X12 X4 a;6'. cosx=- IX1 - ~ ~- -.- &c. I,2 1.2.3.4 1.2.3.4.5.6 MACLAURIN'S THEOREM. 53 5. To develop y = ax. Employing Naperian logarithms, we have Fx = a,, Flx = a.lo.log a, Px = a.lox = a. log3a, &c.. F0 = 1, F10 = log a, 20 = log2a, 30 = log3a, F40 log4a, &c. X X2 X3..ax = + log a -- + loga +- log 3a 1.23 X4 + log14a +- &c This is called the exponential theorem. Cor. If a e the Naperian base, then log a = log e = 1, X x2 X3 X'~ ~-14- - 4- 4- 4 &TO r 1 1.2 1.2.3 1.2.3.4+& and if x 1= also, ex=e ~ + - 1 e 8 2 2 8 4 &c" 1 1.2 1.2.3+ 1.2.3. a formula for the Naperian base. Cor. If x = 1, but a not equal e, then ax = 1 + log ax + log2a 12 lo log g4a + &c. 1.21.23 1.2.3.4 a formula for a numbei in terms of its Naperian logarithm. Prop. To express the sine and cosine of an are in terms of imaginary exponentials. In the series giving the value of ep, put successively z -1, and - z~-l for x..ez l z 1.2 +- /. 4 1. 2 1.2.3.4 1z5 ~i. 2. 3. 4. — &c. 54 DIFFERENTIAL CALCULUS. z -1T i' ^ z2 z3 -1 and eI-Zv- - - i-1 +1 3 z4 z5 -- +13- -- &c. 1.2.3.4 1.2.3.4.5 *. ezV/-+e-z/2[ — 2[ - -j 1.2.4 — &* C ] But the first series within the [ ] is the development of cos z, nid e/z+' + e-V ez — i- e-z2-f COS Z = e 1 sin -, = (B), 2~-1' These singular formulae, discovered by Euler, are very useful in the higher branches of analysis, especially in the development of functions. Cor. If we divide (B) by (A), there will result ez/- - e-z1 —i e2~zT —' tan z (C). e- J - iez + ez/l-] - 1[e2z /=^ + 1] Cor. If we make z = x — in (A), (B), and (C), we can express the sine, cosine, and tangent of an imaginary arc in terms of real exponentials; thus:.__ e-x e- ex + ex sin ({x/ = --.. (D), cos (xV- = -)-.. (I) e-2x_ 1 — e2x tan (x2/- 1) - = ~ — = )' ^(e-2 + 1) y- 1(1 + e2) MACLAURIN'S THEOREM. 55 Cor. If we square (A) and (B) and add, there will result e2V/- + 2 + e —2z -- ee2~-/ 2 - e-2z/-f cos2z + sin2z =- -= 1. And similarly sin2(x -- 1) + cos2(xV/ —) 1; two results obviously correct. 43. The applications of Maclaurin's t;heorem are often much restricted by the great labor necessary in forming the successive differential coefficients. This may sometimes be avoided by expanding the first differential coefficient by some of the algebraic processes. For example, To expand t = tan-ix. di 1 Here d dx 1 - X2' which gives by actual division, the quotient I - x2 + 4 - x6 + x - &c... Fx = tan-lx, F1x = 1 X2 + x4 - x6 + S - &c.'2x = - 2x +- 4X3 - 65 + SX7 - &c. F3x =- 2 - 3. 4z2 - 5. 64 + 7. 6 - &c. F4 = 2.3.4x - 4.5. 63 + 6.7.8x5 -&c. 15= 2. - 3.43.4.5.6x2 + 5.6.7.8x4-&c. 6x -2.3.4. 5.6x + 4. 5.6.7.8X3 -&c. F_7x -2.3.4.5.6 + 3. 4.5.6. 7. 82 -&c. FX- 2. 3. 4. 5. 6.7. 8 - &c. &c., &c...FO = tan-O = 0, F0 = 1, F20= 0, s0 --- 1.2, F40 0, F0 = 1.2.3.4, F60O-0, F70 -1.2.3.4.5.6, 80 = 0, &c. 56 DIFFERENTIAL CALCULUS. Therefore, by substitution in. Maclaurin's formula, Fx = tan-lx = x - -3 + -x - -x7 + -] - &c. 3 o 47 99 If, in this formula, we make u = - == arc of 45~, 4 then x = tan 45~ =1. *' * -=(1-^3 5 7 and. = 4(1 — F- -+ &c.); a formula for determining the ratio of the diameter to the cir. cumference of a circle. This series converges so very slowly, that even a tolerably accurate approximation to the value of if cannot be deduced from it, without employing a great number of terms. 44. Prop. To deduce Euler's more convergent series for the ratio of the diameter to the circumference. If in the trigonometrical formula tan a +- tan b tan + b)- = -) I - tan a. tan b we put a + b _ — ) then tan (a + b) = 1, 1. - tan a.tanb = tana + tanb; I - tan a whence we deduce tan b ta I +- tan a And, therefore, if any value be assigned to tan a, that of tan b can be determined. 1-' 1-~ Let tan a then tan b = - - _' 2 1 I 1'. ta - = - tan — t 4 2 3 MACLAURIN7S THEOREM. 57 But tan-1=-44 - H-1 3 and tan-l 3 - - -- + &c. 3 3 -3'5 \ 7\3/ 1 1 1 1 1 tiplying by 4, we get the common approximation, 4 = 3. 14 16. C(or. We might extend this method, obtaining series still more convergent. For if we take four arcs cl, c2, c3, and c4, such that 1 1 1& c1+a' 5.35 a.37 c + c = tan — and e3+c4 =tan-1-. Then c+C+C+ C4 -7 and if we assume the values of tan cl and tan c3, those of tan c2 atnd y tacing sixbe termsin the first set, d four in t he secon, and multan c3, and tan c4, can all be rendered less than 3, and therefore the series ory deteri4, ig t will be more convergent. 4-3.1416. C45. Pe ight extend this convergent series for the value of re 2 tan a inv ergent. For if wemula tae four as, c, such that cx 4. %,= -tan —' - and c3 4 c4 -- tan-' 3. Then c.-. 1 1 and if we assume the values of tan wc and tan. c, those of tan cp and tan c4 can be determined. Moreover, the values of tan el, tan %,2 ta c and ta ca l be redered less than and therefore the series foi' determining ~ 4 will be more convergent. 45. Prop. To obtain more convergent series for the value of 4,r. 2 tan2a 2 If in the formul tan 2a = -ta I - tal2a' 22 and.'tan4a= - ~ -, h - 1-tan22a 25 119 144 58 DIFFERENTIAL CALCULUS. Now this result is very little greater than unity, and therefore 4a must be slightly greater than 45~. Put 4a - I r = z 4 where z is a very small arc. tan 4a - tan - r 4 Then tan = tan (4a - _4) =- - 1 + tan 4a. tanl- r 4 120 119 1 - _120 239 I +"119 1 I 1 -. 4 tan-l — tan-' 3.53 3 5.55 7.57 9.59 & ) +. _51 _ 1 _ \ 239 3. 2L 5. 2395 /3 By taking three terms in the first line and one in the second, we get the common approximation ir =. 1416; and by taking eight terms of the first line and three of the second, we get 8 = 3. 141592653589793. 46. 1. To expand u sinl x. Ex = sin-1 x. 1 -i Fx = = (1- X2) 2 1 1.3 1.3.5 -1 x2 + - 4 + 1 -x6 + &c. 1.~2 1.2.2 1.2. 3.23 1.2 1.3.4 1.3.5.6 - 2x 2 + 1 4 X3 + 7 X5 + &C. 1 2 2~ 17.2. 22 + 1.2 3.23 la.2 1.32. 4x 1.3 52. 64 X31' 1= -1.- X.. X-&. 1.2.2.22 1.2.3. 2' MACLAURIN'S THEOREM. 59 1 2.2.4 1.3.4.52. 6 4X!.2.22x- 1. X3 ~ &c. 12. 2.32.4 1 32. 4. 52. 6 -F5z = x2 + C. I. 12. 22 + 1.2.32 4.52.6 iFrGx -F &c. 6 1.2.. 3 &c. 12.2.32.4.5E2 6 17. 2. = 3.23 —- &c. Fo = 0, FO 1, ~O = 0, -30 = 12, O = 0, F5O = 12.32, F'60 =, F70 = 12. 32 52, &c. 12. 3 12.32. 5 eI I1 = Sll-= x- + 3 1.2.3 1.2.3.4.5 12.32. 52.X 1.2..4-..' 1.2.3.4.5.6.7 CHAPTER VI. TAYLOR'S THEOREM. 47. Taylor's Theorem is a general formula for the development of a function of the algebraic sum of two variables. Prop. If y = Fx, and if x be supposed to receive an increment h, converting y into y, =-F(x + h); then will dy h d2y 2 d3y h3 d4y It4 = Y + —+-. ~ + + - d x 1 I dx2 1. 2 d' 1.2.3 d x4 1.2.3.4 + dFx h d2Fx h2 or F(x + h) =Fx -t d-' +. d2 d3Fx h3 d'Fx h4 dx3 1.2.3'dx4 1.2.3.4 To prove the truth of this formula, we first establish the following principle: If in the expression y _ = F(x + h) we suppose first that x is variable and h constant, and then suppose h variable and x constant, the first differential coefficient will be the same in both cases; dy__ dy. that is, d d dx dh This is almost self-evident, for when a given increment is assigned to x, or to h the same increment must be imparted to x + h, and therefore F(x + h) - yi will undergo the same change in the one case as in the other. Hence the ratio of the corresponding change.s of x and yl is equal to the iratio of the changes in h and yl. This is true whatever lnay be the magnitudes of the increments im TAYLOR'S THEOREM. 61 parted to x or h, provided that magnitude be the same in both cases. But when we suppose these increments indefinitely small, it is no longer necessary to consider them equal. For since the ratio dy1 -y does not contain dx, it will have the same value whether dx dx and dh be supposed equal or unequal..d:l dy1 dz dh d(y ~^), d]- d2y d2y,( Similarly, 9 d or dy d2y~ dx dx dx dx2 dh2 dAl.yl d-yl And generally, dny d^y dx"- dhWi Now a ssume Y1 = F(x + h) F — + + Ah + Bh2 + Ch3 + )h4 + &c. (1), that being the general form in which F(x + h) can be developed, as shown in Art. 4. The coefficients A, BP C, DZ &c., are finctions of x, but are independent of hI If we differentiate (1) first with respect to h and then wMith respect to x, and place the resulting differential coefficients equal, we shall obtain A + 2Bh + 3 Chi2 - 4Dh3 + &c. dix d dA I dB 12 + dC 3 + &c. dFx dA dB dC which equation being true for all values of h, it follows, by the principle of indeterminate coefficients, that the coefficients of the like powers of h, in the two members of the equation, must be sepa. rately equal. dFx dA, dB d C A A ~-d, 2B = d, 3C — 7, 4D —, &c. dx dx dx dx 62 DIFFERENTIAL CALCULUS. dFx 1 dA 1 d2F..'. A= dx' B 2dx =12- dx - 1 dB 1 d3Fx 1 dC 1 d4Fx 3 dx 1. 2.3 dx3 4d dx 1.2.3.4 dx'' Hence, by substitution in (1), dFx h d2Fx h2 d3Fx 13 F( (x + d) = x + + d -.7 + d-3 -o + dx4 1.2.3.4 + -y dx dx2 1.2 dx3 1.2.3 q~ --- ~ q —-- $g_+& c., dx4 1. 3.4 If we denote the suceessive differential coefficients by F1z, F^x, F3x, F4x, &c., the series may be written Fx~ F h d2F~ h2 d3 F l, 3 or,+ - =F + + F1 F x 1.2 dXF3 1 2.3 ~d4 Y ~4 Cor. The formula of Maclaurin may be readily deduced from that of Taylor; for if we make x = 0 in (2), there will result F(h - F F+ 0 0- l +O0 2 + 30 i h4 + 4o I. 2.3.4 + &C +^r ^1.2.4+ 3.4 which is Macla urin's theorem. EXAMPLES. 48. 1. To expand sin ( m x h), in terms of the powers of the arc h. F(= + ) - - si(.2 + ~), F (.t + h,) = sin (xe + h7), TAYLOR'S THEOREM. 63 Fx -= sin x? Fx cos x, ^ os 2 = sin x, F3x - cos x, F4 = sin x^ &c.. By substitution in Taylor's formula h h.2 h3 sin (x + hl) = sill + cos x - sin x -2 -cos x -~ - + &c. h2 h 14 = ill x (1 - 2 +. __ - + Cos X (.t2 + 2&C.) +-osx(h-~3 1. 2.3.4.5&) = sin x. cos h +- cos x. sin h, a well known formula. 2. To expand cos (x -+ A), in terms of the powers of the arc h. F(x +h) Gcos (x + h),. x. = cos, F1x = - sin x, x - - cos x, F3x = sin x, F4x -- cosx, &c.. By substitution in Taylor's Theorem we have 1 12 1. 2.3 + Cos x - &c. 1 2.3.4 ]h2 h4 = cos x (1- + 1 - &c.,) 1. 2 1.2~~. 3.4 h3 h5 -sin x (h1- - + &c., 1.2.3 1.2.3.4.5- cos x. cos A - sin x. sin h,... a well known formula. 3. To expand log (x + -h), where M is the modulus of the system. Fx = logx, F = -- F2x - —, 1. 2 1.2M 1.2.32 F3x T F - ~ &C, X3 4 X 4 6 DDIFFERENTIAL CALCULUS. Ilrl2 h3 It log (x + h) = log x + - -2 + 3 &.) 4. To expand u1 = tan-(x +- h), u = tan-1x = Fx, Fx = 1 = = cos2u. 1 -2; sec2e F2x - - = 2sin t. cos u - = sin 2u. cos2. dx F3x = (- 2cos 2u. Cos2 + 2sin 2. cos u. sin u) d - 2cos u. cos 3u - 2cos 3u. cos3u. = 2. 3cos2u. sin 4m- 2 2. 3. sin 4u. cos4u, hxh2. tan- ( + h) =z + co s2u - sin 2m u. cos2u I - h3 h4 h5 3 4 5 -- cos 30. COSU?Z + sin 4u. cos4u T- + cos 5u. COS - &c. 5. To expand u = tan (x- + h). x =- tan x, F1x == sec2x, F2x = 2 sec2x. tan x, F3x =- 2 sec2x (1 + 3 tan2x). &c., &c. h h2..tan (x + h) = tan x + sec2 -- + 2 sec2. tan x 1h3 + 2 sec2x (1 + 3 tan2x) + &c. Prop. Having given u = Fy, and y = px, to form the du differential coefficient of it with respect to x, without eliminating y between the equations, in which the characters F and qp, denote any flnctions whatever. TAYLOR'S THEOREM. 65 Let x take an increment h converting y into Yi = p (x + h). Then if k denote the increment received by y, we shall have, by Taylor's theorem, dy h d2y h2 dy h k = y -- - + d 1h23 + &c (1). dx I dx2 1.2 dx3 1.2.o Also when y takes the increment k, it imparts to u = Fy, an increment du kI d2u k2 d3u 1c3. - = -F(y+ k)-Fy = d 4- ++ &c., dy 1 dy2 — 1l.2+ dy.. or by substituting for k its value, (1). du Fdy h d2y h2 d3y h3 U, - u -- L + + + &. U d L 1 dx2 1.2 dx3 1.2. 3 1 d2t rdy h d2y h2 12 + 11- +J+ 1.2 dy2 Lc x2 1 &c. ~&c. Dividing both members by h, and then passing to the limit by making h = 0, in which case 1 we get h - dx du du dy dx dy dx Thus it appears that the differential coefficient of u with respect to x, is found by differentiating u as though y were the independent variable, then differentiating y as though x were the independent variable, and finally, multiplying the first of the coeflicients so found by the second. 49. It might perhaps seem at first view that the equation (2) is necessarily and identically true, and therefore that the preceding investigation is unnecessary. But it must be borne in mind that the dy which appears in the coefficient -/ and which represents the increment given to y by assigning an arbitrary small increment dx to the variable x, is not necessarily the same as d.y which appears in 5 66 DIFFERENTIAL CALCULUS. du since this latter increment of y is arbitrary (though likewise small). du 1. u = aY, y =, to find dx Here d= ay. log a - = b log b. dy dx du du dy du du dcy ay. b. log a. log b. = ab". bx.log a.log b. 2. u = log y. y -logx. du I dy 1 du I1 1 dy y' dx x'''dx y x xlog x 50. Taylor's Theorem may be employed in approximating to the roots of numerical equations. Let Fx = 0 be the given equation, and a an approximate value of one of its roots found by trial; then we may put x = a + h, in which h is a small fraction whose higher powers will be small in comparison with h, and may therefore be neglected without great error. But ih h2 h3 Fx = F(a+h) = Fa + Fa. F + F2a + 3a2 + &c. 0. 1 12 3 1. 2.3 O.. By neglecting the terms involving h2, h3, &c., we get Fa~ +F- aa = 0 and..h= — 1 Fla Adding this approximate value of h to a, we have Fa x = a - a nearly. Call this value a1 and put x - a, + h1. Then by similar reasoning we shall find -Fa1 Fa1 hi = -Fa, and x = al -a =- a2, a nearer approximation, and the same process may be repeated if necessary. TAYLOR S THEOREM. 67 51. Find the positive root of the equation 4 - 12x2 + 12x- 3 = 0, to three places of decimals inclusive. Here we find by trial that x > 2.6 and x < 3. Put a -- 2.8.. a = a — 12a2 + 12a 3 (2.8)4-12 (2.8)2 + 12(2.8) - 3 2.0144 Fa da = 4a3 - 24a + 12 = 4(2.8)3-24(2. 8) +12' - 32. 608. 2. 0144. * A = — 2. 6 08. 062 nearly... x = a + h = 2.862 nearly. 32. 608 To test the accuracy of this approximation, put a1 = 2. 862 and x - a + hl. Fal (2.862)4- 12(2. 862)2+12(2. 862) —3-0. 144674 nearly. Fa, = 4(2. 862)3 - 24(2. 862) + 12 = 37. 083072 nearly. 0,.144674 h =- -- 0.003901 nearly.'' - 37. 083072 --'.'. x = a + h 2. 862 -0. 003901 = 2. 858099 =- 2.858 to three places of decimals. If the process were repeated it would be found that x = 2. 85808; so that the second approximation is true to four places of decimals, and the fifth place is slightly erroneous. 2. Given z = 100 to find the value of x to the place of hundredths. Passing to the common logarithms, we have x log x = log 100 = 2... xlogx 2 = 0. Also x > 3, and x < 4. 68 DIFFERENTIAL CALCULUS. Put a = 3.5, and x -- a +h...Fa= aloga- 2, Fla- d = log a + M, where 1! = modulus of the common system. 43429448 Fa -3.5loog (3.5) -2=.544068 X 3.5 —2 — 0. 095762 F1a. 544068 +. 434294 = 0. 978362.. 095762 ~ h -- - -.098. 978362. x = 3.5 +..098 = 3.598 or x = 3.60 nearly. We shall now apply Taylor's Theorem in deducing rules for the expansion and differentiation of functions of more complicated form.s. 52. Prop. To establish a general rule for differentiating any function of two quantities p and q, which quantities are themselves functions of the single independent variable x. Let u - FT(p. q), where p -./f, and gq =fx, the characters j/; and fi, denoting any function whatever, and let x take the increment A, converting p in p -+ k = pj, q into q - - = ql, and ~, into u.L Then U 1- F( + k, q + 1) F= F(p + k, q), which nmay be developed by Taylor's Theorem as a function of p -- c, observing that qj, which does not contain k, will appear in the development as would a constant: * 21 = F (P + J, q1) = - (p, qj) -+ d p k + d2F (P, q) k ( dp 2 1.2 But F'(p ql) - F (p, q + 1), which developed as a function of q 4- 1, gives TAYLOR'S THEOREM. 69 F(p, q + ) = F(p, q) + dF (p, q) I d2F (p, q) 12 dl dq2 1.2 & And similarly the coefficient of k in the second term of (1). dF/(p, q]) _ d'(p, q 4- 1) may also be developed as a function of q + 1, and will give IdF (p, q+ 1) dF(], q) + d [ (dF( )]l Q ) dp dp dq dp 1+ &C And in like manner d2F (p, q1) _ d2F(, q ~ 1) d2F (p, q) d ld2F(p, q)1 dp2 dp2 dp2 dq dp2'.. By substitution in (1). dF(p,q) I dF(p, q) k F(p + k, q + I) = F(p, ) + dq 1 + -- terms involving 2, k, 12, k3, &c. But cdp h d2p It% But k =2), -p T -+ - +&c., x 1 d ( 1. 2 dq h d2q h2 du [dq h d2q Ih2 ZG1 2^(/ L^~ hcE~ ~ I --- - I.2 +J dq dx 1 d2 1.2 4u[dp A~ * d12p *2 &c.,] + &c. Now dividing by h, and then passing to the limit, by making - = 0, in which case hu =- -, we obtain d dx ddq dx dp d - dp dx (2). du' du dq du dp.'. ~dx = du - - - d' + dx. (3). dx dq dx dp dx 70 DIFFERENTIAL CALCULUS. Thus it appears that we must differentiate u with respect to each function, as though the other functions were constant, and add the results. 53. It is very important that the precise signification of the notation here employed should be distinctly understood. By an attentive consideration of the manner in which the several expressions employed in the formulm (2) and (3) arise, it will appear that the expression ~d in (2), represents the ratio of the chlange in x to the entire change in u, which latter is produced partly by the change imparted to p, and partly by that im-parted to q: du dq that the expression I- "- represents the ratio of the change in x clV CiX to that part of the change in u which is communicated through q: and that d d) represents the ratio of the change in x to Cdp dx that part of the change in u, which is communicated through p. Idu dq idu We must be careful, therefore, not to confound d- --- with dq dx~z dx' or to suppose that the first of these expressions can be brought to the form of the second by the ordinary process of algebraic reduction. This will appear evident, when it is recollected that the du& which appears in ~ refers to the total change in u, while the cd which occurs in d i dq refers only to so much of the dq dx' change in u, as is communicated through q. Similarly, d dp'dp dx' dxt must not be confounded with ~, for a like reason. dx 54. To differentiate u = F(p, q, r, s, &c.) when p, q, r, s, &c. are functions of the same variable x. By attributing to x an increment h, and reasoning as in the last proposition, we readily prove that TAYLOR'S THEOREM. 71 rdlu dp du dq duc dr d ds 1 ~-= + + k+.+~.~+~.Z+.~+&. ~ q Ldph d + * d- * dx- -+ *dx' dx'+ + terms in h2, h3, &c. Tralsposing u, dividing by h, and then passing to the limit, we have du d u d u dq dzt dr du ds - =- -+-* + ~~ + ~+ &c. dx dp dx dq dx dr dx ds dx dud du cdp du dq du dr. *.~dx = du d =- dx+ + - d. + ~. dx dx dp dx dq dx dr dx -- ds d dx - &c. ds dx that is, we must differentiate u with respect to each of the functions, as if the other functions were constant, and add the results. 55. Prop. To diffelentiate u = F(p, x), where p =fx. Here u is directly a function of x, and also indirectly a function of x through p. Now if in the equation u = F(p, q), which gives du dlt dp du dq dx- d dx dq dx we put q = x, there will result du du dp dp du dx u= F(p,x) and i +. - dx dp dx dd dx du du dp du dx or = -- +-y (1), since -= dx dp dx dx dx The formula (1) is that required, but we must distinguish carefully between the differential coefficient d in the first member, and dx the similar expression in the second. The latter, called the partial differential coefficient of u with respect to x, refers only to that part of the change in u which results directly from a change in x, while p is supposed to remain constant; and the former, called the total dif 72 DIFFERENTIAL CALCULUS. ferential coefficient of u with respect to x, refers to the entire change in u, which is partly the direct result of a change in x, and partly an indirect effect produced through p. To distinguish the total from the partial differential coefficient, it has been agreed to enclose the former in a parenthesis; thus we write \I=~ -. -- - and. dt = dclx = -.- cx+ -- dx. LdxJ dpc dxadx' d c dx dx Here again there is a necessity for caution, so as not to conf),und du dx with dut; the former being only a part of the change imdx parted to u by a change in x, while the latter is the symbol of the entire change. Cor. If there were given iu = F(p, q, x) where p and q are functions of x, then [du]- du dp du dq du Ldx - d'p dx clq (dx dx and similar expressions would apply if there were a greater number of functions. EXAMPLES. 56. 1. = sin -l(p -q), where p =x and q = 4x3. du 1 du 1 C lp o d dp / -_- q)2 dq / (~ -q) dx' dx du, du dp dt dq 3 —12x2 dx dp d cq d _ q) 3 - 1 _2,X2 3 /1 -9x2 + 24x^4- 16x 1 2 —2 2. u = pq, where p = ex, and q = 4 - 4x3 + 12x2 - 24x + 24. du qd = P' = e, r= 4x3- 12 + 24 - 24; dp dq dx dx IMPLICIT FUNCTIONS. 73 d, du dp du dq __ -= e_. X4. dx dp dx dq dxz X4p2 X4p x4 3&, f = -- ~3 +,2 when p = logx. 4 8 0 du x4p x4 dp _1 d X3p 3 d-= 2 s' dx x' d p -2 8 Pda d_ v d du I d + = ( X 4 + x 3 2 + a _ I x3(log X)2. ea4. - where p asin x, and q cos x. du eax due eax dp dq - = a1 cos x sill X, dp a2 +- 1 dq a2 + 1' dx a c dx du'aea (p - q) dx- a2 1 dul du dp d dm q + du LdxJ dp dx dq dx dx eax a2 -- 1 (a cos x - sin x + a2 sillx - a cos x) = eax sin x. Difterentiation of Implicit.Functions. 57. In the various cases hitherto considered, we have supposed,he' function to be given explicitly in terms of the variable. It is now proposed to establish rules for differentiating inmlicit functions. Prop. Having given F(x, y) = 0, to form the differential c(,efficient [y without solving the equation. dX Put -- F (x,y): then u will be a function of x directly, and also indirectly through y. dL =u d dy du Fdl dy dx dx LB J I d~ 74 DIFFERENTIAL CALCULUS. But since u remains constantly equal to zero, the total differential coefficient of u with respect to x must be equal to zero also. du.'. - +- 0, whence~ dy dx dx dx du d,/ Thus it appears that we must form the partial differential coefficients I- and ~, then divide the former by the latter, and dx dy prefix the negative sign to the quotient. _. 1. y2 2a.- y + x2 - b2 = 0, to form the differential coefficient of y with respect to x. u =- Y2 - 2axy +- x2, d- = - 2ay + 2x, -= 2y- 2ax..x dy dy -2ay -- 2x ay - cdx 2y - 2.c y - ax 2. Given x3 + 3axy + y3 - 0, to form the 1st and 2d differential coefficients of y with respect to x. dux du.'. u = 3 + 3caxy + y3 d 3x2 3ay, = 3ax + 3y2. 0 — 3ax + 3y2. u-m~0cxy + y3 dT~OX~~ dy dy x' -- ay dx ax + Y2 dy Put d-= p2; then P1 will be a function of x directly, and also indirectly through y. d2y rCIpl dp., dpli J.2 L LdxJ dy dx dx But dpl_ -a(ax+.y2) +2y(x2+ay) d1 - 2x(ax + 2) +a( + ay) dy ( + y)2' d - (ax + y)2 IMPIICIT FUNCTIONS. 75 Ilence by substitution and reduction d2y 2yx2 -t ay2 - a2zx x2 + ay/, a2y - ax2 - 2xy2 dx- (zax + y 2)2 ax + y2 (a + y2)2 2a.xy - 2xy (x3 + axy/ + y3) 2aC13o (O + y42) - (ax + Y2)3 58. Since it is possible to form the successive differential coefficients of y with respect to x, without solving the given equation, it will be possible to expand y in terms of x by Maclaurin's Theorem. 1. Given 3 - 3y -- x = 0, to expand y in terms of the ascending powers of x. cdu d (. dy 1 3=y3~-3y —- -, d o, ~ -3(y2 -1). -_ 1 e U Y' x' dp y d 3(1-~V2) Expanding the last expression by actual division, we have d- I (1 + y2 + y4 + &c.) d2, 1dy I dX -3( = (2y+ 4y3+ 6y5+ &c.) d 32 (2y ++ y l2y54 1 + &c.) d y- 32 (+ 18+6 +& (2 + 20o/2+80y4+ &c.) d4y/ 1 d(y (40y + 320y3 + &c.) = (40y + 3603+ &c.) dxV4 33 CIX 34 dx =- 3_ (40 + S100y + &c.) 5 (40 + 1120y+ &.) a. But when x = 0, [] = 0, y 3' [LJj=' [dJ33' [dy 0 FdYJ1 40 & 3'1 dX2 _:Ldx3-' L3d-3:0, Ldx- 35&. By substitution in Maclaurin's formula, x x3 X5 Y = ++ + + &c. 76 DIFFERENTIAL CALCULUS. 2. To expand y in terms of the descending powers of x, from the relation ay3 - x3y - a3 =- 0. 1 Put xs = -; then ay3v - y- a = 0. v du du dy ay u-=-ay3v-y —a, = ay3, - = 3ay2v, d2 3/2( — 3 ay2j) I + (6ay, - + 3ay2)ay3 d2y d-/v (IV ), &C. &C. dv2 -(1 - 3uy2v)2 & &c. But when v -0 [y] =-a, [L] -a4 [ L - a & a4v 6a.7v2 ~. y-= -— a, &c. I 1.2 or by replacing v by -, a4 3a7 y=-a - - &c. The use of this method is much restricted by the great labor usually required in forming the successive differential coefficients. CHAPTER VII. ESTIMATION OF THE VALUES OF FUNCTIONS HAVING THE INDETERMINATE FORM. 59. It frequently occurs that the substitution of a particular value for a variable x in a fractional expression will cause that expression 0 to assume the indeterminate forln 0 Such expressions are often ca.lled Vanishing Fractions, and they may be regarded as limits to the values of the ratios expressed by these fractions, when the variable value of x is caused to approach indefinitely near to some particular value. X4 1 Thus in the example u _ 1' the value of which can usually be determined when that of x is given, by a simple substitution, we find that it assumes the form - when - 1. But the value of u is even then determinate; for if we divide the numerator and denominator of the fraction by x - 1, before making x = 1, we get,3 + X2 + x- I X2 + X + 1 as a general value of u, and this becomes 1+1+1+1 4 1+1+1 = I- when x 1. 1 + I + 1 3 Here we see plainly that it is the presence of the common factor.- i in the numerator and denominator which causes the fraction to assume the indeterminate form. In this, and in all similar cases, 78 DIFFERENTIAL CALCULUS. the removal of the common factor serves to determine the value of u. But it usually occurs that the discovery of this factor is attended with considerable difficulty, and hence the necessity of some more general method by which to estimate the values of fractions which assume the indeterminate form -, when the variable x takes a particular value. Such a method is readlily supplied by the Differential Calculus. It should be observed, however, that there are other indeterminate 0 forms besides -, such as the following: 0' 1 x, Got X 0, o - O, 00, G ~. 1-, each of which will be considered in succession. 60. Prop. To determine the value of a function which takes the 0 form f for a particular value of the variable. 0 Let u - - =- be a function which takes the form - when Le ux 0 x = a; that is, let Fa - 0, and qca - 0: let it be proposed to find the particular value [it] assumed by u when x = a. Suppose x to take an increment h, converting u, P~ and Q into ul, P1, and Q1, respectively, and let P1 - F (x + h) and Q1 =- (x + h) be expanded by Taylor's Theorem: then denoting the successive differential coefficients Fx by Fix, F2x, &c., and those of px by p1x, 2%X, &c., we have P1 F + (p h) +F F2-~+F29 -~ 3 Cc3 2 + &C. + +1 1.2+ 1 + &c. or when x = a i h h h3 Fa a + F + a + a + F3 + &c. a+ a h h+ 2 1h3 q~a + ql2a - a]' 2a ] 93a -o + &C. I. 1. 2. 3 FUNCTIONS HAVING THE INDIETERMINATE FOtRM. 79 But by hypothesis, Fa- 0, and pa - 0... Omitting the first term in the numerator and denominator, and then dividing each by h. we get l2 Fla + F2a + ~F3a - &c. h hA2 - 2. 1) l1a + P2a 1 + 3a 12 3 + &c. Now making h = 0, we convert u, into [?], and thus obtain pYa Hence it appears that, in order to determine the value of a.Fe~x 0 function - which takes the form - when x a, we must replace (px 0 Fx cand qpx by the values of their first differential coefficients, and then make x = a in each. It will sometimes occur that this substitution will reduce to zero both Fla and cpa, in case [u] -- Fla 0 remains still undetermined. -' a -- 0 we then omit Fla and qgpa in equation, (1) and divide the numerator and denominator by 1h' thus obtaining F2a + F3a, + &c. ZL/ =~.. (2) p9a + (3a - + &c. which becomes [u] 2a P2a when h = 0... when the first differential coefficients both reduce to zero, they must be replaced by the second differential coefficients. If F2a and 80 DIFFERENTIAL CALCULUS. pa both become zero also, we omit them in (2), then divide by h 3, and finally make h = 0, obtaining F3a [a] —' (p3a And since the same reasoning may be extended, we have the following rule for finding the value of [u] = viz.: -p- 0a' Substitute for Fx and cpx their first, secoicd, third, &c., differential coefficients, and miake x = a in eacj result, until a pair (' coefficients is obtained, both of which do not reduce to zero; the fraction thus bund will be thle true value of [u]. EXAMPLES. 5xs —_1 0 61. 1. u= -= when x =1. x-1 0 Fx =-x- 1, and lpx =-x- 1... F1x = 5x4, and (P1 = 1. - Fc 5. Fla 5, pla = 1, and [y] = =- 5. This result is easily verified by division, before making x = 1; thus by actual division s5 _ X4 + x3 + x2 - + 1 = 5 when x = 1. a.X bx 0 2. =- when x = 0. x 0 Fi'x lol a. a - log b.bx Fba logaa ~ ~_ ~logb. b F. _-= log a - log b = [ul. plx 1 (,a This result is easily verified by expanding ax and bx. ax - bx Thus x +1 log a1 q+ log2a — + &c.-1-log b -- ]og26 1 + &c. + $ 1.2 X FUNCTIONS HAVING THE INDETERMINATE FORM. 81. u = log a - log b + (log2a - log2) + &c..*. [u] = log a - log b by making x = 0. a - V/a~- x2 0 x2 0 Fx x(a2 - X2)2 Fla 0 (px 2x la 0 Here the first differential coefficients prove equal to zero, and therefore they must be replaced by the second differential coeffi. cients. But F2x (- X2)4 + x2(a2 -_ 2) 92X 2 F2 (a2)-4 1 2a -2 =2a l ax2 + ac2 -~ 2acx 0 4. U — b2 c + bc2 = - when x = c. bx2 ~ 2bcx + bc2 0 FJx 2ax - 2ac Fla 2ac - 2ac 0 (9x 2bx- 2bc p1a 2bc - 2bc 0 Thenx 2a F2a 2a a Then = —... - -=- [u]. 9Px'20 pa 2b b X3- aX2 - a2x +- a3 0 5. u = - when x =a. 2 _a2 0 = - 3.2 - ~- - 2 =0 [ ] Fx 2-x x-. Fa 0 0=H. ax- x2 0 6. ~ = - when = a. a4- 2a3x + 2ax3 — X4 0 FZl a- 2x la a - 2a p9x -2a3 + 6ax2 - 4x3' a - 2a3 + 6a3 4a3 F1a a or =- =~ ^ oo = [U1. <1p 0 = [ 6 82 DIFFERENTIAL CALCULUS. 7. UZ = l~_ -ogx when x=1. (1 -x) Flx X 2(1- x) Pix 1 (1-) z 2 p-a 1?_ 2(1-1~ oo(1enx - ena 0 8. t =, when x = a, (s being an integer.) -(X -a)s 0 Differentiating s times, we get px s(s - )( - 2)......3.2.1 Fa nsena p'a s(s - 1)(s - 2)......3.2. u tan -sin x 0 9. -. = when x= 0. sinx O0 Fx sec2x - cos x Fa sec20 - cos 0 0 p1 3 sin2x. cos x' a 3 sin2O. cos 0 0 F2x 2 sec2x. tan x -- sin x p9x 6 sin x. cos2x - 3 sin3x' F2a 2 sec20. tan 0 + sin 0 0 q2a 6 sin 0. cos20 - 3 sin330 0 F3,X 4 sec2x. tan2x + 2 sec4x + cos x )3x 6 cos3X - 12 sin2. cos x - 9 sinl2. cos x F3a 4 sec20. tan20 + 2 sec40 4- cos 0 3 1 *p3a 6 cos30 - 12 sin20. cos 0 -9 sin20. cos 0 6 2 0. 62. The method just explained and illustrated, ceases to be applicable when we obtain a differential coefficient whose value becomes infinite by making x= a; for such a result shows the FUNCTIONS HAVING THE INDETERMINATE FORM. 83 impossibility of developing the corresponding function F (x + h) by Taylor's Theorem, for that particular value of x, and therefore the process founded on such development fails. The expedient adopted in such cases, is that of substituting a + h for x, then expanding numerator and denominator by the common algebraic methods, then dividing numerator and denomiinator by the lowest power of h found in either, and finally making h - 0. A few examples will illustrate this method. 63.. 1. == -x = - when x = a. (a- x) Here the first differential coefficients reduce to zero, and all succeeding coefficients become infinite when x = a. We therefore put a + h for x and expand. (a2 - - 2ah - _2)~ (2a + h)2.(- h)i (a -a- h)2 a - a. l (- /I) 3 (a - a - 1,) = (2a + h)2= (2a) + (2 a)h + &c. u ] =- (2a)2. 2. _- - - = 0 when x a. X~ _2 a2 0 a^T^ 0 Put a + h for x (a + h) - a+ a + h - a) + 2 Act - &o. = 1 — = -- (a2 + 2ah + h2 - a2)2 h(2a h)) I + a 2 h2 &c. (2a) + - (72a)7 h &c. 2 ~ (2'a) 84 DIFFERENTIAL CALCULUS. Remark. This method may be used even in those cases to which the method of differentiation is applicable. We will now consider the other indeterminate forms. P v' 64. Prop. To find the value of the function = - - which Q x' assumes the form'- when x -= a. 1 1 Put P =-and Q =- Then we have P q 1 u =-= 0= when x = a. I p 0 q Thus the function being reduced to the ordinary form ~, its value may be found by the methods already explained. Now since dp I dP _ F1x Now since p =? ~ = - -- = ~ ) —P' dx - dxx - (lk)2 And similarly dq = (P1x __?'a - 91a. Fa - F a a. I -=a whence — = = a H. ne it ppers tat the ordinary direct process of substitting Hence it appears that the ordinary direct process of substituiting for numerator and denominator, their first differential coefficients will apply when the function takes the form 1 —. But since when P oo and Q = Go, their differential coefficients will also be infinite, the reduced fraction will still take the forlm -) and therefore will not serve to determine the true value of u, unless we can discover a factor common to the numerator and denominator, or can trace sonme rela.tion between the numerator and denominator of the new fraction, which will facilitate the determination of its value. FUNCTIONS. HAVING THE INDETERMINATE FORM. 85 65. Prop. To find the value of the function u = P x Q =F.x X qpx which takes the form oo X 0 when x = a. Put P = Then u = Q = - when x = a, the common form. p p 0 o 0 1 dp 1 dP F1x Now since p we have dx =- d = - ()2 [..*. [q- (ETa) 2. 1 Fla But since when P - Fx -=, its differential coefficients will also be infinite, the value of u will take the form, unless the infinite factor should disappear by division. 66. Prop. To find the value of the function u = P- Q = Fx - x, which takes the form oo-oo when x = a. Put P =- and Q = Then P q 1 1 q-p 0 u-=~ ~= -- = - when x= a, P q pq 0 and the value is to be found by the ordinary method. 67. Prop. To find the value of the function u = P _- (Fx)Px which takes either of the forms 00, a o, or 1, when x = a. 1st. Let the form be u 00. Passing to logarithms we have log u =- Q. log P = gx. log (Fx) [log u] = pa. log (Fa) = - X o. which is one of the forms already provided for. Thus, having found log u, we have u = el~g u. 2d. Let the form be u=ao o. Then log u= Q. log P=px. log (F). * [log u] =. log (Fa) = O x. a form already considered. 86 DIFFERENTIAL CALCULUS. 3d. Let the form be 1+. Then log u = Q log P -= x. log(Fx). [log q] a. og () = + E og x O, and the form is still the same. EXAMPLES. 68.1. 1. (1- ). tan (x )0= X oo when x = 1. Here Fx= tan( ) and (p = I -x c [- -,,,,(1-co2 \ 2/' -- se 02 9:., 1X 1) —. ~ see2-. se s- 1 _ ~(1 - co-.se2r>J 2^2 iee2 2 2. u = e. sinxz ac X 0 when = 0. 1 1 1 Fx = e,.x = sinx,.'. F x- 2- e, -p1 = cos a. = *[u] —(S )2 F 2 ~ * COS X. - e 02. e Here the function still takes the form so X 0; but the true value is easily found by expanding er. " ^-1 1 1 3) For e. x- (1 2- + &C.) X2 x 1.2.x2 1.2.3.x 1 1 1.2 1.2.3x e.. X 02 = 0 - [U]. FUNCTIONS HAVING THE INDETERMINATE FORM. 87 o n 00 3. u - when x = -. ex oo Differentiating n times, we get [ (n -1) (- )(n-2)....3..1 [l = = 0. log x Go 4. u = = when x oo. xn GO.Fxl =1, (p9x = nx,"-l. [U] = a= 0 [u]- -.'n = O. 1 2 5. u- -1 _ - = (o —Go when x = 1. 1- 1-x2 p= — x, q X2 I 1 1 X2 Ex qpx 2 lp __-x^2)-(l — X) 2q - i o 9 —pX2.. 1 - = 2 (1+ xw -- 1 2 c " 0 1 1. X I 3, u = = c when x = 1. x - 1 log x I x-1 P -= q = -logx. X - q-p xlogx-x+ 1 O..(x —1) g when x1. - pq (x-1) logz O 88 DIFFERENTIAL CALCULUS. logl + 1-1 0 ~[u]-iog I+ -I I1- 0 Differentiating numerator and denominator of the value of u a second time, and making x = 1, we get 1 1 [u]] 1 = + 7. u = x" = 00 when x = 0. log u = x.log x = —O oo, when x=O. log a c, or, log u = = - when x = 0. 1 301 x Then Fx = log x, and px-. X2 8. U = xsin = 0, when x = 0. Since = I when x =,.. in = X == 1 when x = 0. And similarly sin xsin = x" = 1 when x = 0. Again, since sin x. log x sin x. log. og e. * Xsin =esinx.log=, when x= 0. *. sin z. log. = 0, when x - 0. And similarly sin x. log sin x = 0 when x = 0. 9. u = cot Xsi" = o - 0 when x = 0. log u = sin x. log -si = sin x (log cos x - log sin x). =0 —0=0 when x =0,.. Lu]= 1. FUNCTIONS HAVING THE INDETERMINATE FORM. 89 10. u = (1 + nx) = 1 when x = 0. lo log(1 x + ) 0 when x =0. log u x = n By differentiation, [log u] = - =, and [{u] = en. This result is easily verified; for by expanding (1 + nx)a by the binomial theorem, we obtain qb = U + 1 -(jX) + - (-1) (Z) + 1 - + &C. — (x ( x -- 2. = 1 + n + (1 (1 x)- 3x - 2x2) &c. [ —] = +1 n + 1 + 3x + 1C.2 which series is the expansion of e". (cosec cX)2 11. u = (cos ax) = IC), when x= 0. log cos ax 0 log u cosec2cx. log cos ax = - x when x 0. -~ _ F x a nd sin2cx 0 Put log cos ax = Fx and sin2cX = p.. F.x =- a. tan ax, plx = 2cin. si c. cos = c. sinl 2 cx. ~1a 0 Fla 0 Differentiating again we get 2x -- a2. sec2ax, p2x - 2C2. cos 2cx....-=- 2a...a2 F2a a2.- ~. _.. [u]=u = 2c * pa 2 2c2 CHAPTER VIIT. MAXIMA AND MINIMA FUNCTIONS OF A SINGLE VARIABLE. 69. If u2 be a function whose value depends on that of a variable x, so that u = Ex, and if, when x takes a certain value a, the corresponding value u, of.I be greater than the values which immedlately precede and follow it, then the value u1 is called a maximum; but if the intermediate value be less than those which precede and follow it immediately, the value ul, is said to be a minimumn. Suppose for example that when x --- a, the general value i = Fx becomes uI ='Fa, that when x =.t i?I u becomes ut2 = F(a -- A), or?3 =- F(a( - h), and suppose that for some small but finite value of h, and for all values between that and zero, the corresponding values of both?( and ut3 shall be less thanl ul, then will u1 be a maximum; but if %t2 and u3 be both greater than uil, then the latter will be a minimuin. 70. In order to discover the conditions necessary to render a function (u = Fx) either a maximum or minimum, the followihg principle will be established. PropF. In any series Aha -F Bhb + ChC -+ &c., arranged according to the positive ascending powers of h, a value may be assigned to!t so small as to render the first term Aha, (which contains the lowest power of h), greater than the sum of all the succeeding terms. Proof. Assume A > Bhb-a + Chc-a + &c., a condition always possible, since by diminishing h the second member may be ren MAXIMA AND MINIMA. 91 dered less than any assignable quantity. Multiply each member by ha; and there will result Aha > Bhb + Chc + &c., as stated in the enunciation of the proposition. Cor. The value of h may be taken so small that the sign of the first term shall control that of the entire series. 71. Prop. To determine the conditions necessary to render a function tu of a single variable x, either a maximum or minilmum. Let u = Fxr, and suppose x to receive successively an increment and a decrement h. Then developing by Taylor's Theorem, we get dFx h d2Fx h2 d3Fx h3 u-.F( x+ 7 ) =-Fx + + + 2x I +. (1) u:Fx Fxy.dx I dx2 1.2 dx 1.2.3 d1 x h d2Fx h2 d3Fx h3 Now in order that Fx may exceed both F(s + h7) and F(-x h), it is obviously necessary that the algebraic sum of the terms sueceeding the first term in each of the series (1) and (2) shall be negative; that is, we must have by empioying the usual notation, h h2 h3 F1 x 1 x 2 + F3T12 + &2. <... (s). an h2 h3( and -Fz -Fx 2 r - l x + &c. <.. (4). 2 1.2.3 Now the sign of the first term in each of the series (3) and (4) will control that of the entire series when A is taken sufficiently small, and since the first terms of (3) and (4) have contrary signs, it is impossible that both of these series shall be negative, so long as the term Fix. 1 has a finite value. HIence the first condition necessary to render Fx a maximum is that Flx. - = O, or since h is finite dFx Fix = dx =....(5). 92 DIFFERENTIAL CALCULUS. Now omitting the first terms of (3) and (4), we have F2x i 2 i3 13 + &c. < O.... (6), 1.2 l.~iL.2.3 12 h3 and Fix - -F. 2. 4 + &c. < 0... (7). 3x 1.2.3 The signs of the series (6) and (7) will be controlled by those of their first terms, which terms have the positive sign in both series; and therefore each series will be negative when F2x - is an essen1. 2 tially negative quantity, or when F2x is essentially negative, (since h2 is always positive). Thus the two conditions which usually characterize a maximum value of Fx are dFx d2Fx -d = 0, and dx2 < o. dx -dx< On the contrary, when Fx is a minimum, we must have Fx less than F(x + h) and F(x - h), and therefore by a similar couie of reasoning, the necessary conditions are dx, a dx > O. The conditions here obtained are those usually applicable: the excepti6ns will now be considered. 72. The results obtained in the last proposition indicate the following as the ordinary rule by which to discover those values of the independent variable x, which will render any proposed function u a maximum or minimum. du 1st. Form the first differential coefficient -, place its value equal dX to zero, and then solve the equation thus formed, obtaining the several values of x. MAXIMA AND MINIMA. 93 d2u 2d. Form the second differential coefficient - xx, and substitute for x, in the value of that coefficient, each of the values found above. d2u Then all those values of x which render -2 negative, will correspond to maximum values of u; but those values of x which render d2u d2 positive, will correspond to minimum values of u. And when dx2 the proper values of x have been ascertained, the maximum or minimum values of u are found by simple substitution in the equation u - = x. 73. 1. In the application of the preceding method, it may occur that a du d2u value of x, obtained by making d = 0, will, when substitued in ucause that coefficient to reduce to zero also. In that case, the signs of the series, (6) and (7), in the last proposition, will depend on the terms which contain the third differential coefficients; and since these terms have contrary signs in the two series, the value of x du dS't which renders - = 0, and - -= 0. cannot render u either a naxis ~dx' dx2 muml or minimum, unless it should happen to render d3= 0 also. When this occurs, we must examine the sign of the fourth differential coefficient, which now controls the sign of each series, and if this be negative, the value of u will be a maximum; but if positive, a minimum. And since the same reasoning could be extended when other difler ential coefficients reduce to zero, we have the following more general rule for the discovery of maximum and minimum values of a function of a single variable. 1st. Form the first differential coefficient, place its value equal to zero, and deduce the corresponding values of x. 2d. Substitute each of these values in the succeeding differential coefficients, stopping at the first coefficient which does not reduce to 94 DIFFERENTIAL CALCULUS. zero. If this coefficient be of an odd degree, the corresponding value of u will be neitner a maximum nor a minimum; but if it be of an evei degree, the value of u will be a maximum or mininum, according as the sign of that coefficient is negative or positive. The annexed diagram will 1 Y F illustrate the fact that the ~ D same function may have sev- C E eral maximum and several minimum values; and that o P Q R s X one minimum may exceed another maximrum. Thus, if the curve CDiEFGH be the locus of the equation y =Fx, then will D Q and FS represent maximum values of the ordinates y, while CP, EIR, and GX will be minimum values of the same. Also the minimum GX exceeds the maximum D Q. 74. The substitution of a value x = a, derived from the equation d = 0, in the succeeding differential coefficients, will sometimes cause the first of these coefficients which does not reduce to zero, to become infinite. This happens only when the development of F (x +- h) in the ordinary form. (by Taylor's Theorem) is not possible for that pa:rticular value of x. We must then find b' other methods (such as algebraic development) the true value of the term which cannot be obtained by Taylor's Theorem. If it be found to contain a power of h, which will change sign with h, such as A3 or hA, the value of u will be neither a maximumi nor a minimum; but if the power o 8 12 h be sucll as will not change with h, as /l or h, the value of will be a maximum when that term is essentially negative, and a minimum when the term is essentially positive. 75. Finally, it may occur that when x has a particular value a, the first differetil coeiciet will become ifiite, an, therefore the first differential coefficient., will become infinite, and, therefore, dxI MAXIMA AND MINIMA. 90 in order t ) complete the search for maximum and minimum values of u, we ought to solve the equation ~ = c, and if a be a root of that equation, we must substitute a + iA, and a - h for x in u =- Fr. Then if the term containing the lowest power of h be found to change sign with h, there will be neither maximum nor mininmum; but if not, there will be a maximum when that term is negative, and a mninimum when it is positive. 76. Prop. To determine the maximum and minimum values of an implicit function of a single variable x. Let F (x, y) = 0 be the relation connecting x and y, du dy dx Put = F (x,y)-0; then dY d dx du dy But when y is a maximum or minimum, = 0;. = also, dx dx- also, and we have the two following conditions by which to determine the values of x and y, viz.: du dF(x. y) dx d x( -..y 0 O..(1), and u.= F(x,y) 0...(2). Having found the values of x and y which correspond to either a maximum or minimum, we distinguish one from the other by substituting the same values in the successive differential coefficients, and stopping at the first which does not reduce to zero. If this be negative, y will be a maximum; if positive, a minimum. The successive differential coefficients are formed without difficulty from the value of dy already found, and their particular values, when = 0. become much simplified. dx' du7 d(u d2u d2u Thus, put T- p dy q1 -P2, q2, &c.,ad employ the [ ] to represent the particular values of the quantities enclosed, 96 DIFFERENTIAL CALCULUS. du when p = P 0. Then observing that p, q1 &c., are usually funo tions of both x and y, we have dy -= P. 0 dx2 qq d2y 1 ddx dy clx dq dy) d q1 (P2 - p ) p (q2- d —).. _ _ [P21 _ (' L _' I Lq1] L] And in a similar manner the higher differential coefficients can be formed, although the operation is more laborious. 77. The following considerations will facilitate the application of the preceding principles to particular examples: 1st. If a quantity which is a maximum or minimum contain a constant ftctor, that factor may be omitted and the result will still be a maximum or minimum. 2d. If u be a maximum or minimum, then u +~ a is also a max. imum or ninimum, but a - u will be a minimum when u is a maxilnum, and a maximum when u is a minimum. 3d. If u be a maximum, - will be a minimum; and if it be a 1miai, wiul be minimum, - will be a maximum. 4th. If u be a maximum or minimum and positive, then u2, u3, and in general ut, will be a maximum or minimum where u is any positive integer: but if u be negative, t,2, u4, and in general u2n, will be a maximum when u is a minimum; and a minimum -hen u is a maximum. MAXIMA AND MINIMA. 97 5th. If u be a maximum or minimum and positive, log u will also be a maximum or minimum. 6th. If the power u2n be a maximum or minimum, the root u is not necessarily either a maximum or minimum; for it may be imaginary; and even when U2n =0 and a maximum, the corresponding root =- 0, although real, is not admissible as a maximum, because the adjacent values of u are imaginary. 7th. The value x = o cannot correspond to a maximum or minimum value of u, because x cannot have a preceding and a succeeding value; but = oo -may be a maximum provided the preceding and succeeding values of u have like signs. 8th. In determining whether u is a maximum or minimum by d2t du the sign of dx2' when d has the form of a product v1.. v V3....,, and x = a causes one factor v, to become equal to zero, the only dau, dv,, term in d n2 necessary to be examined is that involving cl- sinoe the other terms disappear with v,,. 78. 1. To determine the values of the variable x which render the function u = 6x - 32 - 4x3 a maximum or minimum, and the corresponding values of the function ua. Here - 6x + 3x2 4X3.. 6 + 6x - 12x = 0, dx 1 I 3 1 or x2 —x-...x —- - +1 or 2 2 =4 2 4 2 Hence if u have maximum or minimum values, they must occur 1 when x = 1 or when x =-2 To discover whether these values are maxima or minima, we form the second differential coefficient: thus d2it -_ 6- 24x = 6-24 =-18 when x = dx2 = 6 + 12- = 1S when x = —2 2 7 98 DIFFERENTIAL CALCULUS..'.when x =1, = 6+3-4 = 5 a maximum, 1 3 1 7 when x = -, Z =-3 + + - = 47 a minimum, 24 2; 4' 2. u =x4 - 8x3 + 22x2 - 24x + 12, a maximum or minimum. du 4x3-242 44x - 24 = O or 3- 6x2+ 1lx -6 = 0. dx The value x = I is obviously a root of this equation, and by dividing the first member by x- 1 we have for the depressed equation 2 - 5x 6 = 0..'.x=2, or x =3. Hence the values requiring examination are x =, x = 2, and x = 3. dx2 But ~ — 12x2 —48x~.44= +8 when x= 1, — 4 when x=2, = + when x 3. when x =-, 1 = 3 a minimum, when X - 2, t = 4 a maximum, when x = 3, = 3 a minimum. 3. = X5 - 5x4 + 5X3 + 1 a maximum or minimum du d 5X4- 203 +15x2- 0 or 4- 4x+3x2 =., (1)... x2= 0, or 2-4x +3 = 0, and the four roots of (1) are 0, 0, 1, and 3. 2- 20.3-60x2-30x= 0 when x= —; (. let us examine d 3 -= 10 " x=1; then, = 2, amax. = +90" x=3; then, u = — 26,a min. d- = 60x2 - 120x + 30 = 30 when x = 0..e = - 1 is neither a maximum nor a minimum. MAXIMA AND MINIMA. 99 du 79. In each of the preceding examples, the condition ~ = ao, renders x = o, and therefore not applicable to a maximum or minimum..Remark. In forming the second differential coefficient, it will save labor to omit any positive numerical factor common to every term of the first differential coefficient, and the sign of the second differential coefficient will not be affected by such omission. (X +- 3)3 80. Ex. 1. u = )+ ) a maximum or minimum. (x + 2)2 du 3 (x + 3)2(x + 2) - 2 (x + 3)3 d (X+)= = o, or, —. But, when d = 0 we have 3( + 3)2(x + 2) -2(x + 3)3 = 0.. x + 3 -= 0 or, 3(x + 2)= 2(x + 3),.. = -3,- or, x = 0. d2u 6 (x + 3) (x + 2)2- 12 (x + 3)2(x + 2) + 6 (x + 3)3 d2- (x + 2)4 9 27 - when x = 0 and.. -u = a minimum. 8 4 =0 " x=- 3. Now, without actually forming the 3d differential coefficient, it is easily seen that it will contain one term (and only one) which will not reduce to zero when x =- 3; and, therefore, the corresponding value of u is neither a maximum nor minimum. 27 The value x 0, gives u = a minimum. du 3(x + 3)3(X + 2)- 2(X+3)3 Now taking the equation d - - (X + 2)3~ ), we get x + 2 = 0, or, x -2, and by putting successively x =-2 h and x = -2 Ah, in the value of the original function u, there results 100 DIFFERENTIAL CALCULUS. (-2 - 3+S) ( + h)3 2= F(a + h) = (_ - h2 (-2h+A2)2' 3 = F(a-h) (- 2-h + 3) (-h)3 ( —2-h- +2)2 h2 and since both of these are positive values, and less than that corresponding to x — 2, we have u = oo a maximum. 2. u = (x-1) a maximum or minimum. (X+ 1)" du 2(x - 1) (x + 1)3 - 3(x + 1)2(X- 1)2 du? ^ ~(XTip ~ ~~~ O+ )-.-. —1 = 0, or, 2 ( + 1)-3 (x-) -, or, 1 0... X = 1, or, x 5 or, - 1. d2u 2(x + 1)2 -12( + 1)(x - 1) + 12(- 1)2 dx2~ (x + 1) d2u 1.. x2 = 4 when x - 1, and u = 0 a minimum. dx2 4 1 2 - 3 2 when x =5, and u = -~ a maximum. When x. —1, u- = 0, which is neither a maximum nor a tninimrv I for -F(a. + h)= ( 2+ 4)) > 0o, (2-(+h)3.- P but ^ {a - 3 = F (a-h)=( < o, 3 3. -= b + (X- a)!, a maximum or minimum. du 3 - (X -a) =,.. x =a, and u b. dIx 2 d2u 3 -4 But - = x - a) =-G, when x = a. Bux2 _ _ MAXIMA AND MINIMA. 101 Hence we cannot develop by Taylor's Theorem. Put a - h, for x in the value of u. _8 3. U2- b + (a + - a)= b + (+ h)> b, and u3 = b + (a - h - a)= b + (- h). This last value is imaginary, and therefore, u = b is neither a maximum nor a minimum. 4. u = b + (x - a)x, a maximum or minimum. d-= (x-a)3 0,.. x-a, and u = b. d2u 4 ( dX2 = (X - a)- oo, when x a. Then put = a ~h. A A.. = b + (a + h - a)3=b + (+ 7)3 > b, and iU3 = b + (a-h-~a)= b + ( —h)3 > b, also... x = a gives u = b, a minimum. 5. u = b - (a - x)t, a maximum or minimum. - (a -x)= 0,.. x=a, and u=6b.: 2- -(a - x) i -, when x = a. Then put = a ~ h, dx2 25^ ^' 2 =- b -(-h)'< b, and 3 = b- (+ h) < 5, also..'. = a gives u = b, a maximum. 6. u b + (x - )+ c (x - a)2, a maximum or minimum. d ( - a) + 2c (x - a) = 0. d or + 2c 5 ~ ~ — 0, or, - 2c(x a) =0. 102 DIFFERENTIAL CALCULUS. 125 3125.'. a, and u- b, or, x =a - -- and u = b2!6c3' 466560c5 d2u 10 - But -d2 -= (x- a) + 2c= c, when x = a 2 125 c>O when x=a- - 21 125 3125 Hence when x = a- 2 -1 we have 6- = b - 3 1 9.216c W 46656c5s a minimum. In order to examine the value of x a, put a ~ h for x in the original value of u.. U2 =b +(+ h)( + c(+ h)2 > b, U b + (- h)) +c(-h)2J Ld12j 1.2 L dx,3 1.I3 Now since z = y + xqz,.... (1).. hen x = 0, [z] - y, and.. [] Fy. LAGRANGE'S THEOREM. 123 dul dot dz du du dz Also:- and dx dz dx dy dz dy But by differentiating (1) with respect to x we get dz dqz dz d z qpz = q)z + zx * whence (2). dx dz dx dx dxz dz And by differentiating (1) with respect to y we havo dz dqz dz cdz 1 1= I + x d whence d =-. (3). dy dz dy dy 1dpz dz Dividing (2) by (3) and reducing, we obtain dz dz - = q'z dx dy' du du dz du dz du dx dz dx dz dy dy Hence when x -0, and z=y, ~ Iy ddu dou 1 Now assume ux such that gqpz - = d' d-y dcy dudu dux d2- d2ul d2u1 dx and dx - dy dx2 dydx -dxdy dy But dl du dz dul ~ q)z dz d u 2 d dx dz dx dz dy -= dy'(px) e dy d2u d [(ri) 4] [d2 d[ =y ] ~dx ~ dy~ Ldx2 ~ dy And similarly it may be shown that d2\ d-y' d = 1 dFy] Fdxu1^ dL d 1 L *xdy LdXJ dy2 Ld X Idy3 a2 ~ y 124 DIFFERENTIAL CALCULUS. But to show that this law of formation of the differential coefficients is gellral, suppose that it has been proved that dn-lu d ~' dydn-16 dyn n2 e- dy - dyn d_ dn- dyn-1 Bdut,.n- ddunn-dn-i d,,u dnun~-1 dnun-1 \ dx J dx dy-1 (5). ~ dxdy-(5) Thus the form (4), if true for any value of n, is also true for the next higher value. Now it has been shown true for n = 1 and n = 2; and hence it is true when n = 3, n = 4, &c., or it is universally true. Now making x = 0 and z-= y and the expression (5) becomes LdV -J dyn-I Making Jthe substitutions fo-r [,],d-' d2 ],2 &c., in the expansion (A), we get dFy x dp dy x FY +py wdy 1 + dy 1. 2 "-^+^^1+~d-y-~^ d2 ([).Id 3'].. 3dy &c...(L) This formula is called Lagrange's Theorem. LAGRANGE'S THEOREM. 125 Cor. Let u = Fz = z; then Fy y, ad = 1. dy x d [((py)] X d2[(qy)_] x3 ~..z+ ] dy'. + &c.. (M) a formula for the expansion of z when we have given z = y - xzz. EXAMPLES. 95. 1. Given z3 - az + b = 0 to express z in terms of a and b. b 1 Here z -= - -1 — 3, which corresponds to the form z = y -i- xpz, a a when we make b I -= y, = x, and Z3 = qz. a a b3 Hence by substitution (py = y3 = -; d[(py)2]= d(/6) 56b5 d2[(q)yY3] -d2(y9) W=6- = S.91 =S. 9 -, &C. dy dy a' dy2 d-2 a7 Introducing these values into the formula (i), it becomes b b I bD 1 Ib 1 b9 1 + -3.+6-. ~8.9-. +10.11.12- +&c. a a3 a a 1.2a2 a7 1. 2.3.a3 a9 1.2.3.4a. b [ b2 b4 b6 b8 -[1+ V+3 + + 55 &c.] -a a3 9 + 12 If b be very small in comparison with a this series will converge verv rapidly. 2. Given y = z + z2 - z3 + z4 + &c., to revert the series, that is to express z in terms of y. By transposition, z = y - (z2 + z3 + Z4 + &c.) Put =- - 1. Z2 -t 3 - 4 + &C. 126 DIFFERENTIAL CALCULUS. Then qpy = y2 + y3 + y4 + &c. d[(y)] d[(y2+ y3+y4+ &c.)2] dy dy =-2(2+y 3+y4 +&e.)(2y +- 3y2+ 4y3 + &.) =2(2y 5y4 +9y5 +14/6 + &o.) ((y)3 (y2 + y3 + y4 + &C.)3 - y6 + 3y7 + 6y8 + &c. _[(Y) 3 ] = 5.6y4 + 3. 6. 7y5+ 6.7.Sy6 + &c. dy2 (py)4 = (y2 + y3 + y4 + &C.)4 -y8 + 4y9 + &c. d.3[()] - 6.7.8y5- + 4.7.8.9y6 +- &c. dy3 ((y)5 = (y2 + y3 + y4 + &C.)5 =y10 + &C. d"4[(y)5] d. d -— = 7.8.9.10y6 + &c. &c. &c. dy.. By substitution in formula (M). = y — [Y2 + y + y4 + y4 + y6 + &c.] A+ 1-. [2.2y3 + 2.5y + 2.9yp + 2.14y6 + &c.] 1- [5. 6Y4 3.6.7y5 + 6. 7. 8y + &c.] 1.~2. 3 +1 [6. 7. 8y5 +4. 7. 8. 9y6 &c. -1.. 4.5[7 8s. 9. ly6 + &C.] 1.2.3.4. - &c. = y -?/2 + y3 - 4 + y5,_ 6 - &c. 3. Given 1 - z + e- = 0 to expand zn. HIere z = I + ez. Put x 1, y - 1,pz = e, Fz = -- dllY d ) (Y )' Y = ey. y F= y, py d_ ey = n eY == ne. dy - dy LAGRANGE'S TI-IEOREM. 127 d dFyl d (e2nynyl) = 2ne2Y. y-l + n (n —1) e2. y-2 = (n + 1) e. Ci2 Fy 1[yY 9 d (3Y i ) dElyJ ely2 = 9i,:3y. y^-ll +,ic (n - 1) e. yl -- ( - 1) e( - 2) e3. y -3. = n (it2 - 3n + 5) e3. &C. &c. Hence, by substitution in formula (L), we have (n,(- +1)2 n( ~+3n~+5) Z11 I + 1e + ( + ) + n ( + 323 + ) e + &c. 1.2 1.2.~ 4. Given z =- y + e. sin z, to expand z and sin z. Put x = e, zq) = sin z, Fz = sin z...y = sin y, (py)2 = sin 2y, (y)3 =- sill "y &c. d[(c1y)2 -. *. ('y)] = 2 sin y. cos y. = sin 2y. d2 [(@y)3] d (3 sin 2y. OS sin y. cos 2y _ 3 sin 3y. l2 ~y -dy 3 sin y(1 + cos 2y - - cos2y) 2 2 = Sill + ~ sin 3y -- sin y/ 9., 3 = - sin 3y- siy. &c. Hence by substitution in (M). z=y+siny- +sin2y- 2+ sin 3y- - sin Y 2 + &c, ~Again Fy. sin]. dFy 1 Again y = sin y... dy = sin y. cos y -- sin 2y. Y.Y 2 128 DIFFERENTIAL CALCULUS. dd d (csin y. sin 2y dY' dy dy dy d (cos y- - os 3y) = dy - -= 4 sin 3y- sll y. dy 4 4 d l d(cosy. sin3y) sin 2y -- cosJ 2?. d2 [(dy)1 Ci2 (Cos r:, sin/.) 2y dy2 -) dy 1 dy2 dy2 d2 sin 2y si' 4y) - ~ dy~2 =- 2 sin 4y - sin 2y. &c. &c. dy2 Hence by substitution in formula (L). e e siz = sin y + sin 2y. + ( - sin 3 -- in 2 63 + (2 sin 4y- sin 2y) 1 + &c. da, h d2u h2 dCu, h3 5. Given ~.+ *- - - - & 1-. =.O, to find dx 1 dXx12 1. d3 1.2. 3 h in terms of u and its differential coefficients. tddu dc2 d3 u Pt -Pidx - - -1' d -2 2, - 2p3 CX dd~.= X 2 ^ 2.. h-_ 2 24-s +&c.) P1 \1.2 1 1.2.3'U 1 P 7P2 1 3 I 3.. - (. ~2Z = 2 + 5.1 i. + &c. 2" - \ p-p^ \ ^ ^1.2. Now if a be a root of the equation -- 0, and x an approximate value of a, so tllhat. + h = a, we may use this series in finding a more exact value of x. Thus, if x =- = 1.5 be an approximate MAXIMA AND MINIMA. 129 root of the equation u =x4 - 2x-2 +4x-8 = 0, then 15 + h = a and _/ u 3x2-1 U2 =- 22(x3-+1)+ 24(3 _+ 1)3 1.2 + 21x4 -12x2 -6 3.3 &) 23 Here --.. h=. 11 and a = 1. 5+. 11- -1.61 16 nearly. And if we repeat the operation by putting x — = I 61, a nearer approximation will be obtained. CHAPTER X. MAXIMA AND MINIMA FUNCTIONS OF TWO INDEPENDENT VARIABLES. 96. A function u of two independent variables x and y, is said to be a maximum when its value exceeds all those other values obtained by replacing x by x ~ h and y by y 4- k, when h and k may take any values between zero and certain small but finite quantities; and u is said to be a minimum when its value is less than all other values determined by the conditions above described. 97. Prop. Having given uz = F(x, y), when x and y are independent variables, to determine the values of x and y which shall render u a maximum or minimum. Suppose x to receive an increment 4 h, and y an increment ~: k, the value h and k being small but finite and entirely independent of each other; and denote by iu2 the value assumed by u, so that u2 = F(x ~ h, y k). 9 130 DIFFERENTIAL CALCULUS. Then, by Taylor's Theorem, as applied to functions of two inde. pendent variables, we have du (~h) d+ u (~1 k) d2u ( h)2 +2.='I - + - t - &e U + dx 1 dy 1 dx2 1.2 d2u (~4-h) (~Ic) d2a (~k)2 + ___. \ + d. + &_c. dxd 1 1 dy2 1.2 Now in order that uA may exceed u2 for all small values of h and k, whether positive or negative, it is obviously necessary to have du (~h) du (ik) d2u (~ h)2 d2u (+~h) (+ k) dx 1 I dy 1 dx2 1.2 dxdy 1 1 d2 (~k) + dy2 1.2 &. < o.... (); in which series we must be at liberty to make h and k both positive, or both negative, or one positive and the other negative: or, finally, either may be taken equal to zero, the other remaining finite. Now when k = 0 the series (1) reduces to du (~h) du ( h d ~. — I - +-.+ &dc I.,<. (2); dx 1 d~ 2 1.2 d x3 1.2.3 L +' "') in which h may be taken so small that the sign of the first term du ( — h) which contains the lowest power of h, shall control the sign of the series. But this term obviously changes sign with h, du since - does not contain h; and as we are at liberty to make h dx alternately positive and negative, it is impossible that the series (2) dzu (~_ h) should remain negative so long as du ( - has any value other than zero. We have then, as a first condition necessary to render u a maximum, du (__+h) du du' 1A =0 or simply d=0.... (A). dx I' dxt MAXIMA AND MINIMA. 131 Omitting the first term in (2) we have xd2- (~ h) + h-) (+ 3+ &c. <.... (3 d212 d1.21.2. 3 Here again the sign of the series will depend on that of the first term when h is small, and since that term does not change sign when we substitute - h for + h, the series (3) will remain negative for small values of h, when d2a (~h)2 du d2-t ( -< 0, or simply when d < 0. Hence - <.. (B) dX2 is a second condition necessary for a maxirnum. 98. Returning to the series (1) and supposing h = 0 while k remains finite, we prove, by a course of reasoning entirely similar, that the following conditions are also necessary, viz.: d& d2'% -= 0... (C) and <0.... (D). dy dy Now omitting the terms in (1) which contain the first powers of and k, and which it has been seen must reduce to zero, we obtain d2t ( 2 ttd (f h) (~ Ik) cl2tu (~ k)2 dlu (-: h)3 dx 1.2 dxdy I 1 dy2 1.2 dx3 1.2.3 d3u ( h)2. (- k) d3u (+h)(- k)2 dx2dy 1. 2 dxdy2 1. 2 d3zt (-~ ic)3 dy3 1.2.3 &c or, by making h = r, where r is entirely arbitrary, since A and k have no necessary dependence upon each other. _L. + x 2r + d 2 d 1. 2dz2 dz —-y dy 1.23 ['d3u d d3z( d3 d3C 4~- - d 4-r -- 3r2 --- 3 - ~ d +&c. < 0.... (4) -.2. 3 Ldx3 dxTdy dxdy2 dy3 6' ~~ i~~d~C3- C~~2C~~ C~~C'' 132 DIFFERENTIAL CALCULUS. in which, when h is small, the sign of the series will depend on that of d2u d2.U d2u -~ ~2r, ~+ r2dx" dxdy dy" and this must be negative for all values of r, whether positive or negative, when u is a maximum. We must now search for the condition necessary to rendel d2u2 d2u d2u/ d2+ r + r2 <.... (5) for all values of r. dx"2 d-.-xdy dy2 d2ut d2u d2u Put for brevity d -= A, yd =B, and'- U. dx2 dydx dy2 Then A, B, and C, must, if possible, be so related to each other that A ~ + 2Br + Cr2 shall be negative for every real value of r. Now it is known, from the theory of equations, that if we solve the equation A ~ 2Br + Cr2 - 0 with respect to r, and obtain the values =B + B2-AC B-B - 2AC C1 I and r2' —=- C = ~~ and then substitute in the polynomial A - 2Br - Cr2, for r values alternately a little greater and somewhat less than r1 or r2, the sign of the polynomial will undergo a change. If therefore the proposed substitution be possible, the condition A ~ 2Br + Cr2 < 0 for all values of r will be impossible. And so long as the values of r1 and r2 are real and unequal, this substitution can be made; but if those values of r prove imaginary, it will no longer be possible to substitute for r, real quantities alternately greater and less than such values, and therefore the polyno. mial cannot change its sign. Now by examining the values of r, and r2 it will be seen that the condition necessary to render r1 and r2 imaginary is B' < AC. Hence we have a fifth condition necessary for a maximum, viz. (12 d 2u- d2u (\2 d dy2 dY) >.'.. () MAXIMA AND MINIMA. 133 when this condition is satisfied, the condition (5) will also be satisfied, since (5) is true when r = 0. It ought to be remarked, however, that when B2 = AC, the two roots r7 and r2 become real and equal, and therefore in passing over one of these roots, we necessarily pass over both. Thus the sign of polynomial will not change, so that the fifth condition would be more correctly stated as follows: d2u d2u d2u 2.. da'*d > (__) o.... (E). dx2 dy2 \dxdy) > By a course of reasoning entirely similar, we can prove that the five conditions necessary to render u a minimum are the following: du du d2u, d2ud d2u d2u / dd2u 2 ax:, 0 =o, >~, >o,'\a,>-1y! >o. dx'dy'dx dy d2 dy2 \dxdy > d2% du 99. If d-= 0, when — = O, there can be neither maximum nor dx2 dx d3,U d2% minimum, unless - - 0 also; and similarly, if d- 0, when I dx3 " dy2 du d3u d = 0, there can be neither maximum nor minimum unless d= 0. dy dy3 There are other conditions likewise necessary to render u a maximum or minimum in such cases, but they are usually of so complicated a character as to be unfit for use. EXAMPLES. 100. 1. To determine the values of x and y which render tu = x3 + y3 3axy a maximum or minimum. du du d- 3X2 -3ay- 0,... (1). = 3y2- 3ax = 0,.... 2) dr - 3ay-li- 0, dy' (2). x2 From (1), y = -, and this substituted in (2), gives 4- a3x = 0;.. x = 0, or, x =a, the two other roots being imaginary. 134 DIFFERENTIAL CALCULUS. x2 But when x = 0, y -= = 0, and when x = a, y =a. Now forming the second differential coefficients, we get d2U d2U d2u 6x=O when x 0, = G6y= when y = 0, dcx2 dy2 = 6a when x = a, = a when y a, d2au Cdh, 0 d2u / d2h 2 dxdy -3, dx( = -9a2 when x = 0 and y — 0. dxd T dz dy2 dxcly = 27a2 when x = a and y -- a..The five conditions necessary for a minimum are fulfilled when x = a and y = a, viz. du du d2u d2u d - d2i / d2u \> — = >, > 0, an0d >.2 ( ) > o, dx dy dx2 dy2 dx2 dy2 \.dxdy.. = a3 + a - 3a3 = a3. d2u d2u But when x = 0 and y = 0, and 2 reduce to zero, while -3 and -l do not reduce to zero. Hence the value u = 0 is cdx dy3 neither a maximum nor a minimum. 2. To find the lengths of the three edges of a rectangular parallelopipedon which shall contain a given volume, a3, under the least surface. Let x, y, and z, be the required edges,.'. xyz = a....(1). And u - 2(xy + xsz + yz)= surface = a minimum. a3 a3 But from (1), xz = -, and yz=~-, y x ( a3 a3 \. 2 X + ^ )+....(2). du 2a3 3) and 2a. 2y =... (3).a nd =2 — 0.... (4) dx2y - 3 = ry2,. X y and consequently = xa3, 2.'~. x=y --: y,.a ~. x == y, and consequently xa —a3, MAXIMA AND MINIMA. 135 x = a, y = a, and z = a. ad2u 4a3 d2u 4a3 d2u d - 4 > 0) d 2 =-3 = 4 > ~d - /Xp= x-= — =4>0, dxd= =2. 2 - dy2 - y3 dxdy d2u d2u d2u \2 dx2 dy2 \dxdy = 12 >,.'. u = 2(a2 + a2 + a2) = 6a2 = a minimum, and the parallelo. pipedon must be a cube. 3. Given x + y + z = 17, to find the values of x, y, and z, when cos x cos y cos -- = a maxim.um. Regarding x and y as independent variables, and z a function of x and y, we obtain by differentiating x + y - z =x- r, with respect to x and y successively. dz dA 1 +- 0, and 1 + = 0... (1). dx dy But since u = maximum, log u = log cos x + log cos y + log cos z = maximum, (d log )_ _ tan x - tan z = 0. dx dx dcl ogu\ dz and ( - I- tan y - tan z -- =O. \ dy dy dz dz or, by replacing d and d by their values derived from equations (1). -- tanx + tanz = 0 -tany + tanz = O,. tan x = tan = tan y, and x = y -z = r. u =cos3- r = 4. To find the greatest rectangular parallelopipedon which can be inscribed in a given ellipsoid. Let a, b, and c, be the semi-axes of the ellipsoid, x, y, and z, thb co-ordinates of one of the vertices of the parallelopipedon. 136 DIFFERENTIAL CALCULUS. Then 2x, 2y, and 2z, are the three edges of the parallelopipedon, and, therefore, 2x. 2y. 2z maximum, or u = xyz = maximum.... (1). But, since each vertex is in the surface of the ellipsoid, the coordinates x, y, z, must satisfy the equation of the surface. x.2 y2 z2 a2 +b2 C2 =..... (2). Differentiating (2) with respect to x and y successively, regarding z as a function of the independent variables x and y, we get 2x 2z d 2y 2z dz a2~ + c2' b2 2 dy But, from (1) we have Idu\ ydx jau dz (~ o, \j Z x + xY ~ =, = x+ x = 0, dz dz or, by introducing the values of d and d from equations (3). c2x c2y =.y? z2 yz - xy = 0, and xz - y =. o^ b26.a2z2 c2x2 and b2z2 = c2y2. 2.= 2 a z c2::6"~ 2 2 2 Hence from (2), 2,+ -t= I and x =_: in like r it may be hown that, y c an manner it may be shown that, y:, and z = ~' Thus the edges of the parallelopipedon must be proportional to the axes to which they are parallel. In each of the last two examples, the formation of the second differential coefficient has been omitted as unnecessary, it being easily seen that the proposed question admitted of the maximum or miuimum sought, and also that the values found were the only suitable-values. CHAPTER XI. CHANGE OF THE INDEPENDENT VARIABLE. o01. Hitherto we have employed the differential coefficients dy d2yi dx d2u d-' &c-, &c. or d-' d2 &c. exclusively upon the hypothesis that z dx' cix2' dXs dx i' was the independent variable. But there are many cases in which it is more convenient to adopt some other quantity t upon which both x and y., or x and u depend as the independent variable, and perhaps to pass from one supposition to the other within the limits of the same investigation. di dy f du2y di u It then becomes necessary to express - y &c. or d d&. dx dx2' c. or dx' c. in terms of the differential coefficients of x and y, or those of x and u taken with respect to the new variable t. 102. Pro%. Given y = qx, and x= Fit, to express y and d in dx dt2x dy d terms of' d, -2 dt' d c. -dti^' 6 dcl' dt2' Since y is a function of x, and x a function of t, we have dy dy dy dx dy dt dtYx- dt d....e (1) and dz = d- (A) dt dy Now differentiating (1), and observing that - is a function of t through x, we get d2y d2y dx2 dy d2x dt2 dxz dt2'd dx dt2' 138 DIFFERENTIAL CALCULUS. d2y dy d2x c2y dx d2x dy d2y d2 dx dt2 dt2 dt dt2 d (t ) dx2 - d2 dx3 (B) dt2 i7 The two formuls (A) and (B) resolve the problem. Cor. In a similar manner we might form expressions for d3y' d4 dx3' dx4 upon the same hypothesis, but they are seldom required. Cor. If y be taken as the independent variable, then dy y d d2y \dy t=y, dy-=-dy=l1 and. = 0. "'dt dy - dt- dt d2x dy_ L 1 1 d2y dy2 dx = dx d aX d x3 dt dy dy3 Cor. If x be the independent variable, then dx dx 1 d2x d d = x, dt = d= 1 and d-t = O, and (A) and (B) reduce dy dy d2y d2y to dx and dx - ad dx dx2, the ordinary forms. dr ^~ d ~ " dX2 - ins. EXAMPLES. 103. 1. Transform the differential equation d2y x dy ~Y -O 0 so as to render 0 the dx'2 - 2 dx+ 1 -x2 independent variable, having given 6 = cos - x. dx d2x Here x= cos. = Co dy dy do 1 dy d* d x, s sin 6 dd dd CHANGE OF THE INDEPENDENT VARIABLE. 139 d2y dx'd2x dy d2y dc2'd dO2 dO 1 d2y cos dy dx2 dx3 sin 2* dO2 sin 38 dO dO3 Hence by substitutin in the given equation, 1 d2y cos dyq cos O dy y _ -. -- — O siln 2j dO2 sin 3' c sin 3d dO sill 2 d2y od2YO.rd+ 0. This example illustrates the important fact, that a change of the independent variable will sometimes simplify the form of the differ ential equation. d2u d2it 2. Transform dx2 -+ - 0, so as to render r the independent dx2 dy2 variable, where r2 = x2 - y2. dxe r d2x d (r Here X2 = r2 __y2 d-r -- di2 dr 0 1 r dx 1 r2 y2 x x2 dr x x' X3 dy r d2y x" And similarly - r d2= _ dr y' dr2 - y d2u dx d2x du d2u d dr dr d-2 dr x2 d2ut y2 diu dx2 dx3 r2 d'f2 r3 dr d.3 d2u'y2 d2u X2 CdI And + - dy2 r2 dr2 r3 dr d2u d2u.By substitution in the given relation dxx2 + d =0, and d22C L cU zd d2u 1 din reduction, - +- 0.. dr2 r -Cr 104. Prop. Having given u = F (x, y) when x = p (r, 6) and ydu din y =f (r?, ), to express - and - in terms of r and 6. clx. Jll 140 DIFFERENTIAL CALCULUS. Since u is a function of x and y, each of which is a function of r, we have du du dx d d dy _+.. _...~. (1) dr dxc dr Tdy dr And sirrmilarly, x and y being functions of 8, du du dx d+ d y dO dx dO dy dO dx. dx Multiply (1) by d-, and (2) by - and subtract; then multiply (1) dy dy by d and (2) by d- and subtract. We shall then obtain du dx du dx i du Idx dy dy dx\ dr d dO dr dy \dr dO drl d du dy du dy du dx dy dy d.\ and dro' d d d' d -r d dr d )d dz dy du dy du dx du dx du dr dO dO dr dur drd d dd dr - ^ ~ ~~ ~and dx ddx dy d xy dx a y dy dy dx dr do dr d6 dr d6 dri d6 105. These forimule become much simplified when we have x = r cos 8, y = r sin 6, the common formula for passing from rectangular to polar co-ordinates. For we then have dx dy dx ay d = cos, - = s, =-rsi n, =- r cos. dr' dr' d6'd6 dx dy dy/ dx d' dO d- dO -'-d = r(cos2O + sin2) = r. du du sin 0 du du du cos 6 du. = cos 6 —- and = sin O - + dx dr 2 dO dy dr r dO du - du Ex. Having given x — Y y = a, to transform the equation to the variables r and 6, where x = r cos 6, y = r sin 8. du du' /.du, cos 6 duo\.( du- sin 6 du\ x ~' =rcOs6{sm6-,- + ~'-.*I'rsi6{ cos6-n..~ dy dx dr r dQ dr r d ) 2 SI2 d dx (cos2 + sin2O) - - = a. ^' ~~~d6 d6 CHAPTER XII. FAILURE OF TAYLOR S THEOREM. 106. It has been shown that the general development of F(x + h), so long as the value of h remains unassigned, is of the form F(x + h) = Fx + Ah + Bh2 + Ch3 + &c...... (1), containing none but the positive integral powers of h. But although this be true for the general value of x, it is possible in some cases, to assign certain particular values to x, which shall cause fractional powers of h to appear in the development; and to such cases Taylor's Theorem does not apply, because its proof depends upon the assumption that the series (1) holds true. If, for example, we assign to x such a value as shall cause fractional powers of h to appear in the undeveloped finction, we may expect to find similar powers in the development, and we therefore cannot expect Taylor's Theorem to give the correct expansion. Now when the particular value x a introduced into the undeveloped function the fractional power h7, there must have been in the general expression for Fi (before a was substituted for x) a term of the form (z - a) which becomes (x - a + h)n in F(x + h), and reduces to m h" when x - a. When this occurs some of the differential coefficients will cer. tainlv become infinite, if we make x = a. 142 DIFFERENTIAL CALCULUS. To illustrate this fact, take the example u =Fx b + (x -a)3 + (X - a) and suppose x to receive the increment h, converting u into l x = F(x + h) = b + (x - a + h)3 + (x - a + h). Now, for the particular value x = a, u1 becomes b + h3 + h. But by forming the successive differential coefficients of u with respect to x, we get du = 3(x - a)2 + ( - a) d2. m _1 " dX2 = 23(x - a) + -- 1)( a) d = 1.\ 2. +- — l I-2)(~ a m dx4 -- n - - c., & c. d4uC n I/7In M\/ \/m \and since the exponent of x - a is diminished by unity at each differentiation, it must eventually become negative, renderingL the coefficient infinite when x =- a, Moreover, all the succeeding differential coefficients will likewise become infinite. It may be observed also that if the lowest (and therefore the first) fractional exponent which appears in the develpment, be intermediate in value between the integers r and r + I; then the first differential coefficient which becomes infinite will be the (r +- )th. It appears then that this peculiarity will arise whenever the value assigned to x causes a urdc (such as (z - a) ) to disappear in Fi, while the corresponding slrd [(x - a +h) ] continues to appear in F(x + h) in the form of a fiactional power of h. This inapplicability of Taylor's Theorem, improperly called a failure of the FAILURE OF TAYLOR'S THEOREM. 143 theorem, occurs precisely when the development is impossible in the general form, and therefore does not result from any defect in the theorem itself. Again, it has been shown that the general development does not contain negative powers of i, because we would have, (if there were such a term Ch-C) F(xz + h) = -x =- when h -. 0, an obvious absurdity. But when we assign to x such a value a as shall render,Ex =;, the above argument ceases to be conclusive. In this case Fx = -,n and the differential coefficients will be infinite also. Thus Taylor's Theorem will be inapplicable. Here also we see that the presence of a negative power of h in the development must result from a term of the form (- ) in Fx, (X - a)' B B which becomes in F(x + h) and reduces to -- = Bh(x -a + h) i / when x = a. VWe conclude, therefore, that there are two cases in which Taylor's Theorem is not applicable, viz.: 1st. Wihen x = a causes a surd to disappear in Fx, thereby introducing a fractional power of h into F(x + h). 2d. When x = a renders Fx = -. EXAMPLES. 107. 1st. Case. Given u =b ( + c)2 + (- a) F= Fx, to expand u1 = F(x +- h) = b + (x L c + h)2+ (x- a + h). clzdu 3 d2au 1 3 - = 2(x + c) + 2 (- d-) 2 = 1.2 +2 ( 2 -a) d3u 113 - c., &c. By substittio i Tay s Teoem,.By substitution in Taylor's Theorem, 144 DIFFERENTIAL CALCULUS. u L = b + (x + c) + (x - a) + ( + c) + 2(x-a) 12 hh + 2 1.+ 1 1 3 3 ( a -.2(X-a).-2-3 + &c.... (1). Now this development is entirely true for all values of x, except x = a, which renders the term I. 2 + 22 ( -a) 1 2 and all succeeding terms, infinite; the true development in this case being R1 b +(a+ c + 2 +'I b + (a +c)2 + 2(a + c)A ~ Ae + / 2 +,2 which agrees with the series (1), only so far as to include the term [2(X 4 ) 4- 2(x-) I 2d. Case. Given tu b -4,in x 4 - - = Fx, to expand (v - a)2 F, = (x + h) = b +, )!n(s +.! + (x - a + 7b)' du 1. 2c d2u 1.2.3c = cos x -.. -- - ~l X + - dx (x - a) d3' - - a)4 d3u 1.2.3.4c = -- COS x -- 7~ &c. 3, dX3 (x - a)5.. By substitution in Taylor's Tec.xemn, t-= b + sin - + (~ -+ cos x —( -- a x a) L (v —a)3J1 rF-~.. 1.2.3cl h2^ r 1.2;.3.4 n1.2.3 This development is correct except when x = a, the true devel opment then being (Art. 48) c h2 u b - sin(a + h)- -> b+sin a +ch-2- cosa.h-sin a- &c, 11=i -sna+ h) 2 - 1.2 FAILURE OF TAYLOR'S THEOREM. 140 l[ere the very first term given by Taylor's formula, viz.: C Fx - b + sin a +- a)2' incorrect. (a- a) 108. Prop. If the true development of F(x + h) contain posi. ti-re integral powers of h to the (n - 1)th power inclusive, followed by a term containing hs where s is a fraction intermediate in value between n - 1 and n, the first n terms of the expansion will be given correctly by Taylor's Theorem, but the (n - 1)th term will not be given correctly. Proof. Let the true development of F(x. — h), when x =-a, be F(x + h)= A + Bh + C/12 +h3......+ NJvn-l +- Phs + &c. where s denotes a fraction grealer than n - 1 and less than n. Then, since the differential coefficients of F(x + h), taken first with respect to x, and afterwards with respect to h, are equal, we have dF(x + h) _ dF(x- + 7) - 1B + + 3Dh2.... -n -(- 1)NlVn-2+ sPhS- +- &c. d'F(x + h) cd2F( +- Ih)_ ~~ -~= ~, I 2,,C +. 3-DA...... + (a -2) (? -1i) n7-3 + (S-1 )sPhS-2 + &c. dFx h) 3+ - )D + (n 3)(n- 2)( - 1)h -4 + (s - 2)(s - I)s'Ph-3 + &c. d __ (__ Ic) _ 1.2.3... (2 a - 3)(n - 2)(n - 1)N dxn-' +(s —F)(s -1{+3).... ((s-2)(s-l)sP -n+lI&G), d d" (s-n 1 l )(s-~ (i + 2)(s -n +3).... (s - 2)(s - 1)sPs-n"+ &c. Now when l 0, the preceding expressions reduce to d Tx d2FX dd3' Fx= A, -I B, 2 - 1.2 d 1.2. 3..... I CC2 10dx 10 146 DIFFERENTIAL CALCULUS. _i,1-.:2.3.....( - 3)( - 2)(n -1)N. dxn-1 diFVx P dx =(s-n+1)(s-n+2)(s-+3).... (s-2)(s-l)s - - oo. dFx 1 d2Fx. A Fx, dB- C 1 2 d &c. Thus each of the terms A, Bh, Ch2, &c., of the true development will be given correctly by Taylor's Theorem as far as the term Ih"l-l inclusive (that is to n terms), but the (n + 1)th term of the true expansion is PhS, while by Taylor's series it would appear to be infinite. The results established in thWs proposition are important, because it frequently occurs that the first or leading terms of an expansion, are those only which we have occasion to consider. PART II. APPLICATION OF THE DIFFERENTIAL CALCULUS TO TIlE THEORY OF PLANE CURVES. CHAPTER I. TANGENTS TO PLANE CURVES.-NORMALS.-ASYMPTOTES. 109. In the application of the Differential Calculus to the investigation of the properties of plane curves, we regard the two variable co-ordinates x and y or 6 and r, which serve to fix the position of a point on the curve, as the independent variable and the dependent function respectively. These two quantities are connected by a general relation called the equation of the curve, Such as y = Fx or r = O, F(x, y) =0, or c(r, 6)-=0. When the form of this equation is given, we can readily determine the values of the differential coefficients dy, d2- &c., or d'x dx2' dr d2r ~d- d-2, &c., in terms of the co-ordinates, and these values will be found extremely serviceable in the discussion of the properties of the curves. 110. The first application of the Calculus to Geometry which it is proposed to make, is the determination of the tangents to plane curves. 148 DIFFERENTIAL CALCULUS. Prop. To find the general differential equation of a line which is tangent to a plane curve at a given point x1 y. Y v R 0 T A D. D2 X The equation of the secant line RS, passing through the points X1 Yi and x2 Y2, is Y-/1- -(X -) (1 ) y2 ~- i1 But if the secant RS be caused to revolve about the point P1, approaching to coincidence with the tangent TV, the point P2 will approach P1, and the differences Y2 - y, and x2 - x will also diminish, so that at the limit, when RS and TV coincide, Y2 - Xg2 - xi will reduce to -, and the equation (1) will take the form Yy -- ~ - ~i...'1 which is the required equation of the tangent line at the point x, Yl Ill. To apply (2) to any particular curve we substitute for dL- its value deduced from the equation of the curve and expressed ill terms of the co-ordinates of the point of tangency. Cor. The differential coefficient d-1 represents the trigonometrical dx1 tangent of the angle P1TX formed by the tangent line with the axis of x. Cor. To find the value of the subtangent D1T, we make y = 0 in (2). The corresponding value of x will be the distance OT, and TANGENTS TO PLANE CURVES. 149 therefore x -i will represent the subtalgent D,1T, this latter being reckonedfrom D1 the foot of the ordinate. Thus subtall D1Ti = x - 1 =-' (3). dyl dx1 In the formula (3), x represents the independent variable, but if we take y as the independent variable, this formula may be simplify I I dx fied. For it has been shown that - or -= - Hence dx d,. dy dy dy dx (3) may be written dxx subtan 1 T -y d1 (4). 112. Prop. To determine the general differential equation of a line which is normal to a plane curve at a given point xl y. The equation of the normal v PN, which passes through the B point x1 yl, will be of the form y - Y1 = tj (X - xI)... (5), where t denotes the unknown OT AD N X tangent of the angle PNX formed by PN with the axis of x. But since the normal PN is perpendicular to the tangent PT, we must have, by the condition of perpendicularity of lines in a plane, 1 dy_ 1 + tt1 — O or -- where t -- = tan. anglo PTD. 1 — ~t - dx1 Replacilig t by its value in (5) there results Y Y1 - y - (X -...) dy, dy, dxl To apply (6) we substitute for its value derived from the equation of the given curve. 150 DIFFERENTIAL CALCULUS. Cor. To find the value of the subnormal DN, we make y = 0 in (6) and thus obtain ON as the corresponding value of x.. DNx - = x-1 - Y1 *. (7), when either the subtangent or subnormal has been determined, the tangent and normal can be readily constructed. APPLICATIONS. 113. 1. Let the curve be the common parabola, whose equation is y2 = 2px. dq p dy_ pdX1 y1. - dyl =P-, and * d y dxl y dy1, HI-ence the equation of the tangent is v Y - YyP- (x - 1) - /1 or yYl -- Y2 = -p(x -- X), whence T o D N X YY = p(x - x) + 2px, = p(x + x). And that of the normlal is Y Y1- - (x -1) Also, subtan DT= Y = - 2l i=- - -2x1, 2 i~ and subnorm DN = y - = p. y1 Thus it appears that the subtangent of the parabola is negative and equal to twice the abscissa; and the subnormal is positive and constant, being equal to the semi-parameter. TANGENTS TO PLANE CURVES. 151 2. The Ellipse, a2y2 + b2x2 = a2b2 dy _ b2x dy, b2xl and dx _a2y dx a2y dxj a2y1 dyl bx. The equation of the tangent is 621x )y Y -- Y = y (x-, or, a2yyl + b2xx = a262. a Yi dx1 a2y12 a2 Also subtangent = - y, 1 2 -- X. dl 62 And subnormal = Y dz = 2x - 1 Y 3. The logarithmic curve, whose equation is y = ax. -d= log a. ax. dx Z TO D N _ I.'. subtan = log- a. ax a log a - where m is the modulus of the system of logarithms whose base is a. Y12 a2xT Also subnorm = log a. axTyl = -. m m In this curve, the values of the abscissas are the logarithms of the values of the corresponding ordinates in the system whose base is at. 114. Prop. To determine expressions for the tangent, the normal, and the perpendicular from the origin to the tangent of a plane curve. P For the tangelit P T, we have PT = 0PD +.DT2 N, -= Y1 dxr =Y/l - 1.' ~^v ~^T71 2 152 DIFFERENTIAL CALCULUS. For the normal PN, we have PN1 —V/ +DN2 -- y 1 v i-Y12 For the perpendicular OQ, we have dxt I / ~ 1 ~ 7 OQ=OT.sinOTQ=OT = -OT1/ =_~_ cosec- V + cot (dx12 +- dyl2)2 Ex. The general equation of all parabolas. The general name of parabola is applied to all curves included in the equation ym = am-1x, in which nm may represent any positive number either whole or fractional. When m - 2, tho curve becomes the common parabola. dy am-' dx my-l mX1 Here ym=a-lx,. =and. _7 1 dx vnym 1 dy c. y dx1 yl m. subtan =- - Y, - m - mx, dyl am-l y 2 subnorm y= dx = m,2 -= dx- my-2 1 mx x/ dx2 tan = yI2 1~ +' I +7+ x12, norm = y 1 =+ f + and perp - nV x x^1(1 - nz) and perp [ -- [1 - -,1 1+ y2!%]}- [ + 115. Prop. To obtain expressions for the polar subtangent, subnormal, tangent, normal, and perpendicular to the tangent of a plane curve, when it is referred to polar co-ordinates. TANGENTS TO POLAR CURVES. 153 Let AB be the curve, Q the pole, P the point to be referred, QX the fixed axis N B from which the variable angle PQX is reckoned, QP the radius vector, TQN a line drawn through the.pole Q, perpen- dicular to the radius vector PQ, and limit- T1D ed by the tangent PT, and the normal PV, \ S QS a perpendicular on the tangent fiom the pole. Then QT is called the polar subtangent, and QN the polar subnormal. Put QP= r, angle PQX=, angle QPT=, angle PT1X- i, QD =, DP =y. Then QT = QP. tan QPT = r tan u = r. tan(i - 6) tan i - tan 6 -r. 1 + tan i tan 0 dy y dy Y dlx X But tan i tan 6 =. tan dx x y -- xy — d~x',x' x cd{x Now if we change the independent variable from x to 6, we d cy must employ the formula d = dx dx do dy clx d6 dJ.tanL= d d. (1). xdV+ Y/And from the formula for passing from rectangular to polar co-ordinates, we have x = r cos 0, y = r sin 0, which being differentiated with respect to 0, observing that r is a function of 0, we get dx drc dy dr. d- dO' dos -- r. sin, + c' os. dO d- dO dO 154 DIFFERENTIAL CALCULUS. and these substituted in (1) give r cos ~ c- a for every finite value of 8, the curve will lie wholly exterior to the circle; but if r < a for all finite values of 8, the curve will lie entirely within the circle. 1. Let the equation be (r2 - ar)02 = 1 or P6 r.2 - ar Then 6 = o when r = a. And 6 is real when r > a, but imnaginary when r < a.. The circle with radius a is an asymptote, and lies within the spiral. 2. The curve (ar - r2)2 = 1. 6 = -. = 0 when r = a. -ar - r2 Also 6 is real when r < a, and imaginary when r > a... The circle with radius =a is an asymptote and encloses the curve within it. CHAPTER II. CURVATURE AND OSCULATION OF PLANE CURVES. 122. As introductory to the discussion of the subject of the curvature of plane: curves, the following proposition will be found useful: Prop. To show that the limit to the ratio of the chord and arc of any plane curve, when that arc is diminished indefinitely, is unity, and to deduce an expression for the differential of the are of a plane P curve, in terms of the differentials E of the co-ordinates. 0 A 0, X Let PP h be an are of a plane curve A-APB, whose equation is y = Fx. Put OD-x, DP=?y, DD1-h, DPl-y1, AP-=s, APl=s1. Then F (x + Lh) The are PP1 is intermediate in length between the chord PP1 -'C and the broken line Pl 1P1 = B. If, therefore, we can prove that the limit to the ratio - is unity, it will follow that the limit to the ratio of the chord and are is unity, and therefore at that limit the expression for the chord PP1 will be a suitable expression for the are PP1 wNhich will then becomete the differential of s. CURVATURE AND OSCULATION OF PLANE CURVES. 161 B P T+-PT _ /PE + E7I' + p T B C PoP VP/E2 + EP12 ~v^^ h (Y1- Y) 4 dy2 dy jb2 + (y1 - y)2 ~h + h!Y j(dy/ h2da2 + 3 d3x + ~dY V+h \- 1.2 dxA 1.2.3 dx& h2 + (hd- + ht2d + &c.) and by dividing numerator and denominator by h dy2.l. d2 / h d3y &c. B - X2 12 d +1. 2.3 d + &c ~^-_~~= 1, when h=O. (-I d h2 h cdy d2y dy/ d3y.. /+ --- + -^.^ * d / *d Y Jr 1 ** + &c. dX2 1 dx dx2 1.3 dx. dx3. at the limit, dy2 arc ds chord tan PT d2 chordcl or dx dx d h or d v7 dX2 1.s - d I d- dy. ~ dx dx Also ds dy + l123. In the first of these expressions x is the independent variable; in the second, y. Cor. If we wish to employ some other quantity t upon which sa x and y depend, as the independent variable, we must use the formula for changing the independent variable, viz.: ds dy ds dt dy dt and dx dx dx dx dt dt 11 162 DIFFERENTIAL CALCULUS. ds which, substituted in the value of d' give x' ds / dx2 dy2 dt V l d(2 dt2 124. We proceed now to consider the osculation of plane curves. Let Y = Fx (1), and y =- x (2) be the equations of two plane curves, the first of which is given in species, magnitude, anq position, but the latter in species only. Then the constants or paramleters which enter into equation (1) are fixed and determinate, but those which appear in (2) entirely arbitrary, and may therefore be so assumed as to fulfil as many independent conditions as there are constants to be determined. If, when the abscissa x is sup- s posed the same in both curves, theY D condition y = Y is satisfied, the ^ B curves will have a common point P, but will usually intersect at / that point. o D If the condition dy dY be true also, the curves will have a dx dx common tangent such as SPT; and the contact is then said to be of the first order: if the second differential coefficients be also equal, d2/ d2Y viz., - the contact is said to be of the second order; if dx2 dx2 d3y d3 Y d3Y- d3, the contact is of the third order, &c. &c. 125. In order to show that the contact will be more intimate as the number of corresponding equal differential coefficients becomes greater, let x take the arbitrary increment h, converting y and Y into Y1 and Y1 respectively. d: ]A d2 Y 12 d3 17 h3 Then Y -= Y+ - - -' - ~ -&c. dx 1 dx 1.2 dx3. 2. 3.3 OSCULATION OF CURVES. 163 dy h d2y h2 d3y h3 and y =yy+-. —+- + +&c. Y Ld -dxY l dq.2 3 + d3Y d3y1 h3 L+ dx dx-J 1.2.3~&' Now the value of this difference, which expresses the distance by which the one curve departs from the other, measured on the line parallel to y, will depend, when h is small, chiefly upon the terms containing the lowest powers of h. If, then, the first differential coefficients derived from the equations of three curves (A), (B) and (C) be equal, at a, common point, and if the second differential coefficients derived from the equations of (A) and (B) be also equal, but those derived from (A) and (C) unequal, the curves (A) and (B) will separate more slowly than (A) and (C), because the expression for the difference of the ordinates of (A) and (C) corresponding to the abscissa x + A, will contain a term including the second power of h, but the difference of the ordinates of (A) and (B) will contain no power of h lower than the third. 126. The order of closest possible contact between one curve entirely given, and another given only in species, will depend on the number of arbitrary parameters contained in the equation of the second curve. Thus a contact of the first order requires two conditions, viz.: dy dY = Y and = dx dx dz the first of these conditions being employed in giving the curves a common point, and the second in giving their tangents at that point a common direction. Hence there must be at least two arbitrary parameters. 164 DIFFERENTIAL CALCULUS. A contact of the second order requires three parameters; one of the third order, four parameters, &c. Hence the straight line, whose equation y = ax + b has two parameters, a and b, can have contact of the first order only. The circle (x - a)2 + (y - b)2 = r2 having in its equation three parameters, can have contact of the second order. The parabola can have contact of the third order; the ellipse or hyperbola a contact of the fourth order, &c. The curve of a given species, which has the most intimate contact possible with a given curve at a given point, is called the osculatory curve of that species. The osculatory circle is employed to measure the curvature of plane curves, and its radius is called the radius of curvature of the given curve. 127. Prop. To determine the radius of curvature of a given curve at a given point, and also the co-ordinates of the centre of the osculatory circle. Let the equation of the given curve be y=Fx (1), and that of the required circle (x - a)2+ (y - b)2 = r2 (2), the quantities a, b and r being those which it is proposed to determine. There being three disposable parameters, a, b, and r, in equation (2), we can impose the three conditions dy ddY d2y d2Y y= 1 dx ) do and d2 dx dx TX d. ^ with which determine a, b, and r, and the contact will be of the second order. Denote the first and second differential coefficients derived from the equation of the given curve by p' and p", that is, put dY,a d2Y dx - p and dx2 Then, since the corresponding differential coefficients derived from RADIUS OF CURVATURE. 165 the equation of the osculatory circle must have the same values, we shall have dy d, i2y dy =pl and =P Now let (2) be differentiated twice successively, replacing dy and dY by p' and p". dx dx2 (x —a)+(y-b)p'O... (3), and 1+p'2+ (y-b)p"=O... (4). The equations (2), (3), and (4), will just suffice to determine a, b, and r. Thus, from (4) y-b=-1 +P.' (5) or b==y +,'. (6) pp" and from (3) and (4) x - a y - b)p' = ( +( 2) (7) _____ __...e. -(8) _ P (1x I + p) (8). Now combining (2), (5), and (7), we get 2 (1 + p'2)2 p+ 2(1 + p/2)2 (1 + p'2)3 r + P 1"2 p, 2 - p,2 (1 p'2)" -r = + - (,.2- (9). The equations (6), (8), and (9), resolve the problem. To apply them to a particular case, we form the differential coefficients p' and p" from the equation of the given curve, and substitute their values in (6), (8), and (9). Cor. Since 1 + p'2 or 1 + d2 -- ds2 (Art. 122) the value of dX2 d2X r may be written thus ds3 dx3 (10). dx2 166 DIFFERENTIAL CALCULUS. Remeark. We may omit the double sign - in (9) and (10) and regard the radius of curvature as an essentially positive quantity in all cases. This double sign is sometimes employed to indicate the direction of the curvature, being positive when the curve presents its convexity to the axis of x, and negative in the contrary case. But it seems more simple to consider r essentially positive, and to d2Y fix the direction of the curvature by the sign of d'' It will now dX2 be shown that the sign of this second differential will always be determined by the direction of the curvature. If the curve be convex' Y Y towards the axis of x, as i Y T in Fig. 1, and if an incre- P ~ ment h be given to the ~ E IE abscissa OD = x, the or- Y // y dinate y will take an in- x h x Ih 0 D D, 0 D D, crement p dy h d2y h2 d3y h3 d=x 1 2 1. dx3 1.2.3 + and the ordinate of the tangent will take a corresponding increment dy h ET = - 1- and the former of these two increments will be the clx I greater since the tangent lies between the curve and the axis of x. d2y h/2 d3y h3 EP -ET _ + + &c. > o dx2 1.2 dx 3 1. 2.3 or since the sign of this series depends, when h is small, on that of d2y the first term, we must have d2 > 0. But when the curve is concave towards the axis of x, as in Fig. 2, d2y EP1- T < 0, and.'. < 0 Again, since the arc s and the abscissa x may always be supposed RADIUS OF CURVATURE. 167 ds3 to increase together, - may be considered as essentially positive, clX3 and therefore the sign of r in (10) would be controlled by that of d2y dY It is in this way that the sign of r may be regarded as indidx2 cating the direction of the curvature. EXAMPLES. 128. 1. To find the radius of curvature of the common parabola y2 -- 2px, at a given point. dy = and d2y p dy p2 Y. - Ix2 P -- X 3 ~ "n - p y 3x. p C YI~:2 Y2 dr Y A D? 3~ 8 \ ( + p~,) (y~2 + p2)c (normal) or r- - 2 (semi-parameter)2 At the vertex, y 0, and.. r =p the semi-parameter; and y o, r = Co also. 2. The ellipse A2y2 + B2 2 A2B2, Bx, B2A2y _ A2B2xp' A2y' ~ A4y2 B2(A2y2 + B2x2) _B4 A4y3 A2y3 [L ~ 4 2 [A4y2 + B4X2] 1B4 - A4B4 A2y3 B2 At the extremity of the transverse axis x=A and y=O.. *. r= A2 and " " " conjugate " x=O anldy=B..'.r — 168 DIFFERENTIAL CALCULUS. 3. The logarithmic curve y ax. p' =loga.ax = Y, 1 d-y =_y where m = odulus. =m' m cdx m_ (1 _+_p2) [ Z]d [m:2 + y2] 1 + p12) - --- p" Y my W12 When y = 0, r = oo; and when y = a, r = o also. 4. The cubical parabola y3 = a2x. a2 3a2. 2y a2 2a4 2 3y2 S 9y4 3y2 9y5 [ _ L+ k _]1 (9y4 + a4)~ 2a4 6a4y 9y5 When y = 0, r o, and when y = 4- o, r o. 5. The cycloid, or curve generated by the motion of a point on the circumference of a circle, while the circle rolls on a straight line. Let the radius of the generating circle = a. Place the origin at V, the vertex of the cycloid. pj- ) Put VD = x, DP = y, the point P being that which de- A scribes the curve AP VB, while the circle rolls on the line ACB. Then PD = DF + F P- DF + EC since EP and CF are parallel. Also, since each point of the semi-circumference CFV has been in contact with the semi-base CA we must have arc CFV= CA and similarly arc EP = -EA = arc CF.. By subtraction CA -.EA = CFV -CF or CE = FV; and. PD =D)F+ FV. RADIUS OF CURVATURE. 169 But DF = /2ax - 2, and FV = a versinl- Hence the equation of the cycloid is y = 1/2ax —x2 + a. versin- ~ a 1 aa, a X- a / -X2 * *. p' -, - ^ X~ + 2a - -. 2 x X2 X a/2ax/2ax - x 2a(2a )V a a xa2 Vtx!ax - x2 L' ^J~la-~ P (2a)a v'~ - / - *.... = = - = 2 V/2a(2 - ), a a>/; X 2~CZ -- 22 or, r = 2 chord PE. 129. Prop. At the points of greatest and least curvature of any curve, the osculatory circle has contact of the third order. The condition which characterises these points, is that the differendr tial coefficient - shall reduce to zero, since r is a minimum when dx the curvature is greatest, and a maximum when it is least. But by the general formula for the radius of curvature, (1i + p'2) d3y r = (~,-, we have, by putting = p" 3(1 + p2) 2tpttP - p,"'(l - p2) dr 2 +. dx p=/2 - VP', ft2 pt=..... (1). d^ This is the value of the third differential at the points of greatest and least curvature, of any curve; and if it can be shown 170 DIFFERENTIAL CALCULUS. that the third differential coefficient in the osculatory circle has the same value, it will follow that the contact must be of the third order. I + p'2 But in the circle we have already found y -b =- -~, __ _'p"2 -p p"'(l + p'2) 3p'"2''dx P 1' (). which being identical with (1), the contact must be of the third order. 130. Prop. If two curves have contact of an even order, they will intersect at the point of contact; but if the order of their contact be odd, they will not intersect at that point. If Y F= x, and y = px, be the equations of the two curves, the difference of their ordinates corresponding to the abscissa x + h, will be expressed by Y _ d(Y dY I \(~\~(d2 Y dd2y (~h) Y1- Y1 (\d dx) 1 ) d 2\ dx2) 1.2 (d3T (\~)3~+ &C. + dx ~ dX3 1. 2.3 Now when the order of contact is even, the first term of this difference which does not reduce to zero, must contain an odd power of, h, and must therefore change sign with h, thus imparting a change of sign to Y - y,, in passing through the point x,y. He)nce the first curve will lie alternately above and below the second, intersecting it at the point x,y. But if the order of contact be odd, the first term in the difference will contain an even power of -i A, which will not change sign with h, and therefore there will be no intersection; the first curve lying entirely above or entirely below the second. Cor. The osculatory circle intersects the curve, except at the points of greatest and least curvature. For usually, the circle has contact of the second order-but at the RADIUS OF CURVATURE. 171 points of greatest and least curvature, the contact is of the third order. Cor. At those points of a curve where p" = 0, a P T straight line may have contact of the second order, and it will intersect the curve. If p"' = 0, also there will be no intersection unless p"" = 0, also. 131. Prop. To find a formula for the radius of curvature, when any quantity t, other than the abscissa x, is taken as the independent variable. To effect this object, we must substitute in the value of r, already dy d2y found, the values of p' -= -, and p" -d, given by the formula for changing the independent variable, viz.: dy d2y dx d2x dy dy _dt d2y dt2 dt dt2 dt = and dx dx dx' dX2 dx'3 dt dt3 We thus obtain dy~4 / dx2 dy2 \ d2y dx d2x dy dx2 1\dt2 dt t2 dt t dt d2y /dclx2 Cx3 I 2 dX2 \dtl / dt3 (dx2 dy2\- ds3 d- + di2 __ _____d_______ d2y dx dx dy d2y dx dx dy(1) dt2 dt dt2 dt d12 d d t2 dt Cor. If x be the independent variable, d = dx, ~dt dx ds3 d2x dxa and d-,.'. r = d- the common formula. dJ2 d2y dx2 172 DIFFERENTIAL CALCULUS. If y be the independendent variable, dy = d 1, and d2y 0. dt dy l dt ds3 dy3 * d2x dy2 ds ds If s be the independent variable, dt =-= 1, ^~ d2y dx d2x' y. (2)' ds2 ds d~2 ds dx2 dy2 ds2 But + -d- = ds2 1, which, being differentiated with respect to s, gives dx d2x dy d2y, d ~aand similarly r = ds' - ~ (5).. ds d2 ds ds2 dy d1sds dx d2 d2x' (4). ds2o zd d2x dy ds2 dy ds2 ds dx ds and similarly r (5).~ ds2 And, finally, by squaring the equation (2), and adding to the denominator of the second member, the square of (3), which is equal to zero, there results, by reduction, I 1 r2 - and.' r2y\ i / d2X\ l\ /2W 132. Prop. To obtain a formula for the radius of curvature of curves when referred to polar co-ordinates. RADIUS OF CURVATURE. 173 Adopting the variable angle 6 as the independent variable, denoting the radius vector by r, and the radius of curvature by _R, we have, from the formulae for the transformation of co-ordinates, dOd. dx r x = r cos, y r sin 6,. r sin. + cos d d't dr d d d' d — = r cos 8 + sin d d2x dr d2r - r sin + 2 cos 0 - - 2 sin cos 0 d^y' A d2A' Put d = P and -d =P and substitute in the general value dT dQ 2 of the radius of curvature. rclZ~ r~o 1~dy2 8 L dJ2 dO2 [r 2 + p2 (sinl2 + cos2)] d2y dx d2x dy (r2 + 2p,2 - rp) (sinl20 + cos20) dO2 dO dO2 dO (r2 + p12) 2 N3 r2 + 21_2-rP2 -.2 + 2p2 - r2 where N is the polar normal. EXAMPLES. 133. 1. The logarithmic Spiral r = a0. 0 r JO, e r 1 = log a = a= P2-_a a- 2 3 N_ 23 2 _ (1,2 -2 m21i2).*2 =2 r2(mn2 + 1) 72(1 + qn2) r2+- 2 m2 m2. The radius of curvature of the logarithmic spiral is always equal to the polar normal. 174 DIFFERENTIAL CALCULUS. 2. The spiral of Archimedes r = aO. P1 = a, P2 = 0.'. R- = (r + a ) r2 + 2a2 When r = 0, = —, and when r = oo, r = 3. The hyperbolic spiral r = a. a 7 22 a2 2r3 1i -- -2 = - *2 r4 R - a' -____ _ + 2),,4 2r4 a3 22 + 2 a2 a2 when r = 0, R 0, and whenr = o, R = o. 4. The lituus r28 = a2. 1 -- 3 3 3r5 -_- 2aO P1 = -- 2' P2-= aO 4a - 4a42+ (4a4 + r4)~ r6 3r6 2a2(4a4 - r4) 4a4 4a4 When = 0, r - oo and = —_o; when = —1, r=, / and R=aV/i whhen r= a -2or r4-4a4,R- c and when 9 = ao r =- 0 anil _ -- 0. 134. A curve may be characterized by an equation expressing a relation between the ra dius vector 7- and the perpendicular p from the pole upon the tangent. Thus the equation of the circle referred to the co-ordinates r and p is r =p, the pole being at the centre. That of the logarithmic spiral is r = cp, &c. RADIUS OF CURVATURE. 175 135. Prop. To obtain a formula for the radius of curvature of curves referred to the radius vector and the perpendicular upon the tangent. From the general value of the perpendicular when the curve is referred to the ordinary polar co-ordinates r and 8, viz.: (Art. 115.) r(2.dr2 r4 p ~- ~ we obtain - r = pl2 2-~ dr2 dO2 p2 V,2 + d,_ which, differentiated with respect to 8, gives dr d2r 4r3 dr 2r4 dp ld' d-2 - dO p3 d6 dr dO dO p2 ~dl ftdp. dp dr d.1 Substituting for dp its value c- * and divide by 2 de dr di d b d d2r 2r3 r4 dp dB — * —r.=3p2.' d92 p p dr'dr d2r Now substituting the values of and in that of R, we get - 2 (r2 p12), 2 2.2 +'2p12 - rp92 r4 \ 12r 3 4 d+p r dr dp dp dr Ex. The involute of the circle whose equation referred to p and r is p2 = r2 - a. dr p d R 1 r_ = ~-a2. dp-r' d r CHAPTER III. EVOLUTES AND INVOLUTES. 136. The curve which is the locus of the centres of all the osculatory circles applied to every point of a given curve, is called the evolute of that curve, the latter being termed the involuze of the former. 137. Prop. To determine the evolute of a given curve y — Fx. If in the formule for the co-ordinates of the centre of the osculatory circle, viz.: (Art. 127.) 1- + p,2 1 + p'2 a=zx-p'- -....(1) and b y+~....(2), we substitute the values of p' and p", derived from the equation of the curve y = Fx (3), we shall have the three equations (1), (2), and (3), involving the four variable quantities x, y, a, and b; and by eliminating x and y the result will be a general relation between a and b, the co-ordinates of the required evolute. This equation being independent of x and y will apply to every point in the desired curve. 1`8. In most cases the necessary elimina- tion is quite difficult; the following are com pa-ativPely simple examples. b 1. The evolute of the common parabola. A( Htere we have y2 = 2p'x.... (1)..p' -, and p" =- 11 Iy EVOLUTES AND INVOLUTES. 177 y2 -+-2 y3 y3 y3 = y Y 2Y 2 (3). y 2 2~2 _2 From (2) and (3) we get x - a3P and y =Y 3 b and these values substituted in (1) give p3 b 2 (a - p)3, the equation of the semi-cubical parabola, whose axis coincides with that of the given curve; the distance Aa between the vertices being = p the semi-parameter. 2. The ellipse o A2y2 B2x2 = A2B2.... (1). P — I ^ ^ ^ 1 + 12 _7 a4y2 + 2 _' xA2(A2B2 - A2y2) ~ B4\x3 xf3f or a - ~B2 =where f2 - A2 -B, y(A4,2 +_ Bx2) A4y32(22 2 y3f2 and 6-y A2B —--- -A2 B2 1B +2 6B2 +B -__ P; —3zY' ~a-B- ~ B' f f which values substituted in (1) give -2 _ 1 2 b3o a- A — A9- + - B2 A2B2 12 178 DIFFERENTIAL CALCULUS. or A + BB3b3 = (A2- _ 2), the equation of the required evolute. f2 f2 When a = 0 b = -+-; and when b = 0, a = - -. The curve consists of four branches presenting their convexities towards the axis, and tangent to each other as shown in the diagram. The equilateral hyperbola referred to its asymptotes. y = C2.... (1). C 2c2 1+ c4 + X4 p' x ID' 2' r~ll 3 - __ -- 1. C4 1 X4 C4 + X4..a = x + 2 = y+ 2c-b - 2x3 2c2X g jfte 3 ^ ^ r 7 3 * * a b = — + and a- b- = c — 2 ca 2aLx c2 C3 * ____ ______ t 3-b 3 /2~ and 3a + b - a~ —6-= -/x - Hence by multiplication (a + b)3- (a -b) =(4c). 139. Prop. Normals to the involute are tangents to the evolute. From the equation of the osculatory circle (x-a)2+(y-b)2 —r2, we get by differentiation x - a + p(y - b) =.... (1) a relation alike applicable to the y circle and the given curve, since p X, y and p' are the same in both. Now when we pass from a point x,y to another point on the circle, A aB the quantties x, y and p' must be x 0o EVOLUTES AND INVOLUTES. 179 considered variable, but a and b constant; but when we pass to a point on the curve, x, y, p', a, and b will all vary, and in both cases p" will be the same. The first supposition gives, by differentiating (1) with respect to x, I+ p2 + p "(y - ) =0... (2) and the second gives dard I -da'- d P"(Y -b) =. (3). - +'2 -.p +( Whence by combining (2) and (3) da + db dx dx- db dx db_ 1 da da p' dx Now A represents the tangent of the angle formed by the axis of x, with the tangent to the evolute AB at the point P1, and -tangent of the angle formed by the same axis with the normal PP1 to the involute LM at the point x,y; which normal passes through the point P1. Hence this normal not only passes through the point a,b, but it also coincides in direction with the tangent to the evolute at that point. 140. Prop. The difference of any Y R two radii of curvature is equal to the arc of the evolute intercepted between those radii. Resuming the equation R. (x - a) + (y - )2 = r2 and differentiating with respect to a, as an independent variable, we obtain 180 DIFFERENTIAL CALCULUS. (da ( l6a (Ca/ da But (x - a) +(y - + ( - b)) -a 0.- - But (-ddx ady d +dxO d(G/) dr'. x- a + (y - 5b) -- r; da da db 1 y_ - (- 1 db\ dr or since = -, =,) (I + - —' - (1). da p x~ a -t (l-a^ a Also, (x-.a)2 y b)2 (x- l \ )=2 r or (x - a)(l + )- ~ r. (2). Dividing (1) by (2), there results, ( +2 -, d dr ( dI" _ dd But (1 72)-2= hd where s is the arc of the evolute which terminates at the point a, b. ds ddr.. -= -,a' and ds = dr. dae da Thus it appears that the increment of s is always numerically equal to the increment of r. He-,lce s must always differ fiom r by a constant quantity, or we must have s -= c +- r; and similarly for the are sl, which terminates at the point ab, s- = c - r1,.'. s -s =-1- r, which result agrees with the enunciation. 141. In finding the evolutes of polar curves, it is usually most convenient to employ the relation between r and p, the radius vector and the perpendicular on the tangent; thus, let r - radius vector of the given clurve, p - the perpendicular on the tangent,, l- radius vector of' the evolute, Pi - the perpendicular on its tangent. EVOLUTES AND INVOLUTES. 181 Then since the radius of curvature T PP1 = R, at the point P, is tangent to the evolute at P1, the perpendicu- ular QTI, is parallel to the tangent;, ~ TP. Also QT is parallel to PP1.'.PT-QT=- // and PT = Q T1 = pl..l = R2 + r2 -2Rp,... (1). r2 =p2 + P12.. (2) dr Also R-=r —. (3) (Art. 135). And r =- p,....(4), the equation of the given curve. By eliminating r, 1p, and R, between (1), (2), (3), and (4), there will result a relation between r1 and pk which will be the equation of the required evolute. Ex. The logarithmic or equiangular spiral r = p.. (4). =.. -c, and R-c,.... (3). r p2 = — p2,. p.. (2)o 12 = R2 + r2 - 2op.... (1). Friom (1) and (3), rl2 = c2r2 + r2 2crp, which combined with (4) gives. 2 = C2 —2(1 + c2) - 2c22 = C2p2(C2 - 1).... (5). Fromn (2) and (4), C2p2 = p2 + p2,, or, p2(c2-1) =-p... (6). Then from (5) and (6), r12 = c2p2, or, r1 = Cpj, the equation of a similar and equal spiral. CHAPTER IV. CONSECUTIVE LINES AND CURVES. 142. If different values be successively assigned to the constants or parameters which enter into the equation of any curve, the several relations thus produced will represent as many distinct curves, differing from each other in form, or in position, or in both these particulars, but all belonging to the same class or family of curves. When the parameters are supposed to vary by indefinitely small increments, the curves are said to be conseculive. Thus let F(x,y, a)= 0,.... (1), be the equation of a curve, and let the parameter a take an increment h, converting (1), into F(x, y,a + h) = 0,.... (2), then if h be supposed indefinitely small, the curves (1) and (2) will be consecutive. Moreover, the curves (1) and (2) will usually intersect, and the positions of the points of intersection will vary with the value of h, becoming fixed and determinate when the curves are consecutive. 143. Prop. To determine the points of intersection of consecutive lines or curves. To effect this object, we must combine the equations F(x,y, a) 0,.... (1). and F( x,y,a + A)=0,.... (2). and then make h = 0 in the result. Expanding (2) as a fhnction of a + h by Taylor's Theorem, and observing that x, and y, being the same in (1) and (2), (since they CONSECUTIVE LINES AND CURVES. 183 refer to the points of intersection,) are to be considered constant in this development, we obtain da 1 +d2F(x, y, a) 12 F(x) y, a + h)-F(, yea) + d a. +- d. F&c,. = O. da~2 1. 2 But F(x, y, a) = 0, dF(,'y, a) h (2F(x, y, a) h2 + &c. -. da'1 d2'1&. 2 0. or, dividing by h, dF(, a) d&c, y a 0. I nda dA2 I.,2 F(x, y, )0. (3) And when h_0 this reduces to - ('.d... (3). da The two conditions (1) and (3), serve to determine the co-ordi. nates x and y, of the required points of intersection. 144. Ex. To determine the points of intersection of consecutive normals to any plane curve. The general equation of a normal is (y-yi)P'+ — 1 0.... (1). in which x1, yl, and p', are parameters, all of which vary together. Differentiating (1) with respect to xz, and observing that yj and p' are functions of xz, and that x and y are to be considered constant, we get (y - ylp"- p2 - 1 0,.... or, Y=Y-F +,2.... = P(l 2).(4). o.. -, +...(3). and.....^. =. ( The values (3) and (4) being identical with those of the co-ordinates of the centre of the osculatory circle, it follows that consecutive normals intersect at the centre of curvature. This principle is sometimes employed in determining the value of the radius of curvature. 184 DIFFERENTIAL CALCULUS. 145. Prop. The curve which is the locus of all the points of intersection of a series of consecutive curves touches each curve in the series. If we eliminate the parameter a between the two equations F(x,y, a) 0...(1) and d F(, y, ) (2), the resulting equation will be a relation between the general co-ordinates x and y of the points of intersection, independent of the particular curve whose parameter is a, or, in other words, the equation of the locus. Resolving (2) with respect to a the result may be written a= (x,y), and this substituted in (1) gives F[, y, (x,y)]-0..... (3), which will be the equation of the locus. Now if the differential coefficient t be the same whether dedx rived from (1) or (3), the two curves will have a common tangent at the point x,y, and therefore will be tangent to each other. Differentiating with respect to x, we obtain from (1) dF(x, y, a), dF(x, y, a) dy dx - dy rdx 0.... (4), dx cly,dx and from (3), dF [x, y,(x,y)]+dF[x, y, p (x,y) dy dx dy dx + dF[x, y, (x,) y)] [d ) = (5) d+ (x,y) dx But the first and second terms of (4) and (5) are identical, and the third term of (5) is equal to zero by (2). Hence the values of dy given by (4) and (5), and by (1) and (3), ENVELOPES. 185 are the sane, and consequently the two curves (1) and (3) are tangent to each other. 146. The curve (3) which touches each curve of the series, is called the envelope of the series. 147. 1. To determine the envelope of a series of equal circles whose centres lie in the same straight line. Assuming the line of centres as the axis of x, the equation of one of these circles will be of the form (x - a)2 + y2 0.... (1), in which a is the only variable parameter. X Differentiating with respect to a, we get -2x + 2a = O.... (2) From (2) a x, and this substituted in (1) gives y2 r2 = 0.. y = - r. This is the equation of two straight lines parallel to and equidistant from the axis of x, a result easily foreseen. 2. The envelope of a series of equal circles whose centres lie in the circumference of a given circle. Let x12 + y12 _ 2 = 0.. (1) be the equation of the fixed circle. (x - 1)2 + ( - yl)2 -r2 =.... (2) that of one of the moveable circles. The variable parameters are x1 and yl, the latter being a function of the former. dy, Ir d From (2) we have - 2(x - )-2(y — Y) -I- = 0... (3). Y d.y l d l _ x 1 But from (1) x, + y — - 0 or - - ^' ^i yi V V - 186 DIFFERENTIAL CALCULUS. This value in (3) gives - (x - x) + (y - - - l = 0. or — f + Yi =0, and. x. = —(4) or — + -- ~/ + 2/X2 - 2 -V2 + y2 Now combining (1), (2), and (4), so as to eliminate x1 and y1, we get, - -+= )- r2 = 0, ( 1' y )2 2(-t 2 r r( 2 2 Z )) or y.2 + y2 21(2 + Y) r2 r2. or I r = r X2 y~ - - = r... x2 + y2 = (r, ~ r)2. This is the equation of two concentric circles whose radii are i1 + r and r! - r respectively. 3. The curve which touches every chord connecting the extremities of conjugate diameters of an ellipse. Let Q1P1 and Q2P2 be conjugate dianeters of the ellipse ACBD, x1 and y, the co-ordinates of P1, c %2 and Y2 those of P. Put AO =- a, OC =b, tanl P OB = tan 1 = Y1 t, _ B ta P2 OB = tan 2 - Y2= t2 2b2 Then, since by the property of the ellipse, tt2 =- 2 a12 y2 2. b2x =- aC21y2 and x = - bS_ Also a2y/12 + b2x12 = a2Y22 - 2x2 a22 + a 412 2 2z2 -27/2 (a2y12 -'2X12). I2x ENVELOPES. 187 a2y22 brx a2y/Y2 aYl 1 and.'2 and x b2X 2 2 2.. The equation of the line P1P2 is bx1 Y-Y=x ~ -- x = a x y1yx aya or ( y+ y x (- )-ab = O....(1). Differentiating (1) with respect to x1 we get Y + a ( ab)yd -* 0....(2). - a \ d/x1 But a2yl2 + b2x12 a= c2b.. (3), and. d' 1 b2xdx1 -- a2yl1 Hence (2) can be reduced to a2y (y + a ) + b2X ( b) ay a T-~ a\ I I~\ -.... C(4+ ) Combining (1) and (4) we have 2 (a2y2 + b2) a_ bj(bx - ay) ab62 (bx - ay) and' Y~ a(bx -+ ay) 2 (a2y2 + b22) These values reduce (3) to the form a2b2 [(bx + ay)2 -+- (bx - ay)2] x2 + 2 -l, or + i' 1 4 (ay + b6x2)2 12 I," 2 2 the equation of n ellipse whose semi-axes are a\/ and b\/ and which is, therefbre, similar to the original ellipse. 4. The envelope of a series of lines drawn from every point in a parabola, and forming with the tangent angles equal to those included between the tangent and the axis. 188 DIFFERENTIAL CALCULUS. Let PD be one of the lines. Put DPT = PTD = O, AE -= x1 EP = y. Then PI:E =2b, and the equation of the line PD is y - y1 = tang 2d(x- )..... (1). 2 dy 22 tan dx dy p p But tang 1'2 a ='I; acnd since y2 =2px-, —, yi 1 1ta-26 ^/," - yi. " * Y- yi =^i __p2 (-) tan O = ~ 12 _p2 \ -2/ 2yy + p2 -- 2x, ad (X _i-p y 2p ~~2y' or YY11 - P2y + AY2Y - -pxTY = 0...... (.). Differentiating (2) with respect to yl, we find 2,?/y, + 2 -~ 2p)x = 0, and yj~ This value, substituted in (2), gives 4p2x2-4p3X +p:4 2px - p2 2px -p2 ~4 ~ -~_p2y+ p2 - 2 —— 2x = 0, 4y 2y / 2y or by reduction (2x - p)2 + (2y)2 0... (3). This can be satisfied only by making 2x -p = 0 and y = 0,. (3) represents a point whose co-ordinates are x = p and y = 0. Thus the lines will all pass through the focus; as might have been foreseen from the well-known property of the parabola. 5. From every point in the circumference of a circle, pairs of tangents are drawn to another circle; to find the curve which touches every chord connecting corresponding points of contact. ENVELOPES 189 Let Pi be a point on the first circle P1P2 and P1P3 a pair of tan- Pi gents, P2P3 one of the chords, 0 the origin at the centre of the seeond circle, x1y1 the co-ordinates of / P1, x2y2 those of P2, z3y3 those of P3. OP3 -=, CPl-=r1, OC:=a. Then y - Y2 2_ 2 3 ( - x).....(1) is the ecqation of the cbh-rd P2P3. Also y?,2+-x1x2-r2*......(2) the equation of the tangent P2P1 applied to the point P1. yy3+-1-xx=31r2... (3) the equation of the tangent P3P1 applied to the point P1.'hllen yi (Y2 - y3) + X1 (2 -X3) = 0, and 2 - Y3 X2 - 3 Y1 which reduces (1) to y — 2 - (x - x2) Yi or Y1 + xxl = Y1Yg + x12 = r2..... (4). Now differentiating (4) with respect to x,, we get yy + 0. But 12 + (- a)2 = r2.... (5). d11 dffl a — x *' Y1 -A +x — a = 0 and.y + x = 0, or yx - xy = ay.... (6). Combining (4) and (6) we have r2y - y ay nd r22x + ay2 1 = ~ 2x22 + y d y2 These values substituted in (5) give (+2J y2) (r4 -2a + a2X2)r - ( ~ ) 2 ~-=- r2 or rl2y2+ (l(r2 —a2)x2 +-2cr2x 24. (x' -_k y))2 Hence the curve required is always a conic section. It is a circle when a = 0, an ellipse when a < r, a parabola when a = rl and a hyp erbola when a > rl. CHAPTER V. SINGULAR POINTS OF CURVES. 148. Those points of a curve which enjoy some property not common to the other points, are called singular points. Such are multiple points, or those through which several branches of the curve pass; conjugate, or isolated points; cusps, or points at which two tangential branches terminate; points of infexion, &c. These will be examined successively..l7ulgtiple Points. 149. These are of two kinds, viz.: 1st. When two or more branches intersect in passing through a point, their several tangents at that point being inclined *to each other; and 2d. When the branches are tangent to each other, their rectilinear tangents being coincident. 150. Prop. To determine whether a given curve has multiple points of the first species. At such a point, there must be as many rectilinear tangents, and therefore as many different values of the differential coefficient d dx as there are intersecting branches. Let F(z,y) = O = u,..... (1), be the equation of the given curve, freed from radicals. MULTIPLE POINTS. 191 du Tn ddy d Then since p' — -= - and since differentiation never indx dit' dy troduces radicals where they do notexist in the expression differentiated, the value of p' above given cannot contain radicals, and theref;re cannot be susceptible of several values, unless it assumes the indeterminate form 0 0 Hen-ce the condition p' = will characterise the points sought. To discover whether such points exist, and if so, to find their posidu ciu tions. we form the partial differential coefficients and from dx dy the equation of the curve, then place their values equal to zero, and determine the corresponding values of x and y. If these values prove real, and satisfy (1), they may belong to a multiple point. We then determine the value of p' by the method 0 applicable to functions which assume the indeterminate form -, and if there be several real and unequal values of p', they will correspond to as many intersecting branches of the curve, passing through the point examined. EXAMPLES. 151. 1. To determine whether the curve x4 + 2ax2y - ay3 - 0, has multiple points of the first species. u - x4- +2ax2y - ay3 = 0,....(1). du du d = 4X3 + 4axy... (2). d= 2ax2 - ay2.... (3). - p 4X3 + 4axy 3ay2 -2a 2.. (4). 192 DIFFERENTIAL CALCULUS. Placing (2) and (3) equal to zero, we get (x2 +ay) 0...(5). and, 2x2 - 3y2 2 0..... (6). Combining (5) and (6) we have three x pairs of values for x and y, viz.: x-= 0, and y 0, or X=-ia a, and y = a, or, X =-a, and y=- a. The first pair of values will alone satisfy (1), and therefore the origin is the only point to be examined. Placing x 0, and y = 0, in (4), there results 12x2 + 4a?/ + 4axpa' 0 ( x = when 6(gp' - 4ax 0 w y he 0 or by substituting for numerator and denominator their differential coefficients, 2, 42 + Sap' +- 4axp" Scp' w h = 0 6 ap' + qayp" -a a ()t'2 - 4a y 0...'(6oap' 4c-a)= cap)', and consequently p' 0, or, p' + 2,,or p' = - Hence the origin is a triple point, the branches being inclined to the axis in angles whose tangents are O, + 2, and - /, respectively. The form of the curve is shown in the diagram. 2. The curve ay3 - y - ax3 = U..... (1). d: -. 3Y - 3ax2 = 0,.... (2). d 3ay -3 =-0.... (3). Froin (2) and (3), = O nd y =, or, = Ca, and y = —a. The first pair of values satisfies (1), but the second does not. Therefore the origin is the point to be examined. M ULTIPLE POINTS. 193 3x2y + 3ax2 6xy + 3x2p' + 6ax 0 = 0 Hence p' = ay2- 3 X= - yp-_ 2 -9 when 0 Say2 - x~ 6ayp' - 3x2 0 y = 0. 6y + 12xp' 3x2p" + 6a 6a when 0 (iC6a'2 + 6ayp" - 6x - ap22 y 0..'. = 1, and p' = 1. This being the only real value of p', there is but one branch passing through the origin, and therefore the curve has no multiple points. 3. The curve x4 - 2ay3 - 3a2y2 - 2a2x2 + a4 -0 =... (1). 4(3 - a2x) = 0.... (2). + -- 6(ay + a2y) =. (3). dx dy From (2) and (3) we get six pairs of values, viz.: x =- and y =, or, x0, and y =-a, 0o, x = a, and y 0, or, x = -a, and y = 0, or, x a, and y =-a, or, x =-a, and y =-a. But of these six pairs of values, the 2d, 3d, and 4th, are the only ones which satisfy (1), and therefore there are but three points to be examined. 2x3 -2CCx 6X2a2 4 ( x = a 4' ~ —~aen 3cp y + - 3a2y (Oay -1- 3a2)p' 3p whe, 2 x=0 and p' =,,, when y {y - a. ~". lp =t + (-) at the point where x = a and y = p' " 4 x = 0 and y = - a. Thus the curve has three double points. 13 194 DIFFERENTIAL CALCULUS. 152. Prop. To determine whether a given curve has multiple points of the second species. Here the mode of proceeding is similar to that in the last proposition, but the resulting values of p' prove equal although given by an equation of the second or higher degree. Ex. The curve x4 + X2 2 - 6ax2y + a2y2 = 0 = u. (1). du 4X 2 du d=4x3+2 xy2-12axZy-=.(2), -=2x2y-6ax2+-2a2y=O.(3). From (2) and (3) x = 0 and y = 0, and this is the only pair of values which will satisfy (1). Hence the origin is the only point to be examined. 12acxy-2xy2 —4x3 _12ay+ 12axp' — 2y2 -4xyp -12x2 0 2x2y _ 6ax2 + 2a2y 4xy + q- 2x2p'- 12ax + 2a2p' 2a2p": when x = 0 and y = 0... p2 _ 2 =0 and p'= - 0. 2a2 And the origin is a double point of the 2d species. 153. We may prove directly that at a double point of the 2d kind, the condition p' = - is always fulfilled. Thus suppose the two branches to have contact of the nth order. Then the first n differential coefficients will be the same for the two branches, but the (,a + 1) th differential coefficient wiil be different at the double point. Let P d + Q -- 0.... (1) be the result obtained by differentiating the given equation once, in which P and Q are functions of x and y, the original equation having been freed from radicals. By repeating the differentiation n times, we get d +l _ n dx+^v CONJUGATE OR ISOLATED POINTS. 195 in which P is the same as in (1), and Q1 is a function of, y, and the differential coefficients of the several orders less than (n -+ 1). Now the (ms + 1) th differential coefficient has, by supposition, two different values a and b for the same values of P and Q1..'. Pa + Q1 = 0, and b + Q1 = 0, and by subtraction P( - b) 0... = P-0 since a and b are unequal. This value of P substituted in (1) gives Q = 0. dy Q 0'dx P 0 Multiple points of the 2d species are characterized by having but one value (or rather two or more equal values) for d, but several d2y unequal values for d or some higher coefficient. Coanjugate or Isolated Points. 154. These are points whose co-ordinates satisfy the equation of a curve, but from which no branches proceed. Wihen p' assumes the imaginary form for real values of x and y, the corresponding point will be isolated, as the curve will then have no direction; and since iipaginary values occur only where radicals are introduced, the condition p' - will also hold true in such cases. The converse proposition, viz.: that at a conjugate point p' will be imaginary, is not always true; for if in the development ydF h d2y /2 d3y h3 Y1 = F(x -t- i) = y i I + d* 1 + ]3 dx 1 -dx2 1.2 dx3 1.2.3 any one of the differential coefficients should prove imaginary, y, would be imaginary also. 196 DIFFERENTIAL CALCULUS. To determine with certainty whether a point (a,b) is isolated, substitute successively a +- h and a - h for x, and if both values pf Yi prove imaginary (h being small), the point will be imaginary; otherwise it will not. 155. If the coefficient p' = dy be found to have multiple values, some being real and some imaginary, we may regard the result as indicating the indefinitely near approach of a conjugate point to a real branch of the curve. EXAMPLES. 156. 1. To determine whether the curve ay2 - X3 + 4a2 - 5a2, + 2a3 0 = z.... (1) has conjugate points. d1' dy - 32 + 8ax-5a2 O. ~ (2), = 2ay = O..., (3). 5 From (2) and (3), x = and y = 0, or x = a and y = 0. The first pair of values satisfies (1), and therefore the point (a,0) must be examined. E 3z ~-$( Sx - a2 Gx-x -8 I - -- when x a - 2cap p w yj =- 0...p2_-1, P =./. This result being imaginary, we conclude that the point examnined is isolated. 2. The curve (c2y - x )2 = (x - a)5 (x - b)6, in which a > b. b =: (Yc - X3)2 _ (X - a)5 (x - )5o = o..... (.1) -- 2C2 (Cy- 3) 0. 0 (3), dn,_ -6( --, )~5(r-a)4~)6~6(~-a)_5(*-b)5=0. (2). d"J"~ ~ el CONJUGATE POINTS. 197'The equations (2) and (3) give a3 3 x =a and y= ~, or x b and y,; C C both of which pairs of values satisfy (1), and therefore both require examination. 6._2 (c2y - X3) + 5 (x - a)4 (X -b )6 + 6 (x - )5 ( - )5 P1^~;^~2c2 (c2y - x3) 6X2 (c2y- 3) 5( - a)4(x - b)6 + 6(x - a)5(x - )5 2c (c2y — X3) + 2c2(x -a) (x-6)3 3__ 5(x^ - )^{xa)(x 35 3X2 5( —.)- a(-b)3+6G(x —a) (_-b)2 3b2 = 2 + ) 2: - 2 when x —b 0 2 -= when x -a. 62 Thusp' is real at both points. But if we substitute b ~ h for x in (1), and solve with respect to y, we get (b6~h-a6)3 1 Y - (b -- + W- a)-()3 0 2 ( both of which values of y are imaginary when h is taken less than a -b; so that the point where x=b and y =- is a conjugate point, although p' is rea l. This result is confirmed by forming the succeeding differential coefficients; thus p" - { 3x+* 52 (x-a) 2(x-b)3+15(x-a) 2(Xb)2 5 (- 6b. + 6(x-a)2(-)-] = when = b. 2 c This is a real value also.. But the next coefficient will contain the term 6(x-a)7 —6(b-a) which is imaginary, since a > b. The value x = a does not belong to a conjugate point, as is seen 198 DIFFERENTIAL CALCULUS. by substituting a ~ h for x in (1), and solving with respect to y, thus, (a h)3 1 ( h)i (a which is real when h > 0, but imaginary when h < 0. Cusps. 157. A cusp is that peculiar kind of double point of the second species at which two tangential branches terminate without passing through the point. Cusps are of two kinds, viz.: 1st. That in which the two' Y branches lie on different sides of the tangent, as in Fig. 1. 2d. That in which they lie o o X~ on the same side of the tangent, as in Fig. 2. The test of a cusp is that dy shall have two real and equal values dX at some point (ab), ahd that when we substitute a+h and a-h for x, we shall find, in one case, two real and unequal values of y, and in the other two i v imaginary values. The only exception to this is that offered by the case shown in P Fig. 3, where a cusp of the first kind occurs a at a point P, with the tangent parallel to the axis of y. It will then be more convenient to form the value of dx dX which should be - 0, and to try whether the successive substitution of b ~+ h and b - h for y will render x, in one case, real and double, and in the other imaginary. The condition p' 0= serves as a guide in selecting the points to be examined. CUSPS. 3 99 EXAMPLES. 158. 1. To deterTnine whether the curve (by - cx)2- (x -a) has a cusp, and if so, of which kind. u = (by - cx)2 - (x - a)5 - 0...... (1), d - 2c(by -c) -5( - a)4 = 0....... (2). dx 2b(by- x) - 0......(3). From (2) and (3) we obtain x = a, and y b and as these values satisfy (1), we must examine the point (a, b) 2c(by- cx) +5( - a)4 0 when a 21(ycx when ac 26(y - ex) 0 y=2bcp'- 2c2 - x 20(x-a)3 2b1c'-2c2 w'62p,- 2c 2 2bc when xc a, C ~2 C. 2 2 - 2bcp'=- 2, 2-2 -pi=P - - and p=- c b b2 b. There are two equal values of p', and consequently two tanac gential branches proceed from the point, a, Now put successively x - a + h, and x = a - h, and solve with respect to y. ea + ch t {- (+ hf, when x=-a+h, y- b a+c /+ h), two real and unequal values, ca-ch~ (- ~h)5 when x=a-h, y=- - ~ two imaginary values. Hence there is a cusp at the point a, b, and the tangent at that point is inclined to the axes of x and y. Again, the ordinate Y of the tangent corresponding to the abscissa c ac acc — cAh a + h, is + p'h = - which is greater than one of the corb b 200 DIFFERENTIAL CALCULUS. responding values of y, and less than the other. Therefore the branches lie on different sides of the tangent, and the cusp is of the first kind. Remark. The kind of cusp can usually be found very easily by examining the values of the second differential coefficient; for the deflection of the curve from the tangent is controlled by the sign of d2 I Hence, when the two values of this coefficient have contrary dx2 signs, the cusp will be of the first kind, but when the signs are alike, it will be of the second kind. 2. The semi-cubical parabola cy2 = x3 =y2-3... (1), =-3 0... (2), - 2cy0... (3)... x = 0, and y = 0, and as these satisfy (1) there may be a cusp at the origin. 3x2 (x 0 Y p' = ~==^~- ==^~ when x = 0.'2cy 2cp' 2cp' when p'2 = -0 and p' = + 0, 2c two real and equal values. Now put 0 - h for x in (1), and there will result, when x = 0 + h, y =- ~- two real and unequal values, x = 0- h, y =- - two imaginary values.. There is a cusp at the origin. Also the ordinate Y of the tangent corresponding to the abscissa 0 + h, is 0 + p'h = 0, which being intermediate in value between the two corresponding values of y, the cusp is of the first kind. 159. Sometimes it is more convenient to solve the equation with respect to y before differentiating. POINNTS OF INFLEXION. 201 Ex~. (y - b Cz2)2 = (X-a)y b + cx2 (x - a)', p' = 2cx ( - a). Now y has but one value b + ca2, or to speak more correctly, it has two equal values (6 - ca2: 0) when x =- a, and p' - 2ca ~ 0 has then two equal values also. When x a + h, y b + c(c + h)2 ~4 (+ h)7 two real and unequal values. " x=a-h, y= —+c( —h)2~( —h)" two imaginary values. Hence there is a cusp at the point (a, b + ca2). Iv Also p" 2 c 4-.~.-(x-a) 2 2c -0 P when x a. And since the two values of p" have the same sign, the cusp at the point (a, b + ca2) is of the second kind. The kind of cusp would also appear by comparing the ordinate Y of the tangent with the two values of y. For when x = a + h, Y = b + ca2 + p'h b + ca + 2cah, which is less than either value of y, when h is small. Points of Infiexion. 160. Points of inflexion or contrary flexure are those at which the curve changes the direction of its curvature, being successively convex and concave towards a fixed line as the axis of x. It has already been remarked that a curve is convex towards the cl2y t and when is eJative2y axis of x when d — is positive and concave when d is negative Hence a point of infilexion will be characterized by having the second differential coefficient affected with contrary signs, at points situated 202 DIFFERENTIAL CALCULUS. near to, but on different sides of the point in question. But since a variable quantity changes its sign only when its value passes through zero or infinity, the condition dY = 0 or dy = oo will belong to,dX2 dx2 a point of inflexion. But the converse is not necessarily true, for the sign of -2 does not always change after its value has reached 0 or oo. We must therefore see whether a change in the sign of d2y d- will or will not occur. dx2 We may also recognize a point of inflexion by the consideration that at such a point the tangent intersects the curve, and therefore the ordinate of the tangent will, on one side of the point be greater, and on the other less than the corresponding ordinate of the curve. EXAMPLES. 161. 1. The cubical parabola a2y - 3. x 3x2 6x y =- p' P" - = — =0 when x=0.. The origin is a point to be examined. Put x =0 + h, and y = y, x = 0-h, and y y2. Then d2y Oh Then c - > 0 o x dx2 a2 < 0 Hence the origin is a point of inflexion. The condition p" oo gives x =-, and therefore is not applicable. 162. Sometimes it happens that two of the peculiarities which chariacterize singular points occur at the same point of a curve. POINTS OF INFLEXION. 203 Ex. a2 - 2abx - 2y- x5 = 0 = u... (1), d _ _ —4abxy-5x4 = 0..... (2), dux du 2a3 - 2abx2.... (3). The equations (1), (2), and (3), _ __ are all satisfied by the values = O, y 0. 4abxy + 5x4 when P 2a-3y - 2ab y = 0. 4anby + 4abxp' + 20x3 0 ( = P - 2a3p' - 4abx 2a3p Y he0, 0. p2 - = 0, p'= -- O, and there is either a cusp or a double point at the origin, the axis of x being tangent to the curve. UP /h //~+ ahs If x = 0 + l, y = ~ +- _ — _ two real values, one h2 V 4 C greater and the other less than the ordinate (0) of the tangent. bh2 b'Vh - Coh5 If x - 0h, y =- -, two real values when h is small, but both greater than 0. Hence there is a double point of the second species at the origin, and one branch of the curve has an inflexion at that point. 163. In addition to the singular points already described, two other classes may be noticed, viz.: Stop Points. or those at which a single branch terminates abruptly; and Shooting Points, at which two or more branches terminate without being tangent to each other. Both are of rare occurrence, but the following are examples of curves belonging to these classes. 1. y x. log x. This curve has a stop point at the origin. For, y has but one value, and that is real when x > 0; but the 204 DIFFERENTIAL CALCULUS. value of y is impossible when x < 0, since negative quantities can not properly be regarded as having any logarithms. 2. y r-'ltan-1 -, or, y = ot-x. This cui'\ c has a shooting point at the origin, for d? 1 x tan- = tan-l(+ ) ) X dx I + x 2 o = 2r: = 1.5708 when xz + 0 =tan-l (-o ) =- - = - 1.5708 when x - 0, and whether x be positive or negative, y will have but one value. 164. WThen a curve has the spiral form, and is therefore more c veniently referred to polar co-ordinates, we may distinguish t e existence of a point of contrary flexure by the condition thlt 0= 0 at that point, and that it shall have contrary signs on different sides of that point. This we proceed to show. D P -: Q Q In Fig. 1, the curve is concave to the pole Q; and in Fig. ", it is convex. dp In the first case r and p increase together, and therefore ~ is posi. tive. In the second case, p diminishes as r increases, and therefore dP is negative. Hence, in'passing through a point of contrary dr d flexure, -w will change its sign, becoming equal to zero at that point, dlI ca for dr plainly could not become infinite, since p cannot exceed r. dr CIAPTER VI. CURVILINEAR ASYMPTOTES. 165. WThen two curves continually approach each other, and meet (nly at an infinite distance, each is said to be an asymptote to the other. 166. Prop. To determine the conditions necessary to render two curves asy mptotes to each other. Let the curves be referred to rec- B tLa gular axes, and let the ordinates /'P and EP', corresponding to the / salue abscissa; OE -- x, be express- c by means of the equations of c tha curves in terms of x. The E difference PP' -= y - y can then be expressed in terms of x, and if tbln' difference be reduced to zero by making x -= co, (being finite for all other.values of x,) the curves will be asymptotes to each other. This condition is fulfilled only when the difference (expanded into a series, contains none but negative powers of x, without an absolute terlm, for in such cases only will the difference y - y become zero when x = -. Hence we must be able to express y - y in the form Y - y = Ax-a + Bx-b + Cx- + &c., or the difference x - x of the two abscisse, corresponding to the same ordinate, must admit of being expressed in the form x1- Z - A~1-aL + Bly-bl +- ClY-C~ + &c. 206 DIFFERENTIAL CALCULUS. 167. Cor. If there be three curves, (A), (B), and (C), and if the difference of the corresponding ordinates of (A) and (B), and that of the ordinates of (A) and (C), be thus expressed. 2-~ 1= -Ax-a + Bx-ca+l) + C-(a+)..(1). 3J-;/1 = B1z-(a+l) + Cl\-(a+ + &Gc.... (2). V3 -- B-x(a~~) + Cpx-"+~ ~ &e,.... (2). the three curves will be asymptotes to each other, and, moreover, the curve (C) will lie nearer to (A) than (!3) does. For, by A making x sufficiently large, the term Ax-a, or - may be rendered,xa greater than the sum of the succeeding terms of (1), or greater than the sum of those terms increased by the series (2). 168. Cor. The curve whose equation can be written ill the form y -- D - Axa + Bxb + CxC + Al1-a + B1x-b, +- Clx-Ci + &c., can have an infinite number of curvilinear asy mptotes. For by taking any curve whose equation is of the form y3 = D + Axa + + Cx -6- C+ x + A2x-a2 + B2x-b2 +- &c. in which the absolute term D, and the terms involving the positive powers of x, are the same as in the given equation, the difference y - y will reduce to zero when x - c. 169. Prop. To find the general form of the expanded value of the ordinate in such curves as admit of a rectilinear asymptote. Since the equation of the rectilinear asymaptote has the form y = Alx + B1, the equation of the desired ctrve must take the form y = Aix + B + Ax-a + B- + B Cx- + c &c. 170. 1. The common hyperbola a2y2 b2; 2 = _ a2b2. b I b 1 1 y= -+ (2 - 2) - ~ (x - a2_x - ax —x - &c.) But y = x is the equation of two straight lines passing a CURVILINEAR ASYMPTOTES. 207 through the origin and equally inclined to the axis of x. Hence these lines are asymptotes to the hyperbola. 2. To determine whether the curve y - b(x 2 -a2) has either rectilinear or curvilinear asymptotes. By expansion y = b(x- + a2x-3 + &c.) = bx-l + c bCdx-3 + &c. But y = 0 is the equation of the axis of x. Hence that axis is a recti linear asymptote to the curve. To discover whether there is an asymptote parallel to the axis of y, let the equation be solved with respect to x; thus x _~ (a2 + b2y-2) ~ (a - 2 2-y —2 - &c.) Here it is evident that two lines parallel to the axis of y, and at distances therefrom equal to + a and - a respectively, will be asymptotes to the curve, their equations being x= +a and x=-a. The hyperbola whose equation (referred to its asymptotes) is xy = b will be a curvilinear asymptote, and there may be found any number of other curvilinear asymptotes. CHAPTER VII. TRACING OF CURVES. 171. In this chapter it is proposed to give such general directions as are necessary in tracing a curve fiom its given equation, and in discovering the chief peculiarities which characterize it. The following steps will be found useful: 1st. Hasving resolved the equation, if possible with respect to y, let diTferent positive values be assigned to x friom x = 0 to x c, and let those points be noticed particularly where y - 0, y -= c, or y _- an imaginary value. The first indicates an intersection with the axis of x, the second shows the existence of an infin-ite branch, and the third gives the limits of the curve in the direction of x positive. 2d. Assign to x all negative values from x 0 to x = -- oo, and observe the same peculiarities with respect to y as when x was positive. In both cases the negative as well as the positive values of y mIust be examined so as to include the branches below as well as those above the axis of x. 3d. Determline whether the curve has asymptotes, and determine their position. 4th. Find the value of the differential coefficient - and deterdx mine from thence the angles at which the curve cuts the axes, as well as the points at which the tangent is parallel to either axis. d2y 5th. F'rom'the value of - ascertain the direction of the curdx2 TRACING OF CURVES. 209 vature and the positions of the points of contrary flexure when they exist. 6th. Determine the positions and character of the other singular points, if there be such. EXAMPLES. 172. 1. Let the equation of the proposed curve be 2 3 - a3 x+ b Resolving with respect to y we have 3 - a3 1 = 4 / Y —' — V x + b and since each value of x gives two values of y numerically equal but having contrary signs, the curve must be divided symmetrically by the axis of x. If x be positive and numerically less than a, y will be imaginary, and there will be no point of the curve between the axis of y and a parallel thereto at a distance equal to a on the right of the origin. When x = a, y 0, when x > a, y is real, and continues so for all greater values of x, becoming infinite when x -= 0. If x be negative and numerically less than b, y is imaginary, and there is no point between the axis of y and a parallel thereto at the distance - b, on the left of the origin. When x -- b, y becomes infinite; and when x < - ), that is, negative and numerically greater that b, y becomes real and continues to increase without limit as the numerical value of x increases, bi!,g infinite when x = -o. Thus it appears that the curve has six infinite branches. Again, since x = - b makes y infinite, there is an asymptote parallel to the axis of y, and at a distance therefrom equal to - b. 14 210 DIFFERENTIAL CALCULUS. Also by resolving the given equation with respect to y, and expanding, we get ( a3 - a3)2 a3 2 b\ y= + — 1-) I + (x + b)" — ( - + — ~&c). -- (x — b+ terms involv\=2x8x / 2 9,X: ~x2' - ing powers of x). Hence y = -_(x - 6) is the equation of two straight lines, which are asymptotes to the curve, and are inclined to the axis of x at angles of 45o and 135~ respectively. If we combine this equation of these asymptotes with that of the curve, we shall find that each of the asymptotes intersects that branch of the curve which lies on the right of the axis of y. Forming the value of d from the equation of the curve, we have -dy/ 2x3 + 3bx2 + a3 2(X3 - a3)2 (x + b) which, placed equal to zero, gives the cubic equation 3 1 x3 + bx2 + - a3 = 0, in which there must be one real and negative root, since the absolute term is positive. The other two roots are imaginary, as is easily seen from the form of the equation. Thus there are two points corresponding to the same negative abscissa, one above and the other equally below the axis of x, at which the tangent is parallel to the axis of x. dy By making d- = c, Xwe get x a or x = - b. The first corresponds to a point at which the curve intersects the axis of x perpen. TRACING OF CURVES. 211 dicularly. The second belongs to the point of contact of one of the asymptotes as before seen. d2y By forming the value of dY, we should find that the curve is concave to the axis of x when x is positive, and convex when x is negative. The curve has neither multiple points, cusps, conjugate points, nor inflexions. 0 C A -4 Ca2X2 2. The curve whose equation is y3= a 2x - a When x = O, y = 0, and therefore the curve passes through the origin. When x = y =, when x= +, y=+ ), and when 2' x =- O3, y = o. Thus the curve has four infinite branches. When x = a, or x = - a, y = 0 corresponding to two intersee tions with the axis of x. Since x = renders y = ~ i, there is one asymptote whose 2uatn is equation is x a ~^ 212 DIFFERENTIAL CALCULUS. Also, by resolving with respect to y, and expanding, we get (1 - a1 2 2^/ = (1-3 - &C.) (I +' + &o,). 3 x2'6 2' = —— (x +, -- terms involving negative powers of z) * y =~~(- + ~) is the equation of a second asymptote. Forming the value of the differential coefficient, we have dx dy 6x4 - 2a2x2 - 4ax3 + 2a^x dx - 3(2x - a) (x a - a22) This expression becomes infinite when x =, when x = ~ a, and when x - 0. Hence the curve cuts the axis of x perpendicularly at the origin, and at distances therefrom -= + a and - a respectively. The value of Y becomes zero when 6x - 4a3 - 2a23 - -- 2aX -- 0, which corresponds to a value of x between 0 and - a. The corresponding value of y is a maximum. There are inflexions at the points where x -= a and x =- a, as will readily appear by substituting for x values alternately a little greater and somewhat less than a, and similarly for values greater and less than -- a. For if x be rather greater than a in the equation 24 _ a2X2 y3 =' - a, y will be positive; but if x be somewhat less than a, y will become negative. Thus the curve will cross the tangent at TRACING OF CURVES. 213 the point where it meets the axis. The same will be true when x= -a. There will be a third inflexion between x = 0 and x = - a, for the curve touches the axis of y at the origin, and a parallel asymptote at the distance a from that axis, and, therefore, must change the direction of its curvature between those two parallels. Finally by making the value of =0 we shall find that there is a cusp of the firlt k1ir at the origin. The form of the curve is represented in tha i m PART III. THEORY OF CURVED SURFACES. CHAPTER I. TANGENT AND NORMAL PLANES AND LINES. 173. The consideration of surfaces affords an application of the theory of functions of two independent variables. Thus if x, y, and z, be the co-ordinates of any point in the surface, and z = q(x,zy) the equation of the surface, the values of x and y may be assumed arbitrarily, and that of z will become deternminate. 174. Prop. To determine the general differential equation of a plane drawn tangent to any curved surface at a given point (xl, y1, z1) situated in the surface. Let the surface and plane be intersected by planes respectively parallel to xz and yz, and passing through the point (Xz, yi, z1). The equations of the line cut from the tangent plane by the plane parallel to xz will be of the forms x - x = t(-).... (1), and y = y.... (2), and those of the intersection parallel to yz will be of the forms y —y1 =-s(z-z).... (3) and x = x.... (4). Also the equation of the tangent plane, which contains these lines, will have the form (x - x,) + B(y - y) + C(z - 1) = O.... (5). The equation of its trace on xz is A (-x,) - C(z —zl) By.. (6). 64 4s 44" " yr " B(y-y)= - C(z-z1)+A4xl. (7). TANGENT PLANES TO CURVED SURFACES. 215 But the trace (6) is parallel to the intersection (1) (2), and the trace (7) is parallel to the intersection (3) (4). C C.. t= — and s=- which values reduce (5) to the form 1 1 - z (x - t X ( ) + - (y - Yi).... (8). t s Now since the intersections (1) (2) and (3) (4) are respectively tangent to the corresponding curves cut from the surface, we must dx1 dy1 1 dz I dzl have t = and s = or - = and - = - drz dz? t dx1 s dy Hence (8) reduces to Z-1 =-1 - (x - x1) +- (d y - Y). ~. (9), the desired equation. Cix dr dry The expressions - and d are the partial differential coeffidx1 dy1 cients derived from the equation of the surface, and they will have the same values at the point (xi,y1:z1), as the similar coefficients derived from the equation of the plane, tangent at that point. 175. Cor. If the equation of the surface be given under the form u = p(.x, y, Z,) = 0, the equation of the tangent plane will take a more symmetrical form. For we then have (Art. 57) Fclul du dl dz r ducl du du dz - = -+ = -o,0 and - = 0. ax W"dx chdx LT'/J dy as dy du du Hence - d =dx1 du dy, du dzl dz1 and by substitution in (9) and reduction, we obtain the more sym metrical form ( duz (y- du d = dyl z1dr 216 DIFFERENTIAL CALCULUS. 176. Prop. To determine the equations of a line normal to a curved surface at a given point (x1,yl,zl). The equations of a line passing through the point (x1,y,;zl), have the forms x - = t( - Z), y - =- s(Z - 1) and since the normal line is perpendicular to the tangent plane, we have by the conditions of perpendicularity of a line and plane (A = Ct and B = Cs), the following relations: du du A dz1 dx1 B dz dy1 0 dx= du C d-y d u dz, dz1 These conditions give for the equations of the normal line - +dz r du du z- + -d (z - z1) 0 d1(x - x ) d- ( 1) dx, ror Y - (4 - z) = O y - (Y = d (Z - z1) 177. Cor. If 01, 02, 03, be the angles formed by the normal with the axes of x, y, and z, respectively, or those formed by the tangent plane with the planes of yz, xz, and xy, we shall have dz, du A dx1 dxl cos0!- - - -- _ - - A B +B2+ C2 dz,2 dzl2 du2 d2 d l d2 + — I+1 + -+ +-+ cd/x2'dy12 dx12 dy1 dz12 dz1 du dyl dy1 cos O2 = ~- ~/3~ /dZ2 dz +d du + du2 V dx2 dy2 V dl dy12 dzu2 1 du 1 ____ dzI V d7 + dy + V d2 +dy2 + dz2 TANGENT PLANES TO CURVE SURFACES. 217 178. Prop. To determine the equations of a line drawn tangent to a curve of double curvature, at a given point (xoyl,xZ), on the curve. The curve will be given by the equations of its projections on two of the co-ordinate planes, as xz, and yz; thus F(x z) 0,.... (1). and ((y, )= 0.... (2). The equations of the required tangent will have the forms x — x 1-t(z-z1),.... (3). and y -y1= s(z —z1),....(4); and since the projections of the tangent are tangent to the projections of the curve, (3) and (4) will take the forms dz1 dz1 - 1 = (X- x1),. (5). and z- - 3(y - Y1),..(6). dzx drz in which equations the values of ~ and ~ are to be derived dx, dy, from (1) and (2), the equations of the given curve. 179. Prop. To determine the equation of a plane drawn through a given point of a curve of double curvature, and normal to the curve at that point. The equation of a plane passing through the point (x1,yl,zl), is of the form A(x - x1) + B(y - yl) + C(z - ) = 0.... (1). But, since the plane is to be perpendicular to the tangent line, we must have the conditions A = Ct=-C i and B = Cs -- Cd dz1' dzl' which values reduce (1) to the form ( -1) x + (y - Y) + (z - Z1) = 0, the required equation. 218 DIFFERENTIAL CALCULUS. EXAMPLES OF TANGENT PLANES TO SURFACES. 180. 1. The tangent plane to the sphere whose equation is u = x2 + y2 + z2_ 2 0. du du du Here 2 2 y, ~ 2 dx dy dz Therefore by substitution in the general differential equation of a tangent plane to a curved surface, we get d q- du du (X —) + (y-Y) +(z-z)- = 2x(x-x1)+2y1(y-y1) dxl dy1 dzj + 2z1(z - 1)= O... xx1 + YY1 + zzl = x12 + y2 + zl2 = 2, the required equation. x2 y2 z2 2. The ellipsoid u — +2 + — 1 =0. du 2x du 2y du 2z dx a2' dy b' dz c2. 2. - x( - Y_) + 2Y - zY) + 0. or, 21 + -b2 +~ — = 1, the required equation of the tangent plane. x2 y2 z2 3. The hyperboloid of one sheet u- - + 1 = 0. d 2x 2 du 2Y d 2z dx a2' dy b2 dz C2 2x, x) 2 2 _ b2y xx Yyl zz 4-4- - I 1 - 0, the equation of the tangent plane. a2 42 c2 - CYLINDRICAL SURFACES. 219 4. The conoid u = c2x2 + y2z2 -_ 2z2 = 0. d= 2c2x, - 2z2y, ~ - 2yz - 2r2z. dx dy dz 2c2 (x - x,) + z,2y(y - y,) + 2z(y2 - r2) (Z - ) = 0. or, C2xx1 + zl2yyl + (y12 - r2)zl = y12z2, the ecuation of the tangent plane. CHAPTER II. CYLINDRICAL SURFACES, CONICAL SURFACES. AND SURFACES OF REVOLUTION. 181. Prop. To determine the general differential equation of all cylindrical surfaces. These surfaces are generated by the motion of a straight line, which touches a fixed curve, and remains parallel to a fixed line in every position. Let the equations of the fixed curve or directrix be F(x, z) = 0,.... (1). F, (y, z)= 0,... (2), those of the generatrix, in one of its positions, being x = tz + a,.... (3). y = sz + b,.... (4). Since the generatrix continues parallel to a fixed line, the values of t and s will continue constant for all positions of the generatrix, but a and b will vary with its position. Eliminating x between (1) and (3), and y between (2) and (4), we get one relation between z and a, and a second between z and b. Then combining these equations to eliminate z, we obtain a relation between a and b, which may be written b = pa,.... (5). 220 DIFFERENTIAL CALCULUS. But from (3) and (4), a -= x -tz, and b = y- sz..(5) becomes y -sz = (x - tz),... (6). This is general equation of all cylindrical surfaces, but it contains the t:;lknown function p. To eliminate this function, differentiate (6) with respect to x and y successively, and divide the first result by the second; thus dz d(x - tz) dd(x tz) and 1 — s: X dy d(x -tz) dy dz dz dz dz -s- ~ —t dy dy dz dz whence t- + s.=.... (7), the required equation. 182. Cor. If we denote the primitive or integrated equation of a cylindrical surface by f(x, y, z) = = 0 the differential equation (7) may be reduced to a more symmetrical form. F'or since du du dz dx dz dy - and dx du dy du dz dz we obtain by substitution in (7) and reduction du du du t dy+ d =.... (8) a form often more convenient than (7). 183. Prop. To determine the equation of the cylindrical surface which envelops a given surface, and whose axis is parallel to a given line. The enveloping and enveloped surfaces being tangent to each CYLINDRICAL SURFACES. 221 other, will have a common tangent plane at every point in the curve of contact, and the equation of one of these planes will be z- z = (X - X) d + (Y - dzl du div du or (X - ) + (Y - Y1) + (z — ) — 0, in which x1 yI zi refer to a point of contact. Moreover the differen-.dz dz d du du du tial coefficients ) - or -7 - are the same whetner dedx- dy1 d1 dx- dy-' dz, rived from the equation of the cylinder or from that of the enveloped surface. Hence, if we form the differential coefficients from the equation of the given surface, and substitute their values in the differential equation of the cylinder, the result will characterize the points of contact, being the equation of a surface containing those points. This equation, when combined with that of the enveloped surface, will give the equations of the curve of contact, and thence the cylinder can be determined. 184. Ex. A sphere u = x2 + y2 + z2 -_ 2 = 0 is enveloped by a cylinder whose axis is parallel to the axis of z; to find the curve of contact. Here we have x = a the equation of the projection of the generatrix on xz, and y = b the equation of the projection of the generatrix on yz...t 0, s 0. du du du Also = 2x, =- 2y, = 2z. dx dy d 2 Hence by substitution in (8), 0.2x +0.2y +2z = or z =0, and the points of contact all lie in the plane of xy. Combining the equations x2 + y2 + z2 -.2 = 0 and = 0, there results x2 + y2 2_ 2 o0 222 DIFFERENTIAL CALCULUS...The curve of contact is a great circle of the sphere, as might have been foreseen. 185. Prop. If any surface of the second order be enveloped by a cylinder, the curve of contact will be an ellipse, hyperbola or parabola, or a variety of one of those curves. The general equation of surfaces of the second order is Az2+Bzy+ Cy2+)z+ z Ex- FxyG+ Gz+Hy+Zx+K=O=u.. (1). dzt du. d= z + 2Ex + Fy + Fz + I CB y + Fx + H, = By + 2Az + Dx + G. dtt du du -. +8- + - =t(DZ+2Ex+Fy+I) +s(Bz+2CyFx+ H) + (By+2 z + Dx + G) =0, which is the equation of a plane. Hence the points of contact are confined to one plane. But any section, by a plane, of the surface represented by the equation (1), will necessarily be a line of the second order, and therefore the truth of the proposition is apparent. Conical Sufaces. 186. Prop. To determine the general differential equation of all conical surfaces. These surfaces are generated by the motion of a straight line which touches constantly a fixed curve and passes through a fixed point. Let the equations of the directrix be'(.x,) = 0.... (1), F (yx) = 0...(2); those of the generatrix in one of its positions being x- a = t (z -c).... (3), and y- b = s(z -c).... (4), where a, b, and c, denote tlhe co-ordinates of the fixed point or vertex. CONICAL SURFACES. 223 The quantities t and s vary with the position of the generatrix, but a, b, and c, are constant. Eliminating' between (1) and (3), and y between (2) and (4), we get one relation between z and t, and a second between z and s. Then combining these equations to eliminate z, we obtain a relation betwee- t and s, which may be written s t....... (5). But from (3) and (4), t = - and s -= Y / - b ff.\~ - ^~i ^,, (5) becomes. = ~..... (6). This is an equation of conical surfaces, but it contains the unknown function qp. To eliminate this function, differentiate (6) with respect to x and y successively, and divide the first result by the second; thus y- b dz clp' ] d[] dp[] [ x a czlz (~-~)~' a x - d[ d[ ] [ L ( -a \"7 (z -c)2dx d[ dx [ ]'z-c'd and 1 y-b dz dp[_ d[] x [ cl r a d1 X X - - I z- ( -c)2 dy d[ ] dy d ] L (z- )2 yJ' in which expressios the [ ] is used to signify [1 Now by division dz dx (y-b) - c - -( -a) ddx dx dz dz z - {cy- b) d ~ (x - a)'. z-c-c (x a) + (Y - b) (7) the required equation. 187. Cor. If we denote the primitive or integrated equation of a conical surface by f (x, y, z) - u =- 0, the differential equation (7) may be reduced to a more symmetrical form. 224 DIFFERENTIAL CALCULUS, du du dz dx dz dy For since, and - dx du dy du dz dz we obtain by substitution in (7) and reduction dx du dx (x -a) + (y -b) - (z — c) =.... (8), a form often more convenient than (7). 188. Prop. To determine the equation of the conical surface which envelopes a given surface, and whose vertex is situated at a given point. dz dz du dxb du If we form the differential coefficients and -- or - anddx dy dx' ay dz Y dx fiom the equation of the given surface, and substitute their values in (7) or (8), the differential equation of the conical surface, the resulting relation will characterize the points of contact, being the equation of a surface which contains those points. This equation, combined with that of the enveloped surface, will give the equations of the curve of contact, and thence the cone can be determined. Ex. A sphere x2 + y2 + a2 - q.2 = 0 -= is enveloped by a cone whose vertex is situated on the axis of y, at a distance b from the origin; to find the curve of contact. H-ere we have the co-ordinates of the vertex a - 0, b = b, c = 0. du du du Also, 2, 2y, = 2z. dx dy dz. By substitution in the equation of conical surfaces (x - 0) 2x + (y - b) 2y + ( - 0) 2z = 0; or, x2 + y2 + Z2 _ by --. This being the equation of a sphere having a radius - -b, and its CONICAL SURFACES. 225 centre on the axis of y at a distance -b from the origin, the points of contact must lie in the surface of such a sphere. By combining the equations of the two spheres, we get r2 r2 by = r2 or y = -- and 2+ z2 = b (b2 - r2). IHence the curve of contact is a circle perpendicular to the axis of r2 y, and at a distance b from the origin. 189. Prop. If any surface of the second order be enveloped by a cone, the curve of contact will be an ellipse, hyperbola, or parabola, or a variety of one of these curves. The general equation of surfaces of the second order is Az2 + Bzy+ Cy2 + Dz x+E r2+F xy+ Gz+ HyA+ Ix+K=O =u..(1). d~ du -. d-Dz+Fy+2E+7I, d=Bz+~Fx+2Cy+H, dx dy d = By + Dx + 2Az + G. -(x-a) + (Y -b) J + ( - c) - =[Dz ++Fy t 2Ex+I] (x- a) + [Bz + Fx + 2Cy + H] (y-b) + [By + -Dx + 2Az + G] ( - c) = 0, or, 2LAz2 [ + Cy2 + Ex2] + 2[Bxy + DSz + Fxy] ~ [G - Da - Bb - 2Ac]z + [H - Fa - Be - 2Cb]y + [I- Fb - c - 2Ea] - [c + I-b + a I] 0... (2). By combining (1) and (2), we get [G + Da + Bb + 2Ac] z + [f + Fa + Bc + 2 C'] y + [I + Fb +- Dc + 2Ea] x + 2Ki 4- Gc -- JIb + Ia = 0. This is the equation of a plane, and therefore the curve of contact is the intersection of the given surface by a plane, and consequently an ellipse, hyperbola, or parabola. 15 226 DIFFERENTIAL CALCULUS. 190. Prop. To determine the general differential equation of all surfaces of revolution. Let x t a.... (1) I be the equations of the axis. y s + b.... (2) F(x,z) = 0.... (3), and F(y,z) = 0.... (4), those of the generatrix. The characteristic property of this surface is, that every plane section perpendicular to the axis is a circle. Now the equation of a plane perpendicular to the line (1) (2) is z + tx + sy = c, and the circle cut from the surface by this plane may be supposed situated on the surface of a sphere whose centre may be assumed at any point on the axis, and whose radius will be determined by the value of c, when the centre has been chosen. Take the centre of the sphere at the point (a, b, 0), where the axis pierces the plane of xy, and the equation of the sphere will be (x - a)2 + (y - b)2 + z2 r2 But r and c are mutually dependent upon each other, which fact may be indicated by the equation c = (r2(). Hence z + tx sy = p [(x - a)2 + (y - b)2 + 2].. (5) To eliminate the unknown function p, differentiate (5) with respect to y and x successively, and divide the first result by the second. dz d [ ] d[] d d d[] *' —t~y+s = ~q X - and - + t= X d[] dy dx d[] dx dz dz d +s y~ b -z- ~ dy T-i dy dz - dz + t x-a + z dx dx or (x-a-tz) — (y-b-~sz) +(x-a-)s-(y-)t=0... (6), which is the required equation of surfaces of revolution. SURFACES OF REVOLUTION.. 227 Cor. When the axis of revolution coincides with that of z, we have t=0 and s = 0, a = and b =0. dz dz..(6) reduces to d — Y d=....(7) dy dx 191. Prop. A given curved surface revolves about a fixed axis; to determine the surface which touches and envelopes the moveable surface in every position. The required surface will obviously be a surface of revolution, whose generatrix will be the curve of contact of that surface with one of the moveable surfaces. Hence if we determine the values of the differential coefficients d and A from the given surface, and substitute them in the gene1x dy ral differential equation of all surfaces of revolution, the result will characterize the points of contact, being the equation of a surface containing those points. This equation, combined with that of the given surface, will give the equations of the curve of contact or the required generatrix. 192. 1. A right cone with a circu- z lar base, whose vertex is at the origin, A and whose axis coincides originally with the axis of x, is caused to revolve about the axis of z: to deter- mine the form of the enveloping surface. Put the semi-angle A O C of the Lone = v and tan v = t. Then the equation of the cone, in the position A OB will be Z2 + y2 t2X2 or Z2 t22 y2.... (I) dz t2x dz y and --- dx z dy z 228 DIFFERENTIAL CALCULUS. which values substituted in the differential equation of surfaces of revolution, viz. dz dz xy t2xy - -Y -= 0, gives -- - = 0. dy dx x z..x-0 or y = 0. Combining the first of these results, x = 0, with the equation of the cone, we get +2 y2=0... z =0 and y, which conditions apply to the origin exclusively; but the second result y = 0, gives by combination with (1) 2 = t2x2 or z = 4 tx and y = 0, which are the equations of the lines OA and OB. Hence the required envelope is a double cone generated by the revolution of the lines OA and OB about OZ. 2. A sphere (x - a)2 (y - )2 1- 2 = r2 revolves about the axis of z; to find the enveloping surface. Here we have dz x- a dz? _ - d x z dy z cdz dz xy - bx xy - ay. _. X - + -- - 0. dy -dx z z. bx- ay = 0, the equlation of a plane passing through the axis of z, and the centre (a,b) of the sphere. This plane intersects the sphere in a great circle, whose equation, in its own plane, is (rl - a1)2 + Z2 = r2, in which r12 = -2 + y2, and a2 = a2 + 62. [(x2 + y2)- _ (a2 + b2)] + Z2 =. 1 1 or. x2 +?2 + 2 - 2(a2 + b2)2(2 + 2)2 = r2 - a2 - 62. the equation of the required surface. SURFACES OF REVOLUTION. 229 When a2 + b2 r2: this reduces to x2 + y2 + 2 - 2r(x2 + y2)=- 0; and when a. - 0, b = 0, x2 y2 z2 -r2 the equation of the sphere. 2 y2 z2 3. An ellipsoid + 2 + - 1, revolves about the axis of y; to determine the enveloping surface. The differential equation of the surface is, in this cas:: dy dy xz x -. dy b2x dy b2z Alsogdx a2 dz c-y b2x b2z.'. z — + x- -- = 0~. xz = 0, ay cy and consequently x = 0, or, z = 0. y2 z2 But when x = 0, +- - 1, an ellipse in the plane of yz. 2 2 x2 y2 And when z -: 0, + - 1, an ellipse in the plane of xy. Hence the required envelope consists of two ellipsoids of revolu. lution, whose equations are y2 x2 _ z2 y2 x2 + z2 +- =-1 and -+ 1. o" c" b~2 a" CHAPTERT I1I. CONSECUTIVE SURFACES AND ENVELOPES. 193. In the last chapter we have presented some examples of surfaces enveloping a series of other surfaces, but in the only case considered, the enveloped surface was supposed to be of invariable form, and its change of position was effected only by a revolution around a fixed axis. In that case, the enveloping surface was necessarily a surface of revolution. It is now proposed to consider the envelopes to any series of consecutive surfaces. 194. If different values be successively assigned to the constants or parameters which enter in the equation of any surface, the several relations thus produced, will represent as many distinct surfaces, differing from each other in form, or in position, or in both these particulars, but all belonging to the same class or family of surfaces. When the paramiieters are supposed to vary by infinitely small increments, the surfaces are said to be consecutive. Thus let F(x, y, r, a) = 0,.... (1), be the equation of a surface, and let the parameter a, take an increment h, converting (1), into F(x, y, z, a + h) 0,.... (2); then if h be supposed indefinitely small, the surfaces (1) and (2) will be consecutive. Moreover, the surfaces (1) and (2) will usually intersect, and their intersection will vary with the value of h, becomling fixed and determinate when the surfaces are consecutive. CONSECUTIVE SURFACES AND ENVELOPES. 231 195. Prop. To determine the equations of the intersection of consecutive surfaces. To effect this object, we must combine the equations F(x, y, z, a) = 0,... (1), and F(x, y, z, a + h) = 0,.... (2), and then make h = 0. By reasoning precisely as in the case of consecutive curves, (Art 143) we prove that the two conditions F(, y, z,a) 0,... (]), and dF(x, y,,a) _= 0 (3) must be satisfied at the same time. By conmbining these equations, so as to eliminate first y, and then x, we shall have the equations of the projections of the required in. tersection on xz, and yz. 196. Prop. The surface which is the locus of all the intersections of a, series of consecutive surfaces, touches each surface in the series. If we eliminate the parameter a between the two equations F(, y z, a)= 0,... (1), and LF(x y a) -. (2) the resulting equation will be a relation between the general co-ordinates x, y, z, of the points of the various intersections, independent of the particular curve whose parameter is a, or in other words, the equation of the locus. Resolving (2) with respect to a, the result may be written a -= p(x, y, z), and this substituted in (1) gives F[x, y, z, y, z ( )] - 0,.... (3), which will be the equation of the locus. Now differentiating both (1) and (3) first with respect to, and then with respect to y, we readily prove, precisely as in the case of 232 DIFFERENTIAL CALCULUS. dz dz consecutive curves, that the values of - and - are the same dxr dY whether derived from (1) or (3). Hence the two surfaces (1) and (8) will have a common tangent plane, and will therefore be mutually tangent to each other at all points common to those surfaces. 197. The surface (3), which touches each surface of the series, is called the envelope of the series. 198. Exr. To determine the envelope of a series of equal spheres whose centres lie in the same strlaight line. Assuming the line of centres as the axis of x, the equation of one of these spheres will be of the form (x- a)2 + y2 + - 2 _ 2 =0. (), in which a is the only variable parameter. Differentiating with respect to a we get - 2x + 2a = O.... (2). From (2) a - x, and this substituted in (1) gives y2 + z2 _ r2 = 0. This is the equation of a right cylinder with a circular base, the axis of which coincides with that of x. 199. When the equation of the proposed surface contains two parameters a, b, independent of each other, we must have the three conditions dF(x, y, z, a, b )'(x, y,,a, b) 0... (1), d y - 0.... (2). and - e in d be. A(3), And by eliminating a and b between (1), (2), and (3), the equation of the required envelope will be obtained. Also, if the proposed equation should contain three or more parameters a, b, c, &c., two of which, a and b, are arbitrary, and the others connected with them CONSECUTIVE SURFACES AND ENVELOPES. 233 by given relations, such relations will enable us to eliminate the additional parameters and to obtain a final equation between x, y, and z.; y z 200. 1. A plane whose equation is - - -f — 1, is touched in a - c every position by a surface, the variable parameters a, b, and cbeing connected by the relation abc =- m3: to determine the equation of the surface or envelope. x y z From - - + - - 1 - 0.... (1) we obtain by differentiation, a b c regarding a and b as independent, and c dependent upon them, x z dc y a dc 2 C= 0.... ), ancd -=.... (3) a2 c2 da b2 c db But the condition abc = n3.... (4) gives by differentiation dc dc bce +ab - 0, and ac + ab = 0. cia db cdc c de c.'.C = —, and --, da a db b which values substituted in (2) and (3) reduce them to the forms x -d Y + - =0, and -, b a2 c2 a b2 c2 b x z y z whence - =- and - _ -* a c b c These values in (1) give - + - =1, C C ~ 3z or -=1..'. c = 3a. c And similarly b = 3y, a = 3x. Finally by replacing a, b, and c, in (4), by their values just found, we obtain xyz = as the equation of the enveloping surface. 2. To find the envelope of all the spheres whose centres lie in the 234 DIFFERENTIAL CALCULUS. same plane, and whose radii are proportional to the distances of their centres firom a fixed point in that plane. Assuming the plane of the centres as that of xy, and the origin at the fixed point, the equation of one of the spheres will take the form (x — a)2 -+ (y - b)2 + z2 -_ -2 — 0.... (1), in which a, b, and r, are variable parameters, a and b being independent, and r connected with them by the relation r2 t2(a2 + b2).... (2) where t is a constant. Eliminating r between (1) and (2) we have (x - a)2 + (y )2 + t2(a2 + 62) =.... (3). Differentiating with respect to a and b successively, (x - a) -t2a =... (4), and - (y -b)- t2 O... (5). x Y. a -- and - t22-; which values in (3) give -1 ) (- 1 )- 2 (1- t2)2- ~1 - 12~~~1 z (X + y2)(t2 - 14) = 2 (1 _- t)2 or 2 +y 1- t2 This is the equation of a right cone with a circular base, its axis being coincident with that of z, and its vertex at the origin. CHAPTER IV. CURVATURE OF SURFACES. 201. Two surfaces are said to be tangent to each other when they have a common point, (x, y, z,) and a common tangent plane at that point. Let the equations of the two surfaces be F((X, Y Z,) =.... (1), and (, y, ) =0....(2). The analytical conditions necessary for a simple contact, or contact of the first order, are dZ dz dZ dz X x, = y, Z z, dX - dx' dY dy If the second differential coefficients, derived from the equations of the two surfaces be also equal, viz.: d2Z d2Z d2Z d2z d2Z d2z and dX2 dx- 2 d2 - dy2 dXdY ddY dxy' the contact is said to be of the second order. If the third differential coefficients be also equal, the contact is of the third order, &c. 202. In order to show that the contact will be moe intimate as the number of equal differential coefficients becomes greater, let the arbitrary increments h and k be given to the independent variables, X: x and Y = y, converting Z and z into Z1 and z,, we shall then have (Art. 82) cdZ h dZ k d2Z h2 d2Z hk d2Z k2 Z~.Z-X Y- -'-2 1. dXdY'- - &C. 1~7 1 1.2l~2 d~ddY I d Jr2 1. 2 236 DIFFERENTIAL CALCULUS. dc h clz k d2z h2 d2z hk d2z k2 Z+ = ~+ + -d 4- -~ —+ - &c.'dx' I dy 1 dx2 1.2" Cddy 1 dy2 1.2 and when Z z. r(l Z Cl h dZ dz-h d Z d+k 2 z C12Y zi- LdY 6Ldx2 — + -- - -x- I.2 Y+ &c. Z1 = N ~ IX Id + LdY dy1 L+ dX dxT2 " 1.' Now the value of this difference will depend (when h and k are very small), chiefly on the terms containing the lowest powers of h and k. If, therefore, the first differential coefficients, derived from the equations (A), (B), and (C), of three surfaces, at a common point, be equal, and if the second differential coefficients, derived from (A) and (B), be also equal, but those of (A) and (C) unequal, the surfaces (A) and (B) will separate more slowly, in departing fromi the common point than will the surfaces (A) and (C). 203. The order of closest possible contact between one surface entirely given, and another given only in species, will depend on the number of arbitrary parameters contained in the equation of the second surface. Thus a contact of the first order requires three conditions, and therefore there must be three arbitrary parameters. A contact of the second order requires six parameters; one of the third order, ten parameters, &c. Hence the plane, whose equation has three parameters, may have contact of the first order. The sphere cannot, except at particular points, have contact of the second order, since its equation hs but four parameters; but of two tangent spheres, one may have closer contact than the other. The ellipsoid, hyperboloid, and paraboloid, can each have contact of the second order. 204. Prop. To determine the radius of curvature of a normal section of a given surface at a given point. Assume the tangent plane at the given point as that of xy; the normal coinciding with the axis of z. CURVATURE OF SURFACES. 237 Let OX1 be the trace of the se- cant plane on that of xy, forming B with OX an angle 0. AOB the normal section, and P a point in that section. Put p OE —x, ED-=y, DP=z, OD=-x1 E~ The co-ordinates of the curve / AOB, estimated in its own plane, are x1 and z; and the general value X of the radius of curvature of a plane curve where x, and z are the coordinates, and any quantity t the independent variable, is (Art. 131.) rds 13 L _I d'z dx1 d2x1 dz dti2 dt dt2 dt which, applied to the present case, making t = x, and observing that cds dx~ dz at 0 O - 1 and - 0, reduces to dx dx dx Fdx112 dx2 ddz In this expression, the coefficient 2 has reference to those points ClX2 of the surface which lie in the curve A OB, and therefore it differs d2z from the partial differential coefficient d derived from the equation of the surface, which latter refers to the change in z produced by a change in x only, while y is constant. Let z -= cp(x,y) be the equation of the surface; then (Art. 55) Fd] dz dz dy dz dz t dy. -- -d —~ ~ d x= — + tan 6, since tan 6 in the preslt dy x casey x serlt case. 238 DIFFERENTIAL CALCULUS. Fd2Z d2z d2z d2z'LdJ - d —2 + 2 - tan + tan2. dx1 I Also d - =Hence by substitution in (1) and reduction, dlx cos d2 d2zz 12Z [P]. cos2 -4- 2 cos 8. sin 6 + - sin2i dx2 dxcy ady2 205. Prop. The sum of the curvatures of any two norinal sections of a curved surface, drawn through the same point of the suriace, and perpendicular to each other, is constant, those curvatures being measured by the reciprocals of the radii of curvature. Let 6 and 61 be the inclinations of the secant planes to the plane of xz;? and R1 the radii of curvature of the two sections at theii common point. Then, since the sections are perpendicular to each other, i = 2^ q + 8, and. cos = sin a, sin = -cos 6, and by formula [P] 1 d2 d2z d2z -= -2 *cos20 + 2 * cos 6 sin l +6 sin2. R Vx dxcldy dy2 1 d2z d2z d2z - - d- sin2 - 2 sin 6 cos 6 + - cos26. Ri dx2 cldxdy dy2 Hence by addition and reduction 1 1 d2z d2Z - + - - d + -= a constant for the same point. JLI JLI- dcX2 cly2 Cor. The normal sections of greatest and least curvature at any point of a curved surfiace, are perpendicular to each other. For since - + -? is constant, - will be greatest when - is leas t, and it i be east when is greatest. least, and it will be least when -is greatest. Ri CURVATURE OF SURFACES. 239 206. The sections of greatest and least curvature are called principal sections, and the corresponding radii are called principal radii. 207. Prop. To determine the principal radii of curvature at a giveni point of a curved surface. By differentiating - with respect to 6, as an independent variable, and placing the differential coefficient equal to zero, we get d R d2z d2,k -l - 2 ~ cos 6 sin 6 + 2 (cosO - sin28) ad ax2 dxciy dsz +2 sin 0 cos 6 = 0. dy2 d2Z d2z d2zl d2z. ot - - +oot La _ -.( Q)c clxcly Ldyy dx2J dxdy. From which we obtain two values of cot 0, viz.: C12z d2z 7- 2z /12 2 4 6X2 dy/2 ~~ / dy2 1dX2 + (T- 4 cot= -~ d2dxdy Substituting this value in the formula (P), which may be written thus 1 t- cot_2 d2z d2z d2z' - cot2 +'2 cot 6 + - adx"r2 ddxdy dy2 and denoting by R1 and R2 the least and greatest radii of curvature, there results R= ~2~ ~ -~ - - (I?). d^2 d'z /ri ~ d2rl 2 / 2z \2 /g ~ ~~l:~~.... ~ ~ (S). d2z d2z [d 2+ (Id 1(S). dx2 dy Ly2 dx2 dy 240 DIFFERENTIAL CALCULUS. 208. Prop. To express the radius of curvature of any normal section in terms of the principal radii R, and R2, and the angle ( formed by that section with the principal section of greatest curvature. If we make successively 0 = 0, and 8 = -r in [P] we obtain 1 1 R_, and R -, d2z' d2Z' clXz dy2 and these will be the values of R1 and R2, if the planes of xz and yz be supposed to coincide with those of greatest and least curvature. Thus we shall have, upon this supposition, d2z 1 d2z 1 and _ I dx2 =-? 23-, ~ d2z The same supposition/renders - 0, as a ppears when we put = 0 in (Q)O., -:.. ~. These conditions reduce (P), when 0 is replaced by q(, to the form pi __I__ pi1 -[... T]. 2, cos2 + - F sin2 [ the desired formula. 209. Po02p? If the two principal sections of a curved surface; at any point, have their concavities turned in the same direction, then every normal section through that point will be concave in the same direction. in the formula (T), the siglns of R1 and PR depend upon those of and'; and the si^ns of these coefficients indicate the directions of the curvature of the principal sections. In the case under consideration, the signs of R1 and R2 must be alike, and therefore if both be +, the sign of R will be + also; but if both be -, then the sign of P will likewise be negative. Froml which the truth of the proposition is apparent. CURVATURE OF SURFACES. 241 Nlt0. Cor. If R, and R2 be also equal, then R == R R-2 for every varlde of p, and every normal section, through the same point, will hbve the samne curvature. This occurs at the vertices of surfaces of revolution. 211. Prop. If one principal section of a surface be concave, and the other convex, it will be possible to select a value q1 for p, which shall render R infinite, or the section a straight line; also, between the values p = - -p and p -~ 1-u qp, the signs of R and Ri will be alike; but from g = q9 to p - -- p1, the signs of R and R2 will be alike. In the formula [T], suppose -1 negative, and it will become R2 cos09 - Ri sin2p in which transformed expression, the quanrtiies 1R1 and R2 are to be considered essentially positive. Now suppose (p so taken that R2 cos2p - R jsia-L -_ 0, a condition that will be fulfilled when P = 1 -- tanl[-l ]2 or ~i or, a Then pR = 2= Thus there are two sections corresponding to the argles P, and - qg which give straight lines. Also, if qg >- p1 and q < qpl; then _2 cos2 p — R1 sinl2'p > 0, and.'.. K < 0. But if qp > 1 and (p < 7 - 91, then 2 cos2qp - R, sinp9 < 0, and R > 0. Hence the surface may be divided into four parts by two planes, and if the first of these parts be supposed concave the second will be convex, the third concave and the fourth convex. 212. PIrop. To determine whether the principal radii at any point have the same or contrary signs, the co-ordinate planes not being coincident with the principal sections. 16 242 DIFFERENTIAL CALCULUS. The general values of R, and R2 may be reduced to the forms 2 R1= -. p" -F q +- (^ t+ ")2 - 4(p"q" - s"2) 2= _ 2 __ __ p p + q"/ - (p_ + q)2 - 4(p"q"/_ S!2) d2. d2,z d2 d in which p" = - q" =-, and s' - - - X2 - y2 dxcl@ and these values will have the same sign when p"q" - s"2 > 0, and contrary signs when p"'q" - s"2 < 0. 213. Prop. At every point of a curved surface, a paraboloid (either elliptical or hyperbolic) can be applied, with its vertex at that point, which shall have contact of the second order with the given surface. Assume the point of contact as the origin, the normal being taken as the axis of z, and the planes of xz and yz coincident with the principal sections of the surface. Take the normal as the axis of the paraboloid, its vertex being at the point of contact, and turn the paraboloid about its axis until its principal sections coincide with xz and yz. The equation of the paraboloid when in this position will be Ax2 _+ By2 = C, X2 Y2 which may be written z ~= 2- -4 2P- 2PC C where 2P — A and 2Pl -= B which represent the parameters of the principal sections, are entirely arbitrary. x2 y2 Take P= R,, and P1 = R2. Then z = 2- - - d2z I d2z 1 Hence = and - - -- dx2 R dy2 2 2 and therefore R1 and R2 are the principal radii of curvature of the paraboloid also. Then, for any other normal section of the parabo CURVATURE OF SURFACES. 243 loid, we shall have R- = s~ ~ 1R1 - the same value as that Jl sin2 4- ~ 12 cos2"p of the radius of curvature of the corresponding normal section of the surface. (Art. 208). Cor. It appears that when the principal sections of two tangent surfaces have contact of the second order, every other normal sections made by the same plane drawn through the same point will likewise have contact of the second order. 214. Prop. To determine the radius of curvature of an oblique section of a curved surface. Take the point of contact as the origin, and the tangent plane as that of xy. 2 A I/ /5 Let OX, be, the trace of the(secant plane on xy, caOb the section Let OX1 be the trace of the/secant plane on xy, a Ob the section of the surface by that plane, A OB the normal section by the plane Z O, R the radius of curvature of AOB at 0, r the radius of curvature of aOb at 0. Draw OZ1 perpendicular to OXr, in the plane a Ob, and refer that section to the rectangular axes OXr and OZ1. Put Od - xl, dp -zl X = angle between aOb and AOB, DD =, DE = v, OE = x. 244 DIFFERENTIAL CALCULUS. Then at the point 0 we shall have [ds,]2 ds 2 r = R = e d2 z1 d2z dx2 dx2 d2z d2z ds~ d1x ds But z = zos. cos.' Also = ~ d dx2- dx2 d dx dx.. r =R.cosX, and consequently radius of the oblique section = projection of the radius of the normal section, on the plane of the oblique section. This result is known as Meusnier's Theorem. Cor. If a sphere be described whose radius shall be identical with that of the normal section, and if through the tangent to that section any plane be drawn intersecting the sphere and the given surface, then will the small circle cut from the sphere be osculatory to the curve cut from the surface. Lines of Curvature. 215. If, through the consecutive points of any curve traced upon a given surface, normals to that surface be drawn, such consecutive normals will not usually lie in the same plane, and therefore will not intersect; but when the consecutive normals do intersect, the corresponding curves (which enjoy peculiar properties) are called lines of curvature. 216. P'rop. To determine the lines of curvature passing through any point on a curved surface. Let the equations of the normals passing through any point (X1, Y1, z1), be 2x-, 1+t(z-Z1)=0=P... (1) and y-y1+s(z —z)== Q....(2), and suppose the independent variables x and y to receive the incre. ments h and k. LINES OF CURVATURE. 245 Then the equations of the normal in the new position will be dP h clPdk P +'+ 1- + &'E. = O..... (3), dx1 1 dy1 and Q + +- + &c -0.... (4). If these two normals intersect, the equations (1), (2), (3), and (4), will apply to the point of intersection; and if the co-ordinates y,, and z of that point be eliminated between the four equations, the result will be a relation between the increments h and k and con. stants, it being observed that t = I and s = 1 are constant for dx1 dy dP dP the same point, and the same is true of dy, &c. This relation between h and lk implies a necessary relation between the new values of x and y, in order that an intersection of the normals may be possible; and when the normals are consecutive, A -- 0, and kl = 0 and - -d. Thus by omitting P and Q (each h - dxcl1 of which is equal to zero) in (3) and (4), then dividing by h, and finally making h = 0, those equations become P +dP dy (5), an dQ d Q dy. (6) dx+ dy, dx, dx1 dy, d1x 1 Or, by forming the values of the partial differential coefficients, dP dP dQ dQ d' dy' -d and dQ fronm (1) and (2), dx1 dyI dx I dy1' ^ d72zl cd12 d2z, cyl (1, dl d(1 + (+ I(Sz1) - *- =, (Z — 1) +( 1) r; — I dx+dy, dy- dcx( dxc1 dy12 dx dy1 dx1 J and by eliminating z - z, putting di p d =, i2zl d, z = i t d and ~ ^ ^, T-^ ~ -~ p~> ~d, x,dy9dx^ ^ dy3 dx^ ^ dyC 2' dxzdy,' 246 DIFFERENTIAL CALCULUS. we obtain [ S +"(l+ q2) -pq"2] -t dy- [p(1l +q') — q(1 + )'2)1 —S'( +'2) + pl'q'p" -O. (U). This is a quadratic equation, giving two values of dy the tangent of the angle between the axis of x and the projection of the tangent to the line of curvature passing through (xi Yl l), upon the plane of xy. Hence there will be two lines of curvature passing throngh each point'of the surface; and if p', q', &e., be replaced in ( U) by their general values derived fiom the equation of the surface, the result will be the differential equation of the projection of every pair of lines of curvature upon the plane of xy. 217. Prop. The lines of curvature at any point of a curved surface intersect each other at right angles, and they are respectively tangent to the sections of greatest and least curvature. If we suppose the plane of xy, (which in the last proposition was assumed arbitrarily) to coincide with the talngent plane at the point under consideration, we shall have p'- = - O, and q = d 0. cldx dy1 Hence the equation (U) may be reduced to the form dy p" — q" dyl - ~1i dx 2 3"- dr1 1 =. y) Hence if 61 and 62 denote two angles determined by the condition that tan 0 and tan i2 shall be the roots of this equation, we shall have, by the theory of equations, tan 6 tan 62 - -1, or 1 + tan l tan -2 = 0, which is the condition of perpendicularity of two lines in the plane of' xy forming angles t6 and 62 with the axis of x. Thus the tan. gents to the two lines of curvature intersect at right angles. LINES OF CURVATURE. 247 Cd?/12 218. Again, if we divide equation (V) by dl2 = tan26 and replace t by cot 6, the result will become identical in form with equation tan^ - (Q), which serves to determine the two angles formed by the principal sections with the plane of xZ, and hence the directions of the lines of curvature are tangent to the curves of principal section. 219. Propl. The consecutive normals to-a-sui —facetdrawn through points in the lines of curvature,,iintersect at the same points as the consecutive normals to the principal sections to which the lines of curvlature are tangent. Regarding the tangent plane at the given point of the surface as still coincident with that of xy, we shall have dz- d, dz ~ d = ad - 0 an d the equation (7), gives dx, dy I tan 6 Ixd2 dx1Cly dx!dyl dy11 Now if the plane of xz be supposed coincident with a principal section, these expressions will be still further simplified, since d2Z dx dy, will then be =-0; thus, 1 1 Z = d2 or z d'xz - d2Z1 d1l2 dy12 But these expressions are precisely the same as those previously found for the radii of curvature of the principal sections, and hence the centres of curvature of the principal sections must coincide with the points of intersection of consecutive normals to the surface thtough points in the lines of curvature. INTEGRAL CALCULUSO PART I, CHAPTER 1. FIRST PRINCIPLES. 1. The object of the Integral Calculus is to determine the function from which any proposed differential has been obtained. The process by which this is effected is called ittegration,, and is indicated by the sign f, the result being called the integral of the proposed differential. 2. Whenever the given differential can be reduced to a known form, we may return to the function by simply reversing the rules for differentiation. 3. Since d(a. Fx) -- a. d(Fx) -= calx, dCx we infer that f aF1x. dx = a f Fx. cx, that is, we may remove any constant factor from under the sign of integration, placing it as a factor exterior to that sign. Pa 1 4. Again sf F1.x. d - x.dx = -f a. F x.dx. Therefore we may introduce a constant factor under the integral sign, provided we write its reciprocal, as a factor, exterior to that sign. 5e To differentiate the algebraic sum of several functions, we differentiate each function separately, and take the algebraic sum of the ALGEBRAIC FUNCTIONS. 249 several differentials. Hence, in order to integrate the algebraic sum of several differentials, we have only to integrate the several terms successively. Thus f(adx + 6dy - ccd +- edv) = fadx + f dy — f cdz + f edv = ax- - by - cz- ev. 6. Again, since differentiation causes all constants connected with the variables by the signs -- and - to disappear, it follows, that in effecting an integration, we should always add a constant, in order to provide for that which may have disappeared by differentiation: thus we write Jadx ax + c, in which the value of c will be arbitrary, unless fixed by other conditions. Suppose, for example, that the general value of the integral is X, so that X = ax+ c; and that for a particular value x, of x, the integral assumes a known value X1: then X1 = ax, + c, and.. c = X -ax1. And this value substituted in the general integral, gives X= a(x- x) + X1. Integrcation of the Form (Fx)ndFx. 7. Prop. To integrate the form (2Fc)xdFx. Here we have f(Fx)" dF = 1 (n + 1)(F)'dFx n+1 = 1 x(F)+ z - c. The same process can obviously be applied, whenever the quan. tity exterior to the parenthesis, can be rendered the 6xact dif 250 INTEGRAL CALCULUS. ferential of that within, by the introduction or suppression of a constant. Hence we have the following rule for the integration of this form, viz.: Divide the given expression by the differential of the quantity within the ( ), then increase the exponent of the ( ) by unity, and finally, divide by the exponent thus increased. EXAMPLES. 8. 1. To integrate ax3dx. a ax4 f axzdx = aJ'x3dx = X f4x3dx = — + c. 2. To integrate 23 -+ ~. 3cx3dx. _1_ ~ 3c /2 nu3 x4f 3d. f(b2 + x,) x3. scxx *-. 2 (b2 + x4)2. 4x3dx = (b + X4) — 3. To integrate dy = (2a + 3bx)3dx. This may be integrated in two ways; thus y f(2a + 3bx)3dx =f(8a3 - 36a2bx -- 54ab2X2 + 27b3x3)dx -- f Sca x + f36a2bxdx + f54acb2x2dx + f27b3x3dx 27 = 8a3x - 18a2bx2. + 18alb2x3 + 7 b3x + c (1). Again y/ (2a + 3b)3dx b f 4(2a+3Ib)3. 3b (2a+36)4 - f42b (a -. - (2a-3bX)4cl, _- - + Sca3 18 + 188abc + 82x3 +' + x.... (1). 4 The formule (1) and (2) are identical. For if y1 denote the particular value of y when x =- 0, we shall have from (1) y2 =- c; 4a4 4a4 and from (2) y-= - + c,.' c -- + C,. 8b 60 ALGEBRAIC FUNCTIONS. 251 4. To integrate dy - 3(4bx2 - 2c3) 3 (4bx - 3cx2)dx Y = 2 J(4bX2 - 2cx3 (Sbx -- cx2)cz = (4bx2 -2cx3) + c. 2/ 8 9. In each of the preceding examples the proposed differential has neen brought to the required form, viz.: that in which the part exterior to the ( ) is the exact differential of that within, by introducing a constant factor. To ascertain when this is possible, take the last example, and denote by A the required unknown factor: then 1 1 Y = J'(4b2 - CX3)3 (2Ab - 9Acx2)dx, and if this be of the required form, we must have d(4bx2 - 2cx3) = (12Abx -- 9Acx2)cdx or Sbx - Gcx2 = 12Abx - 9Acx2, and since this condition must be satisfied without reference to the value of x, we must have, by the principle of indeterminate coefficients, the two separate conditions 8b = 2Ab.... (1) and - 6c - Ac.... (2). Sb 2 Sc 2 From (1) A 1- b = and from (2) A The values of A derived from (1) and (2) being identical, the pro. posed reduction is possible. The next example will illustrate the contrary case. I 1. dy = (4b2z + 3a2)4 (2b2 + 8axz)cdx. If possible, let A be the required factor. Then 1 1 y = -/ (4b2X +- 3ax2) (2b2A + SaAx)dx, and.. d(4b2x. + 3ax2) = (2b2A + 8aAx)dx, or 4b2 + 6-ax = 2b2A + 8aAx, 252 INTEGRAL CALCULUS. which gives the two separate conditions 4b2 b2A.... (1) and 6a = 8aA.... (2). W,2 6a 3 From (1) A =b 2 and from (2) A - -4 2b2 from (2) A 8- - 4 These values of A being different, the desired reduction is impossible. acdx 2. To integrate dy = x. a a (-3 a a fx-CX-y421JX — X 21 +c — - f21 3-c. ctdx 3. dy A - x 3 + 4c2x2 y = afx-1(3bx + 4c2x2) X = a~ (3bx-l + 4c2) -. 3bx-2d -2 (3bx_1 +- 4c2) + c = 2- (3bx - 4cX2)::- 3b 3bx axcx 4. cy = (2bx + x2) 3 ___ y = a (2bx + 42) 2. xcix = aS(bx-6 + 1) 2 (X2) 2. X(d = 2b/ (2b-1 1) +. -2. 2bcl (26-1 + ) c cF ~2bx + -x2 - aCx b L J2 b126x + x2 3x4(x3 - a3) 5. dy = -a dx. x-a y = 3 f4(x2 + ax + a2)dx = 3 (x + ax5 + a2Zt)dx X7 6 25) + CI APTER 1I. ELEMENTARY TRANSCENDENTAL FORMS. Loyarithmic Forms. 10. Prop. To integrate the forms ad and ad(),, adx padx a Since d(a log x) =. = a loo + c x a clx a. dFx F a. dFx Also since d(a. log Fx) =. a. lodg + c, EXAMPLES. adx II 1. To integrate dy = b c y c = log (b + c) + C =log [(b + cx) + C c b + ex c 8XVX 2. To integrate dy ^ ^ ^" a a — 2x4 v - 8 4-I =log(a + 2x4) + C=log(a + 2x4) - log c-log[c(a + 2x4)]. In this example the constant introduced by the integration is put into the form of a logarithm (which is always admissible) for the purpose of simplifying the form to which the integral is finally reduced. 254 INTEGRAL CALCULUS. 3. To integrate dy = 7 _ Sa - 3x - 7xdx 7 f- 6xdx y - f- 8 - - -'e/ s =8 - log(8a - 3X) + a logc - log (8a - 3x2)6 log (8a - 3x2) 6(.8,, b{x - a)dx 4. To integrate dy = ). _b (81x4 — 10Sx3a2 + 54.x224 - 12.;a6 + a8)clx C X3 b r 540 120 as or, y - 81x - 108a2 ~ - + x x J..... +-l cix = [1 X2 - 108a2x + 54a4 log x + - - 12- c L2 x 1 ^>vX Circular Forms. dx 12. Prop. To integrate the form dy - /a2 - b2X2 Taking the upper sign, we have'' /- 6X 2 d F! 2 dx - x b 2 -b2x2 ~ a bb2''2 6b2x2 U a2 1 a2 Let the quantity under the sign of integration be compared with dz the well known form d(sin —l) -, and it will be found identical therewith, provided we make - x z. dx But / ~z = sin-lz + c,. S. sin- 6+c -x z Jv - 622 a Y. = -sin — +- c. 5 a TRANSCENDENTAL FUNCTIONS. 255 -- COS — q- C. Similarly, since - = cos + c. z2 - dx I1 bx' "1 2 _ 2b 2 b a dx 13. Prop. To integrate the form dy = 2- 62 a2 + b2X2 Taking the upper sign, we have _+ _ - i-c. ~' y = f62 -b22 t- b2X22 a2 a2 Comparing the expression under the sign of integration with the well known form d(taz-l) z = +- they become identical by bx making - = Z. a b dx But a -- = tan-lz + c... j - -+ 2 = tan- - x + c. ^1 -1-+ I &Z^2 a J a 1 bx y. - tan-1 - + c. " ab a And similarly, since 1 +' = cot-lz - c. - dx 1 bx * Y. 6 y + /- = - cot-l- + c.'2 c- 2X2 - a a dx 14. Prop. To integrate the form dy = -- X. 62X2 (it2 Taking the upper sign, we have I - - d -x r / + dx a 1 a -a b, X2 b2x2 a- I /2 a bx 22 aV 2 aV Xa2 256 INTEGRA-L CALCULUS. Comparing the expression under the sign of integration with the dz known form d(sec-lz)= - they become identical by making bx a b d daz a bx But/ ~ = sec-,z + c..- =- sec- C+c. BZ] _2 a b /62,3 a JV -- a / a2 1 -'bx.. y= - sec -r c. a a And similarly, since f- - 7 =1 cosec"lz + c. -- Clx 1 bx.*''. y -=-~ _ cosec1 -- + c. Jx16%2 _62 a a Ca 15. Prop. To integrate the Iforrm dy? = + ~ /ax.2 ~ b2X2 Taking the upper sign, we have 26 dx 262 dx + | (2 1 a2 c/a^2./6- b62x /42x 4b, 2 b /42 4-,2 2 a 4. V 2 - a 2 dx 1 -8 (02)2a2 Comparing the expression under the sign of integration with the known for'm dc(versin~-z), they become identical by dz2 r.a 22 mnaking --- 2 But f l ~- versin-lz + c. a 2, 22x.'. I a versin ~ + c. TRANSCENDENTAL FUNCTIONS. 257 1 2 2x. *. y = - velrsil-~ 2 + C. b a - dzAnd similarly, since / z = coversinz.' ~ -dx I 2b2x.. y =_- / ~.==, coversin7l - + ec xv/a2x-b2X2 a2 EXAMPLES. xdx 16. 1. To integrate dy = - -Va2 62x4 2bx 2. To integrate dy = - - - d - + X6 I f3xdx I Y= s f +, = 3 tn-l(z3) + e. x 2dx 3. To integrate dy =- ~'2z-1 - 6, 2,_2. T i n r dxa t 2dy="x6ed 2.6x -6. 6x 2.6 - 6. 6 = 4 /6. versin-1(6x) c. 17. Since each of the trigonometrical functions can be expressed in terms of any other, all the circular forms must apply, whenever one is applicable. To illustrate this, take the example 1 x dx dy ='2- 4x3 17 258 INTEGRAL CALCULUS. x cll 2 1 2 _ -.3 2 f 1 2 3 2 1 3dx 1sinl2 13 2 31 or y = _- ~ = ~ S " 3 + C1 Again, x2dx 1 12X2 1. X~dvy = 12x~d, == - versin-l (4x3) + C.' y - - J 4X6 J 4x3 - (4x3)2 6 1 - 12X2dx 12.,d or, y 6~. ~-3 _ I (4 3) coversin- (4X3) + C3. xj' ~ 7r" -1 /T x ~ 1=4 -- / -3 1 - — 3 -- r 2 tz -1 - 1 W (l ) — -x3 -a -- 1c6, I 3 -333 I or y = co sec- xa + cc' ==^tan-x X-3 - I + fd, or Y cotai3 2 TBIGONOMETRICAL FORMS. 259.Triyonozetlricca Formns. 18. Prop. To integrate the forms sin xdx, cos xdx, sec2xdx, cosec2xdx, sec x tan xdx, and cosec x cot xdx. Since d(cosx)= -sinxdx,.'. sin xdx= -f-sin xd.= — cosx+c. d (sin x) = cos xdx,.'. /cos xdx = sin x + c. d(tan x)= sec2xx,.. fsec2xdx = tan x + c. d(cot X) — cose2xdx,.'. Scosec2xclx - cot x + c. d(sec x) sec x tan xdx,.'. fsec x tan xdx = sec x + c. d(cosecx) =-cosec xcot xdx,..fcosec xcot xdx= -cosec z - c. EXAMPLES. 19. 1. To integrate dy = 2 cos 3x. dx. y = 2 cos 3x. dx =- cos 3x. d(3x) = sin 3x + c. 0 2. dy = 5 sec2 (3). x2dx. 5 r 5/5 =f5 sec2 (x). x2dZ _ se2(X3) 3 x2-d= - /ec2 (X3) d(x3) = tan (X) + Co 3 3. dy = 6 sec 4x. tan 4x. dA y - / jsec 4x. tan 4x. d(4x) = - sec 4x + c. 4. dy = 2 sin (a + 3x)dx, 2 = - cos (a + 3x) + c. 5. dy = 2cosec2(2x). x dx. Y =- fcosec2(e ).V. x d = cot2 x - c. V i - c xc 260 INTEGRAL CALCULUS. 6. dy =2 coseo (nx). cot (2x). dx. 2 i* 2 y - cosec (nx) cot (nx). d (nx) - - cosec (nx) + e. Exponential Forms. 20. Prop. To integrate the form dy == axdx. Since dax = log a. axdx,.. fadx - l log a. axdx ax + c. log a EXAMPLES. 21. 1. To integrate dy = 3exdx, where e is the Naperian base. 3ex y=3f edxx = + cc 3ex + c. log e 2. dy ba3xdx, y = bfa3xdx =,-g. log a. a3d(3x) = 3log+ c. 3 logo a 3 log a c. 3. dy = menXdx. y = - en^d(Znx) = en + c. The cases which ha e now been considered. include all the elementary forms. CHAPTER III. RATIONAL FRACTIONS. 22. Having disposed of the simple and elementary forms, or such as admit of being brought to such by some veiy obvious process, we shall proceed to the consideration of more complicated expressions, endeavoring in each case to resolve them by a systematic process into one or more of the elementary forms. 23. The first form, in point of simplicity, which we shall have occasion to consider, is that of a rational algebraic fiaction, and in such expressions we may always regard the highest exponent of the variable in the numerator as less than the corresponding exponent in the denominator, since the fraction, when not given originally in that form, may be reduced by actual division, to a series of monomial terms and a fraction of the desired form. 24. Prop. To integrate the form bX11-l + cx -2.. + lzx + k dy = - dx. alx11 + blx-1- + Cl,2 —2..+ 11x +- k1 1st Case. When the denominator of the proposed fraction can be resolved into real and unequal factors of the first degree. To illustrate this case, take the example ax + c ax + c dy -2 d7x = a dx X2 + bx X(x + b) a:+(c A B Assume x - -+ - where A and B are unknown x" -\ ox x x + b' 262 INTEGRAL CALCULUS. constants whose values are to be determined by the condition that this assumed equality shall be verified. Reducing the terms of the second member to a common denominator, we have ax+ c A(x + b) Bx _ Ax + Ab + Bx x2 +- bx x2 + bx x2 + bx x2 + bx Hence ax + c = Ax + Ab - Bx; and since this condition is to be fulfilled without reference to the value of x, the principle of indeterminate coefficients will furnish the separate equations. c = Ab, and a = A + B. Thus we shall have two equations with which to detelrmine the values of the twAo constants A and B, Resolving them, we find C c ah c A - and B = a - A = a - - b b b Hence by substitution ax -+ c rA f B cdx ab- cb / dx x2+ bx dx =Jdx +J dx i+b Y J c + bx J x J x b bJ. b b'J ~+b Cl ab-c =b og x + log (x + b) +. As a second illustration take the following example dyi x2= dx.. 2 x- bx a A B Assume =- + x2 - bx x x + b Ta A(x +- b) Bx A A-+ Ab + Bx Th.en x- 2 + X2 + 6^x -^2 -+ b+x 2 h bX x2 + bx.. a = Ax -- Ab + Bx, and consequently by the principle of indeterminate coefficients a a a = Ab and 0 = A- B, whence A- = and B- A —-- b b RATIONAL FRACTIONS. 263 And by substitution /acdx r adx a c dx a f dx ~b b(x Jx+b) bJ bJ x+b = log x log (x + b) + logc = log (x ) - log [(x + b) ] + log c - log c[( b 1 x ( + 3 b 42)dl Ex. To integrate dy (2 - 4x-) d 4x - x3 Here the factors of the denominator are x, 2 + x, and 2 - x, and we therefore assume 2 + 3x- 4x2 A B C 4x-xX3 2 - x 2 -x 4A - Ax2 + 2Bx - Bx2 + 2 Cx + Cx2 4x - X3... - 3x - 4x2 =4 - A A2- 2Bx -- 2 x2 + 2Cx + CZ2, and by comparing the coefficients of the like powers of x, we have 2 4A, 3 =-2B + 2C, -4 =-A-B+- C. These conditions give 1 7 A=- B+C=, and B-C= 4-A= A A- 2 B = C- = -1. 22 2 I dx 5 dx dx e 2 2 2 -+ x - log x +. log (2 + x) + log (2 - ) + c. 25. A similar decomposition into partial fractions, each integrable by the logarithmic form, will be possible whenever the denominator 264 INTEGRAL CALCULUS. can be resolved into simple and unequal factors. For if the num, ber of such factors be n, each constant numerator, as A, B, C, &c., will be multiplied (in the reduction to a common denominator) by all the denominators except its own; and since each denominator contains only the first power of the variable x, it follows that there will appear in the numerator of the sum of the reduced fractions every power of x to the (n - 1)th power inclusive, and also an absolute term. Hence the number of equations formed by placing the absolute terms, and the coefficients of the like powers of x equal to each other, will be ai, and therefore just sufficient to determine the n constants A, B, C, &c. 26. When the factors of the denominator are not immediately apparent, we may place the denominator equal to zero, determine the roots x, X2, &c., of the equation so formed, if practicable, and employ the factors x - x,, x - x2, &c. (4 + 7x)dx Ex^. -dy 2x4 -- 4x - 10 Put 22 - 4x-10 0 or x2 2x- 5 _ 0. Then x = I -6, and the factors of the denominator are x-1 I +64 and x -- 1 — 6. I l (4 + 7x)dI7 f I (4 + 7.x)dx' * 2 -2 X - 2x - 5 2 J -- 1 - +-l /'(x - 1 - _ 1 IdX I Bdx -2 _ I + 6 2 x - 1-6. 4 + 7 -- Ax - A - A 6 — +B -B + -B whence 4 =-A-A -B-BB B and 7= A B, from which we deduce A 7 and B = -+ 2,/6 RATIONAL FRACTIONS. 265 7 _6_-11 dx 7_ ~+ 11f dx 4 * 6 46 J - I + f4/V 46J x - 1 -I 7- +-llog -g(-1+ +llog(I-l-)+c. 4^67 4/6 27. 2d Case. When the denominator of the proposed fraction contains equal factors of the first degree. a + bx + cx2 To illustrate, take the example dy =- 3 dx. ^x - I 3 If we attempt, as in the first case, to resolve this into three frac. tions having denominators of the first degree, by assuming a + bx +cx2 A B C (x h)3 - + Z 3 + x -q: - + there will result a + bx + CX2 (A + B + C) (x + 1)2, and... a(A+ B+ C)h2, b=(A+B+ C)21, and c(A +B+ C), b aC whence c a 2t, 1 h2 Thus the assumec condition would establish a necessary relation between the constants a, b, c, and A, where none such should exist, those constants being entirely arbitrary. It is easily seen that such a result might have been anticipated: for since + + A = -+, the proposed ex- I x + A + A'- pression.a + bx - Cx can only be reduced to this form when the (x + 1A)3 numerator is divisible by (x + h)2. Hence the decomposition of the proposed expression into three fractions of this form is not- usually possible, and when possible it is not necessary because the form of the fraction can be modified by reducing it to simpler terms. 266 INTEGRAL CALCULUS. But if we put x+h=z, we shall have dx=dz, and by substitution (a + bx + cx2)ZX [a + b(- A) + c(z ~- 2z + h2)],z (x + 7h)3. - -' + + dz a - bh + ch2 b - 2ch c L ( +A)3 + (X -/2 + )2 + J d. Hence the proposed fraction can be resolved into three fractioils having the forms A B C (X+ )3' ( )2 and h and since the same reasoning would apply if the number of'equal factors were greater, we may in general assume a + bx + cx2... + ixn_- A B. I (x + 1hx)n - ( + h) (x + h) "- x - h' the number of such fractions being n. EXAMPLES. 2- 3' 28. 1. To integrate dy = —-- 2dx. (x~- a)2 2 -3x A B Assume 2 A ~* (X - a)2 (x - a)2 x - a 2 - 3x A B(x - a) A + Bx - Ba (X - a) (x - Z a) - (Xx - -)2 -.*. 2- 3r =- A + Bx - Ba, whence 2 = A - Ba, and -3 B... B=- 3 and l =-2 -- Ba = 2- 3a, and consecluently di Pdx 1 y_(2 — 3a) f d_- 3 / - = (2 -33a) — -3 log (x —r)+c, - ( a)2'-a' a- x When the denominator contains both equal and unequal factors of the first degree, the two methods must be combined. RATIONAL FRACTIONS. 267 x2- 4 4-+ 3 2. dy = x2- x 9x dx Since x3 - 6x + 9x = x(x - 6 + 9) = x( - 3)2 we assume X2 - 4x 3 A B C A(x-3)2Bx Cx(x-3) + _ + x3-62+9 x (x-3)2 (x-3)- x3 - 6X2 + 9x.. 2 _ 4x + 3 = - ( - x + 9) + x+ C(x2 3w), whence 3 —9A -4 -- 6A + B-3C, and 1 A - C. A.A -, C =; and B-0. 3 0 I fdx 2 r dx 1 2 Y 1 log + x = dc + _ log (x - 3) + G - -3 x — o3 -3= 3 3 - log x + - log (x - 3)2 + loc log [cx(lo - 3)2] 1 = log [x(- 3)2]3. dx (X - 2)2(x + 3)2 A 1 A B C D Assaume+ + -F + (x 2)2(x + 3)2 = (- _ 2)2 x+-2 (z+3)2 (x+3) I. A(Xz3)2-+B(x-2) (x+3)2+ C(x-2)2+ -)(x-2)2(x+3), or i -1 A(x2+ 6x + 9) + B(x3 +- 42 - 3x- 18) + C(x2 - 4x - 4) + D(x3 - x2 - X + 12)..O = B -D, O=A+4B +C —D, O= 6A-3B-4C-8D, and I = 9A - 1S8B 4C + 12D. These equations give, by elimination, 1 2 1 2 A =- B =- C = and D = 2' 25 25 125 I_ d 2 clz +1 d x I2 f dx "25f(x-2)2 d 2x-2 251(x+3)215 + ^- ~~25(^2)- TX-^-1) 2 1 5 fx+3 1 2o 1 2 + - 2 (-2) 1 25log(x2) 5 (x + 3) ~ log(x+3)+C. 255x2) 25 125 268 INTEGRAL CALCULUS. 29. Case 9d. When the simple factors of the denominator are imaginary. These factors, which correspond to the imaginary roots of an equation, e: ter in pairs, and are of the forms x + a + b --, and x z a —b / — 1. alnd their product gives the real quadratic factor x2 _ 2ax + a2 + b2 (x ~ a)2 + 62 I-ence, if there be but one pair of simple imaginary factors, or a single quadratic factor, in the denominator, the corresponding partial fraction will be of the form in which the numerator (x ~ j1)2 + 62' must consist of two terms, one containing the first power of x, and the other an absolute term, because the denominator now contains the second power of x; and, therefore, if we introduced a constant only into the numerator, we should not provide for having the explolnent of the highest power of x, in the numerator, only one less than the corresponding power in the denominator, But when there are several equal quadratic factors, the denominator being of the form [(x ~ a)2 + b62]q the partial fractions will be of the forms Ax + B Cx + DE + F [(x ~ a)2 + b2] [(x ~ a)2 + b^2?'-l (x a)2 + b2 the number of such fractions being n. That such a decomposition is possible in all cases, will appear more clearly by the following illustration. Let the proposed fraction be cx5 + eX4 +- f23 + gx2 + hx + i [(xPut an)2 + b2]3 Put x 4 a = y, and Y2 + b2 = 2 RATIONAL FRACTIONS. 269 Then the fraction can be reduced successively to the following forms c(y p- a)5 + e(y T at)4 +f(y f Ca)3 + g(y T=F a)2 -- h(y il a)+ i 26 cy5 + ely4 +fiy3 + gly2 + hly + i1 z6 (cy + el)(z2 - 62)2 + (f1y + gl)(Z2 - 6,2) + /hy + i1 z6 [c(x ~a) + el](z4-2zb2+ b4) + [fl(x ~ a) + gl](z-62) + l(Z ~a) + i, z6 (cb4 -f12 + A)x + b4(e2+ a)c -flb2(g2 - a)+il Ala [(x -+ a)2 + b213 (- b 2 +fi)x - 2(2 ( e2 a) +i(g2 ~ a)c + x + c(e2 + a) L(X + a)2 + b62 (x ~ a)2 + b2' which is of the form Ax + B Cx + ) Ex + F [(x ~ 4 a)2 62]3 [(x -~ +a)2 + b2]2 (x+ ~ a)2 + 62 And a similar decomposition would evidently be possible, if there were n equal quadratic factors in the denominator. 30. It appears therefore, that when the denominator contains simple imaginary factors, the general form presented for integration, will be (Az + B)dx dy= [(x a) + b2]n where it may be any integer. Put x + a z, then (Az T Aa + B)dz (x2 + 6>b2)n y f zda (fB - T Aa)dz _ A _' Y ZJ 2+b2 -t n W2(, + b2)n 2( _-1 ) ((2 + b2))n-1' /_ Al (Z2 b2)~ 270 INTEGRAL CALCULUS. by making B T Aa= A1. Thus the proposed integral is found to depend on the more simple form, -( A2+d - It will now be shown that this latter can be caused to depend on the form 1( z2 -2)-l, in which the exponent of the parenthesis is diminished by unity. Thus we have dz (r2 b62)dz z2dz b 2cAz ("2 + 2)n-]- (z2 (2 + z2 + (2 + b2)n (Z b)n 2(z2 + b2) 2 (Z + b2) (1) +') -z __)- 6I')'-'(n - I Z g dz (ut ( 2 + b)-') (Z2 + b2)n-1 (z -+ b2)7 z 2dz i ~ c~ 1 z (',2 + b2)n- 2(-n 1)( 2(n- 1) (Z2+b2)n-' which value, substituted in (1), reduces it to the form z + 2n 3 _ cSi.ilar l y, 2b2(n - 1)(z2 + b2)-1 2b(-1)J (2+ b() -) Similarly, 1(2 + b2)l~- can be rendered dependent upon (-~ n2 &C., SO that eventually, the original integral will depend on the form / - t- -- an1 - 31. We infer, therefore, that the integration of a rational fraction can be effected whenever its denominator can be resolved into simple or quadratic factors, and that the integral will be expressed in the form of logarithms, powers, or circular arcs. RATIONAL FRACTIONS. 271 GENERAL EXAMPLES. 83. L dy' Since (X3- 1) (2 +- X + 1) (X -1), and x2 + x + = ( + +3) (x + -3), 1 Ax + _ C we assume = 2+ X + - X3-1 +2 - X+ x~ - -.. = Ax2 + Bx -Ax - B + Cx2 + Cz + C. whence 0= A + C, O = B + C-A and 1 = C- - B. 1 2 1 A=- B= —- and C-=. 3' 3 1 2),d Idx 0 -. dy x2+x+ 1 + -1 1 +1 1 and if we put x + -= Z, or x2 + + x, + x -— and'dx=c dz, 2 4 2 there w ill result 1 1d (d + 2,)dx I 1 1 )-2 ) > 2dz 2zdz 1 - V/ _-log ( x- 1) — 4e 1 1og(__l)1 3 log ( - 1)-6 log (2 + ) 1z tan- + c 1log (x 1) 2x + 1 =. Z1 (a- 1( + -x+ 1B) —-.ta- +/c +. x (a + bX2)2 272 INTEGRAL CALCULUS. I A( B5x + C Dx - E Assume - + + - x(a, J bx+2)2 x (a (+ bX2)2 a + bx2 1 A(a2 ~ 2ab+ 2 + b2X4! + Bx2+ CX + D(ax2 + bx4) + E(ax~ b3),. l=Aa2, O=2Aab+B-+Da, O=Ab2+Db) O=C+Ea, 0O=Eb,.'A I B= —- C= 0, D = D - -- O. 1 fdx b xdx b pxdcx HenTe, = - a — - a2 Jx a 6 2' a x a' (a + bx)2 a" 2 = log z + 2a a+ - 2a log (a + x) + c. 1 1 x2 ~a( log + c. -2 (ax + b2) 2a2 a +- b - x2d~x X4+ x2 - Put - - x2 - 2 - 0, and resolve with respect to A2. 1 3.2 _ _ 1, -or 2 _- 2, (4 _ — 2 _ = (X + ) (,.2 - 1) = (:2 - 2 ) (z - -1) and we may assume, ___ A + A B - T'en 4 + 4 + 2 + I ~ - x2 + T2;2- A( _X X2+2x'-2)+B(x3+x2+,2x~2)+ (x3-.x)~D(x2~1). O-A + B + C, 1 = -A + B + D 0 2A + 2B - C, O = -2A + 2B- D.. A-= B= 6 =0, - D A =~ --, B- -— C — -. I "Y dx I x =- logr (x + 1)6+ log(x - 1)6+ tan — + c RATIONAL FRACTIONS. 273 4. dy Since -- x6 — 6=(1 x3) (1- +3)=(1-x) ) (1 ++x ) (+) (1-x+2), put 1 A B Cx D Ex F 1_6 -1+ -- x 1 + 2 1 — + X2.'. 1=A(1-x- x-34- 3 5+ 4 — )+-(l+ + x2+ + 5) + C(x -- x2+x.x4 5) - (1-x + X3 —x4)+E(X+X'2 —x4-X5) +F(1 + x -x -- x4)..'. 1 = A + B + D + F, O = —+ B + C-D + -E -+ } O = A B-C + E, O =-A+ B +D — F, = A+ B + C —D-E- 0 = - A +B-C- E. 1 1 1 1 1 1. A = — B = C — D- = +,,E -6 F=. 6) 6 6 3 ~ In + 1d I d z, I (x + 12)d. 1 x (- 2)dx y' 6~ + 1 "x 6 i-~ 6 J 1 + z+ 6J 1 -- t+2 6 6 g12-1l+x+x2 1-2 ~ 1)23 X+2) +i 1 f(x2 - )dx Ij 3dx ]21 11-+x2 2J / 2 3 ( ) +4 >5 1t )+8 1 4 log (1-) -log (I -+2) og(+ + 2 - (- log (- 2) ~d1 21 18 1 A/o 1 + tan- _ + c. CHAPTER IV. IRRATIONAL FRACTIONS. 33. The differential form next to be considered is that which is still algebraic, but which involves irrational or surd quantities. As the general mode of treating such expressions is the same in principle, whether presented in the entire or fractional form, they will be considered in the latter, which is of very frequent occurrence, and which presents some difficulties peculiar to itself. 34. When an irrational fraction, which does not belong to one of the known elementary forms, is presented for integration, we endeavor to rationalize it, that is, to transform it into a rational form by suitable substitutions. The following are the principal cases in which this is possible. 35. Case 1st. When the fraction contains none but monomial terms. As an example to illustrate this case, suppose n, in, dy =- ax + zx Cl el alzcx + blxze Put = znmce or x = zl where I is any common multiple of the denominators n, m, c, and e. nl nl nll Then x = znlmce, or x' =zn where 1- is an integer since I is a multiple of n. mi ce el Similarly X = Zmlnce, xc -= zcnme, and xe znme. IRRATIONAL FRACTIONS. 275 Also dx = nmce. z~mce- dz. aznlmce + bZmlnce Hence dy =- (nmce. c ce-1ld), aIZclmlie + b6 Zelgnn which is a fraction entirely rational. I 1 2A - 3x6G Ex. To integrate dy = - 8 dx. 3x- + x 1 1 2 Assume x = z2: then x2 = XG, x2, =-, - z and dx 12z l. dz. 2z6 __ 32 1 24z - 36z5' d= - 3z8 q_9 12. 3- (24z8 - 72z7 + 216z6 - 648X5) d -+ 1908(z - 3z3 + 92 - 27z -81 - ) dz. o*. y = 12f(2 - 6z7 + 18Z6 - 54z5)dZ -' 1908J -3- 3 +r 9-2 27z h- 81 — d=12 l2z9 - _ 3Zs l 87 - 9Z6) LO 4. L; 22 12 3-1 18I,2_I L 4 36. 2d Case. When the surds which entet the given expression contain no quantity within the ( ) but one of the form (a —bx). As an example, take dy = (a + bx) +~ (a ~+ ) dl el (a + bx)c + h 276 INTEGRAL CALCULUS. Put a + bx -= zn"c wNhere the exponent of z is a multiple of all the denominators n, m, and c. nL1 m C 1 Then (a + bx)" = znmc, (a + bx) mc ( + bx)'C = Znm d n nm c-dz, and dx x ~n c-ldz, and by substitution Znlmc _+ Z n17l dy - b c nmc xnmcl-d, b(zc1 + h)7 which form is entirely rational. x 3dx 37. 1. To integrate dy (1 + 4x) Assume 1 + 4x =- 2. Then -2 - zd I1 z - = dx= zz 3 (Z63x 4 +3Z21), and (1 +4 z) (' +4 ) _ * O1 o ( z — 3z -- d-y = 1 ( —j l 3 + 4) ~2.0. cr -0dy = O l x 1~, - +x Put 1 A- x - z2. Then x = z- 1, dx = 2zdz; and 1 + x = 2zdz 2dz dA dz Y (. =log ] )log(z + 1)+ c, or y -- l (1+4 3(1 +4c. Pvut 1-1- z2. Then xz = ~ - dx- = 2zdz ancd /1 +- +g'. y = logz -I) -~log (z+ 1) -lc, oz - I, -v - I y lo zg i-l + c - og = /- - + C. IRRATIONAL FRACTIONS. 277 38. Case 3d. When the proposed fraction contains no surd except one of the form /a b V xa + bx 2 -= /~ x +_ x 2 When the last term is positive, assume aZ b a /~2+ ~ x- x2 =-z + x; then - +- x + $- 2 = z2 -+ 2z + x2. a - c2z2 2C2(a - bz + C2z2) 2c2z _- -- (2c2z -b)2 an~d 62bxL. a~-cz2\ c(a - b- + c2z2) and ~ca-+x+=-+-c)-xC — ccz+1-CZ 224-' The values of x, -j bx + c2x2, and dx, being all expressed rationally in terms of z, the proposed differential when transformed will also be rational. Again, if the term involving x2 in the surd be negative, denote by x1 and x2 the roots of the equation bx a bx a2 2 6x' bx -_-or 0; then bx - (x-xl)(x:-2) and therefore 2 + 2 - = (2 - ) (X - ). C C Now assume - 2 - x) - x) = (x - 1).. z. 2X2 +- X1Z2.'. x- x= (x - xl)z2, whence x = -+ Y, -2(xl - x z and A1/ + bx - - - cZ( -)- cz(zx-x1) (1+ 2)2 + z2 Hence the several expressions which enter into the proposed differential will be rational, and therefore that differential will become entirely rational. dx 39. 1. To integrate dy dx /1 + x + x2 278 INTEGRAL CALCULUS. Assulme 1/l + X 2 = Z + X; then 1 + + + X 2 = z2 + 2zX+'2 1-2 Z 2(1-+-2), ~ l - +^ X = -, d= — - dz, andl++ - - 2Z I ((2z - 1)2 2 - I f2~-~ ~(1-~ + 2Y= 2 -1 2(1 -+ z) 2 log(2z-)'. - J r - - D = - -x 1- -og - -)+ 1 1 = log ~ + - o l/ + - ( ~~ + c l - 2 _-1 + clog 2+1+ +a2-(2a +1) = lo [ + a2 ++ (2 + 1)] log 3 + c = log [Q2 / x +' 2 4- 2 +- 1] + Cl. dx 2. dy -/0+ ~ _ 2 Put I + x -- x2 = O, and find the roots xa and x2, thus I I I X 1 1 1 1 W n, Now assume 2 2 ~11^, 11~2- 11' o (xV —(-a) (2-a) z2 and = ~- Zo 2 1 1 2 1 ( 5 2 1+25 zdz 2 n /.z, d, - and 1f+-i-a'- 1+.? (1 + z2)2= - + Z2 I + ZX 2 05 z A, _ = f~2 (l)2 Z =21 z =- 2ta-=-2tan -- — = - 2 tan- + c. -2 -2 /5"-2 -22 IRRATIONAL FRACTIONS. 279 dx 3. dy = d Xa2 + b2x2 a2 Assume + x=z+x; then -b + x2 = z2 - 2zx + x2 a2 b2 -2 2 a- 2 b2(a2 52( b2z2) and 2 + 62z2 2b2z dx (2b2)2 dz, ad l 2bz 2b2_ 26z 2b2(a2+b2) / 2bd * Y J a2_ b2 2X a2+ )2z2 (26z)2 - J a2 b2Z2 I bdz 1 - bdz I a-bz - -~ ~- / -= -log + c. aa a —bz a a-b0 a a+ bz 1 a+bx- ^2 —2+b2 1 /2 ~ b2X2- a = log a+ b - + c =- log ~ — +_a c. a a-bx+~1-2/c b2 bX a bx 40, The other irrational forms which admit of being rationalized, are chiefly those belonging to the binomial class, which it is proposed to consider carefully in the next chapter. CHAPTER V. BINOMIAL DIFFERENTIALS. 41. Prop. To determine the conditions under which the general form inl?1 ri dy == x (a + bxnn) "dx, can be rendered rational. n'l nl If we put x=Zmn, there will result; xZ -zml xn = dzmm and dx _ zzn. + -l b n. dCy - Zrln(1 a ~ bznrln)r I2lzn2m-ICdz so that the form will be equally general if written thus dy = xm(a + b))Pdx..... (1), in which p is the only fractional exponent. Assume a + bxn = z, then =( x -- m+1 m —97G1l (Z a).cZz, m~I_ z n)~ and x"'dx =-, ~ +a nbn and by substitution in (1), -1 — m+l dy = -+ (Z -a) n z dz. nb n Hence, if + be a positive or negative integer, or zero, the nw+ quantity (Z —a) can be developed in the form of a series of momials (with a limited number of terms), a rational fraction, or BINOMIAL DIFFERENTIALS. 231 anity, and thus the value of cy can be rendered entirely rational. For, although p is a fractional exponent, the expression can be so transformed as to remove the fraction, by the method explained in the first case of irrational fractions. Again, since xzl(a -- bxn)P = xm+nP(cx-?" + b), if we put ax-, +- b, there will result, X +npll -n a a (Z -b) ~~i mn-+-n+l1,m+Ttp+l *a thi =-n (z r e a- ratio li we dy l ( w b) p oP.sdt. n And this can be readlily rationalizedl when m + p is a posltive or negative integer or zero. We conclude, therefore, that there are two cases in which it will be possible to rationalize the general binomial differential, viz.: 1st. When the exponent m of x exterior to the parenthesis, increased by unity, is exactly divisible by the exponent n of x within the ( ); or 2d. When the fraction thus formed, increased by the exponent p of the ( ) is an integer or zero. 42. These two relations are called the conditions of integrability of binomial differentials..43. 1. To integrate dy- x5(a + x2)3dx. gn + 1 Here m =- 5, n = 2,.'. = 3, an integer, and the expression can be rendered rational. 1 Put a + x2 = z,.. x =( - a)2, = ( - a)3 x5C —= (z-a)2cz, and dy= ( c-a)2. a (Z_ -2S 28)dz 2 2 282 INTEGRAL CALCULUS. e. *y=f(z3- + c2)z = 2 (L 3-I I + 1 3 6 3 3 aR4 &g, -- zTI a2Z3)Z =- 3 - 3 - ( + X) a + + a + (a + x. 2 + 2. dy d ~ = x-2(1 + x2) 2dx. 3 2(i + X2) Here m — 2 2, = 2 and 2 -3 rn- +1 1. e ~ = -, a fraction; m+ — 1 1 3 but n - p=- - - - 2, a negative integer, n' 2 2 and the expression can therefore be rendered rational by the second transformation. Put x-2 + 1 = z,.. Z(z-1) 4 (z - 1)2 ~-5. clx - (Z- 1)dz, and dy - ( -( 1)z dz. 1- 1 -- dy=-UZ clr4 —z 2dr or, dZ= 2 2Z 2. z / +1 IX-2 1 /1,'. ~ —C z +~= z -— (-+C ~~ ^-~2x)+c. CHAPTER VI. FORMULAE OF REDUCTION. 44. When a binomial differential satisfies either of the conditions of integrability, it is possible to transform it into a rational expression; but, instead of applying this process of rationalization directly, it is often more convenient to employ certain formzlac of redlctcion, which render the proposed integral dependent upon others of simpler form, or such as have been previously integrated. 45. Such formula are deduced by employing another known as the formula for inltegration byparts. Thus, since d(uv) -udv + vdu, we have fS dv = d uv - vdu... (1). By this formula, J'fudv, is made to depend upon fvdu, which latter integral may be more simple. 46. Prop. To obtain a formula for diminishing the exponent m of x, exterior to the ( ), in the general binomial form y =- fm (a + xn) F dx. Put (a.+ bx)pXn-x d- dv, and mn-nl i. (Then v (+ bxn)P+1 Then v = ~( ~-j' a(ndc = (mn q + 1)x'ldx. But'=.f'm-+(a +- bx n)Pxz-ldx = S- Zdv- uv - jvdu. ~X.....(C.+..1 m - ~1'x m-na + bx1)P- dx (2) ~ ~ v= m(+ ) -Fb( ~+ 1) — b 284 INTEGRAL CALCULUS. The formula (2), effects the object of diminishing the exponent m of x, but it increases by unity the exponent 1 of the ( ), and as this would ofteii Ihe an inconvenience, we must endeavor to modify (2). Now Jsxi('-a + bx)+ldx = f x^n~(a -+ bxn)p(a-+ bxn)(x = a xm-7l(a + bxn)Pdx-b If 67(a + bxdlx)Pd. xm-n+1(a + bxl)P-l *.y = Jfsx(a + bXn)Pdx = -' 1). b-(p + 1) ~ ~ ~ -fI xz-n(a + bx')Pdx - ~ fxM(ca + bx )Pdx, b -1) (+-) Transposing the last term to the first member and reducing, we have'p, m +- 1 n — 1 -1((o + bXn)p+l 1) fxm(a A- 6xn)P 9 n ~ -~1) ~ -n A-'- ) x -1 + bx )Pdx. n_(p+' )'- + )1 Hence y = fxm(a + bxn)Pdx xa —&+1(6 + bxa)r~1 -( - A- 1 ) f xn-l(a + bxr?)P dx. b(,p ~ )-a+ )1 47. By this formula, the proposed-integral is malde to depend upon anotlher of a similar form, but having the exponent m ~ nz of x, exterior to the (), less than the original exponent m, by n, the exponent of x within the ( ). 48. Prop. To obtain a formtula for dimilishing the exponent p of the ( ), in the general integral y- f=Sx(a + bxn)Pd'x. Silnce xm(a A-,bx2)Pdx = fx"'(a + bx.l+)P-l(a -- bx"-)d-x =a cJxcm(a + bxl)P-lC.dx + b J'x7n+^(C A- bx+)P-dxC; and since by applying formula (A) to the last integral, replacing m by (mA -n), and p by (p —1), we get fm4(^c A )d xn+l(a+- bxO)P- (mq — 1)c fI xc(a- bxn)P-ldx f xl+7a +\ bxn)P-ldx = -~~^~ -- ~ - ~'' ^'' ~~~~~~b~np + m). + J)~ FORMULA OF REDUCTION. 285'+fl(a + I x")p'-. m f am(a + bxr)cldx =- a fxnCL(a + bxn)P-clx +r-( 1(in + 1)a tp + -d- 1, xrxm(a + bzn)r-'dx xn+l((a + bx n)P + peia f xm(a + bX7L)P-ldx. (B). 1p + ~2 -+ I 49. By the use of this formula, the proposed integral is made to depend upon a similar integral, but having the exponent of the ( ) diminished by unity. 50. When the exponents mn and p are negative, and numerically large, it is generally convenient to increase them, so as to bring their values nearer to zero, and hence we require two additional formulm, on1e for increasing the exponent of the variable exterior to the ( ), and the other, for increasing the exponent of the ( ). 51. Pro.o To obtain a formtula for increasing the exponent - m, of the cpareli iesis in the general integral y f= X/ -(a + bxn)Pdx. From formrula (A), we obtain, by transposition and reduction, xm-?-}l(CT + abl)p+l -b(np -F) m - 1 )Jx"L(a + bx1)Pdx J:i'-'~(a"'n)Pclxza (ns - n - 1 ) Now rnaking' i - n1, or ono -= -,n- 1,. there results fx-'-l(a + bxn) Px X-xl~i+i(a + zbxl)P+1 - b(n2p - n - i^, + 1) fr-^,,+n (a. + bxl)Pdl a( -n + 1) or by omitting the subscript accents and reducing, y = x —(a + bx.)cPdx x2-"1+1 (a + bx' ) nl +b (- -{ p - n-i1) fx-m+n (a - bxdn)Pdxz (c). - a(? - l) 52. By the use of this formula the exponent - m of x exterior to the ( )is increased by n the exponent of x within the ( ). 286 INTEGRALt, CALCUI US. 53. Prop. To obtain a formula for increasing the exponent — p of the ( ) in the general integral y = fx.'(a + bxn)-Pdx. Fromn formula (B) we obtain, by transposition and reduction, x+,.- (a +bxz) d - (np- + - I )fx_ (a-+ bxn)Pdx Vfm(a+bxflIPl-dx = - -pna Now making p- 1 = -- p- or p =1 - p, there results Sxm( + bx')-.Pldx xm+l(a + bxn)-Pl+l + (2np -- - 2n - 1)/fxz(a + bxcn)-Pi+ldx -na(- p1 + 1) or by omitting accents and reducing y =- Jfm(a + bx+)-Pdx n-l+ (a -V bxn)-P+l + (np - m -27-1 ) S z'l(a-V bxn)-prlclx -a (5). na(- 1). *' 54. By the use of this formula, the exponent -p of the ( ) is increased by unity. Applit ctt(ion of Fornmulce (A), (B), (C), and (D). 55. 1. To integrate dy = where mn is an odd integer. Put m successively equal to 1, 3, 5, 7, &c., and apply (A). Thus, I xdx f- -:x2 l~ — in + C by the rule for powers. Aix 1 2 1 V cl I/ 3 =i _ n ~+ i i/ n by formula (A), in which n, 2, nc p which rn= 3, n = 2, and 1 -~ APPLICATIONS OF FORMTULA/. 287 =- _ ~-x/' x + - by formula (A), in Jy- 5 - 5 -X/i which n = 5, n = 2, and P -2 x6^ ^/, ~,,6 f xcl+ I,~/ =- 1 X6- /l Z2 x _ -6 / d; and generally x17 x- I 1~~ 1 2 1 /x Cn-cdx /_-_.- = _ - - x-1/X- 2 -+ ~ —-- = ==.' J^T~.^ m — "12 m ^^ Hence by substitution, f Xd _,1 1.6 14..l.2.4.6\ - t$1-x2 (5 3 5 X23 5+ C / -- 57 2 + 3.-57. 71.+ L^57) - 2+ and generally xdx 1 n (l._ — 1). (m~- 3)(m~-1) //_ 2 mI (t (1~-2)., ()n —-4)(7n-'2),1 2. 4. - ( - 1) ~ Q e * + l717577 (7~ ( )- x 2(,- + C) ] 2. To integrate?dy -, where n is an even integer. Put m2 = 0, 2, 4, 6, &c., and apply (A) thus A -7 sin-ix -+ C0 by one of the circular forms..X I ~ I6/" dx ~ = ~ 1- + ~x2+ -2 -12 f X by formula (A), in which mn = 2,; n 2, and p —. 288 INTEGRAL CALCULUS. pX4lx I ^ 1 - 3 f p 26.X, _ X - 4 1- x2 + (_ by formula (A), in which nm 4, n 2 = and p=and-<2 =-6XI -2 + and generally IzlX'_clX 1 r3n' -_ 1.,-a(X'm-2dX f dx I m-1~ / -2+ + - Jy-~2 122 2 m 1 j X2 Hence by substitution, _ z____ 1 1. x — +2x; In,~ I -lx sin-1x + U4..__ - - (6I x1. 7 (x++ I sin -s -nX + C & J. /l (?,s)'7, ~ o. \ "1.3 -' - 4 *~7 —— 4 6 8 (.),- 2)<.4.X 1.3.5.7. (-)( -1 )4 - ) + ) 2.4.... (m -2) s- s + C) 13. dy...-x-2(1 - 3 )(d -1). $ 1 Make r = - 2, p - -' and - 1, and apply (C) then V sin-l.j )!fl( + x)Td J ~x^l A..x 1. x11. (2 1) - T I+ j dx __ ____ T+ APPLICATIONS OF FORM.ULA.. 289 Now put 1 + x = z2, then x = z2 -1, dx = 2adz and l- + x= z. r dx f/)2zdz r f2dz -- dz _ d'* + + (Z2-1)2-)z J -z-1 J -1 Jz+ x — /1-+ x (Z _ — y — lo + x~1 + U. = -- + C=lO g l + z ~ +x 1,/ix-l+I -- 2 log x - 1 8 4. dy x= (~^ + dx - x -(a + bx)jdx. x Put m= — 1, n=1, and p, = - and apply (B); then f (a-p + x)kl ~x(a J.- h)x) fa (a + bx) dx x (a bx) + fx1 (a ~ bx)idx. 2+1 —1 2 Now put m = ~ 1, ^ 1, and p == - and apply (B) to the last integral; thus 2 z2 - Q 2zdz Now put a + bx — 2; then x =- dx = — and. v.f1(a +bz) 2 dx f4~- =_' a =-d Va - + y^+f 6^x =1-V xVa =-log + C — / log f a Z +a v -r bx + a/ and by substitution, 19 290 INTEGRAL CALTCULUS. 2 -. y. y (a + bx)2 + 2 (a b + +Clog-C + +. A~ 4 + -^ 5. dy= - = -(1 + 2x) 2dx. X(1 + 2x)" 3 Put m= —, n=l, p —, and apply (D), then dx(1 +x2( _) 2+i-( OX )-2 J ~ = ~70 V - - ~ fx-(1x + ) dx..(1 + (x) 1 -1) l But /frl(l + 2x) dv= log -- 2z 1 + C, by the last ex. 1 - +(1x ~ 1 ample. e= y - 2 + log 2- + Ci. -2 y =/iJ+2 a —i +l CHAPTER VII. LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 56. We shall now proceed to the integration of those forms which involve transcendental functions, beginning with the case of logarithmic functions. 57. Of the logarithmic forms, only a very limited number can be integrated, except by methods of approximation. The principal integrable forms will be examined. 58. Prop. To integrate the form dy r= X. logn. dx, in which X is a given algebraic function of x. Put Xdx = dv, and lognx - u, and apply the formula for integration by parts. Thus v = fXdx, and dzu n=. log"-lx. -, and since'fudv = uv - fuvdzu X. lX. og2x. dx = logl. fXdx - {. log-. J (Xdx). -( X or, by making f Xdx- = X1 Y X, lognx - n X. log-rx. dx. If, therefore, it be possible to integrate the algebraic form Xdz, the proposed integral will depend upon another of the slame general form, but having the exponent of the logarithm less by unity. Now put - 2dx = X2, and by a similar process, there will result log-lx. dx X2 lggn - (n - 1) f log-22. lXe X X 292 INTEGRAL CALCULUS. If n be a positive integer, the repeated application of this formula will cause the proposed integral to depend ultinately upont the alge. braic form - dx, provided we can integrate Xdx,, dx, 2 dx, sx'x/x x &c., obtaining in each an integral in the algebraic form. lofg x. dx Ex. To integrate dy- (1 o a.)d (1 + x)2 Here X - -)2 (1 + )-2 of f Xdx (1 - x)-d =_ =I -+ ~ = X2d I +x x 1 _ 2. f -~ f\i + x)o- x y dx - Also, n 1, and.'. y =- + But f - -2 f d- o log (- - (I + 0) -+ C y. - - -l lo x-log( (+ )+ - - 1ogx-log(l -x)+. 1+x in j~x 59. Prop. To integrate the form y -- xWi. log"x. dx, in which n is a positive integer. Put x r:Xn; then XY =: Xdxc = Sxdx - ~- - n -+ 1 And, theirefore, by the last proposition, xm+l ~ 1 Y?/ fX /. logx. dx - logx' -~- ~ f xm log0-lx, dx lf~ ~1 - and similarly, m~ +1jg ____ f1x'nlo~g,-2xdx~ fx"\logn-2 xdx -- loggn-2x -- f.I x\logn-3ox.dx. &c. &c. ~ &Ce Hence by successive substitutions, LOGARITHMIC FUNCTIONS. 293 n+ 6 n(n- 1) y= fxmlognxdx = [~g - + 1ogt - + ( - )g 2 n( -1 )( -2) - ( +;;~ i log- -x+ &o. ({+ 1)3 n.n-) (-2 -3) + C.2.1 Cor. This formula ceases to be applicable when m =- 1, as the terms become infinite; but we then have f x-logx. dx 7-flogxz. - f logx. d(log x) = lg+ + C., + —Ez. To integrate dy -. log3x. cx. He1re ma = 3,? _ 3, 4 - 1 4 =, n - I = 2, ~-2 = 1. Here m —. n3 and-4 n- =n 5[23.2. 1 2 y =. log 5 2 log 4-x 5. 4lo. 2o: -t- 5logx 5.. 2; 43 2 24og -F. 4 3. 2.x. 25 2 Remaric. If we suppose n to be a positive fraction, the same formula will apply, but the series will not terminate. xmdx 60. Pqrop. To integrate the form dy = ~-, in which n is a posilogq 54o tive integer. 1 dx Put fxlnfl = tu and si, - =a dv, then =logld x du = (m + I-)xmdx, and v = l, o- I l'' - ( )log'-xx 294 INTEGRAL CALCULUS. Applying the formula fudcu uv - f vdclt we obtain /,xcl'dx x mn-+l — 1,. x7. o log2'x (n - 1) logl-lx + lon^ f* x~dx x,,*)+ m +_1 x. dx (o- - - 2) 1nogn-2x a2 _- 2 JloO'-2x XL'n'61,.:/ $q- i M, -C 1,12 V Xm. J.o - - (^ 3) - -; 2- log &- & c& x xmd x" + l r [1 __ 1 I -, ~ - ~ -, ~- ~ ~ / ~ ~'&e. &c. &C. _ ^xd Xm+ I"I m +1 I g"Jloo'% L~n - M (m 4- 1) ____ 1 1 -2) ) ( -3) ]... (,~ -I) ( (- 3) - 4).... 2. llog; (m + I)"- _ ( - ) (2 -2) ('-).. log x The last integral admits of only atn approximate determination, but its form may be simplified; thus, put z = Xm"+', then dz = ( + 1)xdmcdx, and (m 3+ 1) logx -.log z. fx x f ck J log Jlog This, also, can only be integrated approximately by expanding the expression under the sign of integration into a series, and then integrating the terms separately, a method which will be considered more at length in a future chapter. x4dxr 3. To integrate approximately dy = -log Here n = 4 and n- = 3,.. m 1 = 5, n~-1 = 2, n- 2 = lo 1" 1 5 1" -- ]+_t/.' J - -O2g ~2 log + og x 2 logx' Now put x5 --, then g l og....... ^ ^~~ log x log' EXPONENTIAL FUNCTIONS. 295 and, making log z = t, we have z = et, dz et. dt. fldz Tetdt f(1 - t2 123 d ic t2 t3 -1. x22 J'2 =logt+-t-1 22 + 1.2.82 -&c. =log [log5] +logx5-4- 12log2x5 + 1 2 og32z5 &c. x 5X 25 1 *x 5z~~g 2lg + -_- [log (logx5) + logx5 + 1 g + log325 + &C1. 1.0.3s Exponential.Fuctions. 61. To integrate the form dy = a.,m.dx, when m is a positive integer. Put axdx=dv, and xm-u; then v= *. a and d= —-nmxm1-dx. loga f. xmdvax. Xa z _ - ax,. fa'. xm. c dx -o - ~~ a". a fa xm-X, and similarly log a log a log f axxm2d X.''~m —2 1 2 f ax. x~-2d=x = f ax. xm-3dx, &c. &C. &C. log a log a Hence, by substitution, ax Menm-1 y =f a".xm.d x d = l a x m, log a _ log a m(nZ - 1 )x'n-2 m1Z(m72-1) (m -2)x)m-3 log2a log3a (- log)a.. 2.. ** -.llogma...... +'"("-L~-:' 21]+ 296 INTEGRAL CALCULUS. ax 62. Prop. To integrate dy = ~dx, when?m is a positive integer. Xm71,-VI+l Put x-mdx=dv and ax=u; then v=- 1 and du = log a. axdx. f axdx ax lo^ a r c6xci r axdx ax logoa fcxfd x- x (- n ~ — -2 - -2 m-i raxr,,va$ log Ca f a dlX xm-2 - ^ 3)-3 in 3 J Xm-3' ( )M- q- _ - m &C &C 0 Hence, by substitution, f adx ax,_ log a log2a Y-/ 2m - (n - 1)Xm L"mn - 2 ( -2) (n -3)2 8 lon x-2a m-2l ~ (3 2-) (I - 3).. 2.1 X 1. To integrate dy = ax. 3dx. Here m = 3, rn- 1 = 2, I 2 - 1. Hence a P. 3ax 6 C Y log a [ log a loga log^a 2. To integrate dy = ex. X4. dx. Here 4, n -1 = 3, m -2 = 2, n - 3 = 1, log e = 1. y ==e(x4 - 43 + 122- 24 + 24)+ C. 3. dy = e-x2rdx = e-(- x)2dx = e-(- x)2d(- x). Here m - 2, - 1 = 1, log e = 1, and x in the general formula is to be replaced by - x..y =fe-.. 2. dx- = eX(2 + 2x + 2) + C. EXPONENTIAL FUNCTIONS. 297 ax 4. dy = dx. X4 Here m 4, nz - 3, &c. ad ax r log a log2a lo 1a Iax **'- I+^ -=/ x-J 2 + X2 t-+ 6 dx the last integral being found, approximately, as in a previous exanm pie, by expanding ax. 1 +- x2 5. To integrate dy e". dx. (1 + x)2 Put 1 +- x= z. a =z-Z-1, 1+x2 —1+z2-2Z-+1-=Z2-2Z+2, dx=dz, ex-=ez- - * y_ (^ + 2z )6e —^ l d 2z 2 f ~ Ozz2 /f-~)e. dz i[ez -2 d +2- C ], =e - - 2 -- z ]t J or by integrating the last term by parts e e ezez ez e ez-1 y = eZ -,, dz- 2 d - c=ez-l -2 + C - (l + C e(- )- )+- C CHAPTER VIII. TRIGONOMETRICAL AND CIRCULAR FUNCTIONS. 63. Since the tangent, cotangent, secant, cosecant. versed-sine. and coversed-sine, can all be expressed rationally in terms of the sine and cosine, it will only be necessary to investigate formlule for the integration of expressions involving sines and cosines. 64. Prop. To obtain a formula for diminishing the exponent m of sin x, in the general integral y = Ssin". cosxn. dx, when in is an integer. Put cosnx. sin x. dx = dv, and sinr-lx = eu cosn+lx then v =-, and du = (In -1) sin-2x. cos x. dx. 2z -- I and by the formula for integration by parts y = fsinmrx. cosX. dx.= -- - + fsilnm-2x.cosl"+2x.dx. But cosn+2X -= cos2. cosnX = (1 - sin2) cosnx. Sillmn-1. COSn+lX *'eY —-- n- +1 + — sinm-2X. coSX. dx -- fsinmx. cosnx. dx.'z n +1 n -n+ I Transposing the last term and reducing, we obtain fsinmx. COsnZ. dx = -- - Silll-s 2. cos". dx In + -n 7 + o-. TRIGONOMIETRICAL FUNCTIONS. 299 And similarly, sinll3X. cOSn+lX 3 - fsilm-2x. cos1x. dx- -- + - fsinm- 4xcos.x.dx, m7 + it 24 m+n-2 fsinm —4. cosLx.dX = - ~ + 22 - 4 + m ~ /4fsinfl.lcos7^X dx, &c. &c. &c. -Hence by successive substitutions, y = f sinlmX. cosX. clx = - [sin~-lz + - sin"'-3x (^2-1 ) (77-3 ). + (7 -2)(n -3) Si) 7-5x + &c. (7, + n-2)- )(1 + n - 4) (+ n2-1) (-;-3) (2 -5)..... 4 or 3 (r2 +?-2)(n + n —4)(:L +n -j).. ( + 3) or (2) - ) X sin2x or sin xl ( (m-l) ( 3 - j) (7-5).. 2 or I (, + Il)(,t + n~- 2)(,b +, -4)... (n +3) or ( — + 2) x sin x. cosnx. dx or fcosnx. dx..... (E). 65. This formula renders the proposed integral dependent upon that of the form sin x. cosnx. dx or cos. cdx, according as mz is odd or even, the effect of the formula first obtained being to diminish by 2 the exponent 2m of sin x, at each application. Also the first of these two final forms is immediately integrable by the rule for powers: for cosn-lli fcox. si. si x - fcosn. d(cos x) = +. Hence we have only to obtain a formula for the integration of the form cosnx. dx, in order to effect the complete integration of the proposed differential. 300 INTEGRAL CALCULUS. 66. Prol. To integrate the form dy = cosnx. dx where n is an integer. Put cos x. dx = clv, and COSn-1X = z; then v = sin x and du = — (n - 1) cosn-2x sin x d. Hence by substitution in the formula fudv =:- uv - fvdu, we obtain fcosnx. dx = sin x. cos71-x q- ( — 1)fos-2. n2. i. cl = sil X. COS~1-1 + (2- )f os —"X(1 -Cos2x) d sin x. Cosn-1 + (t-1) f osC -2X. d -( - 1 )fosx. dx. Transposing the last term and reducing, we get sin x. cos~-lx n - 1 f cos"x. dx — = s - + - f cosSn-2X. dx, and similarly 9` n sin x. COSn-3 n — 3 fcCOSn-2X. = - 2-. -4 fcos-2~X~.dx = i x ~ cosn-x.dx. n 2 n~2 Sinl X. cosSx-5X n - 5 fcos-4x. d - + fcosG-6x.dx., - 4 n -4 &e. &c. &c. Hence by successive substitutions, sin x -- I y = feOSnX. dx= — [cOS~-s G-t Cosn-3X (n-l2)(-4) + ()~ - ) cosx-a+ &c. (n-_1)(_o-3)(r-35)...4 or 3... + ~CO-S~~- 01o cos 23 ('n —2)(n~-4)(2 —6)... 3 or 2 (n-1)(n-3)(n-5)... 2 or 1 + -2) 4) Scos x. dx or fdSx.. (17F). -[ (, — 2)(.n — 4)......3 or 2 This formula renders the proposed form dependent upon one of two known forms, viz.; fcosx. dx = sinx + C, when n is odd, or, fdx = x + C, when n is even. TRIGONOMETRlCAL FUNCTIONS. 301 67. The two propositions just given effect the complete integration of sinm'. cosx. dx, when m and n are integers, by first diminishing the exponent m? of the sine, and then the exponent n of the cosine. But it is often preferable to reduce n first, and for this purpose we require the following proposition. 68. Prop. To integrate dy = sinmxz. cos"x. dx, by first dimin ishing the exponent n of the cosine. If in the formula (B), we makie z = - - x1, mn = n, and n -= mz, then sin x = cos x1, cos x = sin x, dx =-dx1, and by substitution we shall obtain fsinmx. cos7x. dx -: cosx~x1. sinmlxdx1 = — ~i + [~-Cos-nl1;1 -r &c.], or by omitting the accents and changing signs, sin+ l - + n- 1 fsinmx. cos"x. oX. dx = [cos-lx -- - COS-X n'b + m n + sm - I 2 (._- i)(-_ - 3) (it+,t2 -+ )(2 +t n 4) + &C. (1-1 )( 3) (,-5).......4 or 3 (n -?7' —2)(n q e+m~-4)(2 +?n- 6).. (2t Q+ 3) or (in +2) X cos2x or cos z] J (b-1l)(n —3) ( —5)......2 or 1' (,m+m)(n+,,~-o~2)(,+m~,-4).. (,,m+3) or (m7+-2) X fcos x. sinmx. cx or fsin' x. Z....... (G). sinrmelx But fcos x. sinmzx. dx fsinmx. d (sin x) -- + C. In + which will be the required form when n is odd. We have therefore only to provide a formula for the integration of the form sinex. dx, which will be necessary when n is even. 302 INTEGRAL CALCULUS. This may be readily effected by substituting in fbrmula (F), mn for n, and -C - x for x, and changing the signs. Thus f sinmx. cx - [silm —x + - sinm-3x + (9 _1 j -) Silnl + &c. (n-1) n)( — 3)(in-5)... 2 or 1 x o... (m 2)(rn-2)(n-46)...3 or 2.m(m~-2)(m~-4)....3 or 2 - 69. The formulte (G) and (H) effect the same object as (E) and (F), reducing the integral fsinmx. cosnx. dx to one of the known forms J'dx -1= + — C, f cos x. cl = sin x — C, or, fsin x. dx =-cos x+ C, the exponent mi or n which is first reduced being an even integer, and the other exponent an even or odd integer. But if mn be odd, (E) alone will effect the integration, whether n be an integer or fraction; and similarly, if n be odd, (G), alone will suffice. SinImI x cos"X 70. Prop. To integrate the forms s z dx, and c7 dx, where cCOSX sinmC mi and n are integers. By the formula (E) the first of these forms may be reduced to dx sin x. dx dz, or -, and by (G), the second may be reduced to COSSnX cosnx dx cos x. dx or SinlmX si nmx sin x. dx cos-n+l, cos x. dx sill-m-t But ~ = C, and +c = -. cos"X n — 1 smmx - m 1-1 Hence there will remain to be integrated the forms cos-x. dx..... (1), and sin-l. cdx.... (2). TRIG(,NOME''RICAL FUNCTIONS. 303 Put in (1), cos x. dx - d, and cos-"-1x - 1it then v = sin x, and du = (n + l)cos 2..2. sin x. dx, and by substitution in the formula for integration by parts, f cos-nx. dx = sin x. cos-n-1x -- (a - + 1) J'si12x. cos-n-2x. dx = sin. eo~-n-'x - (, + ) f os-1-. dx -i (n + ) f cos-~x. dx. Transposing and reducing, we get idx silln /x dx D s+L = -- I( e -- 1o+-o; and by analogy co(s)i+2,' ({n + l)cos"+I. n + cosl dc. sin x n -2 os(-~ - )- o= sX + _ C o -, and similarly GcosXv {n - I)COS"- x n - 1 co"S-^ 2,* dx _ sin x n- 4 dx sJ c'1 2 3)C(Os n- J cos4x &c. &c. &c. Hence by substitution r_ dx _sin xl 1 n-2 co J s n - 1 Lcosn —~x (? - 3)cosl-3x + (n —2)(-4n ) &4) ( -- 3)(n-5)co, +- C. (n-2)(n-4)(n-6)..... 3 or 2 1 (T- 3)({~-5)( ~-7)...... 2 or. os2x or cos xJ + (n —2)(n —4)(1n-6)....- or 0 cx or or lx. M...(). 1(e-1)(n-3)(rb-5).... 2 1 co The second of these integrals, f dx = x + C, will never be required, because its coefficient is zero, and therefore we stop at the preceding term. For the first we have / dx c F cos x. dx rcos x. 1 I cos x. dx 1 cos x. dx cos x J cos I -sin, 2 2J 1 +sin x I - 1 sin o- o in) log(1-sinn )+ 1log[ ] i 1 -t-sin xJ 2 og( sin( + x )os )1 =log L2 i -i - l 1 ( +C= log ta(4 2-/ ~+ C' 2 s in ^ ^^-^) - COSI _'n + 2X'J ttv ). 4, a, 4 2) 304 INTEGRAL CALCULUS. 71. To obtain a formula for z =- 1 replace in (I), by sm, x by -x, and y by. Then / dx cos x [ 1 -2 Z I + it 3) Sijj —2 - sinmx 3 - 1 sin^- (m7-3) silm-3x + (-2)(-4) + c. (m~-3)(a~-5) simS -5S (m-~-2)( —- 4)(m- -6).. ~ 3 or 2 1 -(m —3)(v7 —5)( -- 7)... 2 0or 1. sinl or sin x (e -2 )( —4)(m-6)''' 1 or 0/ d. * ( ~ (~-1)(s-o)(m~5)...2 or 11 sini o The second integral has a coefficient equal to zero, and therefore will never be used. For the first we have, by replacing x by -I - x in (I1), and changing signs dx 1 1 f/'cl log cot x - lo - log tan -x- +. sm Sial 1 2 cot - x dx 72. Proop To integrate dy-.x where n and n are siajnX. cosnX integers. Since sin2X + cosx- =- 1. n drx p /(smintx + cossx)dr J sin"^'x.CSx Silcos ^ sinm x. cosx. p dx cx cox J sinm —2x. Costx sinnlx. Coss'-2 f (in - + co s2x) d + (Sins + COS2x) d Sillm-2X. COSnX SillsnX'X. COSn-2X dx 2 rdx J sinm-4x. cosnx^ sinm-2;. cosn-2x sinmx. cos"-4x and by continuing to introduce the factor sin2x +- cos2X = 1, TRIGONOMETRICAL FUNCTIONS. 305 we obtain finally one or more of the following known forms y dxJ dx sinx.dx f cosx.dx fSilnx.dx pCosx.dx smilxJ OSX C OS'1X sinmx Cos x sin x. Applications of Fonrmulc (E), (F), (G), (H), (I,) and (K). 73. 1. To integrate dy = sinx. cos5x.dx. Here m- 5, and tz = 5, and since both are odd we may apply (E) or (G) with equal advantage. Employing (E) we have Cos6X 4 4.2 y —= - 1 [sin4x + - sin2xz +. 08 fsin x. cosx. dx I0 10.'C Cos6X sin2x] 1 -sin4X + sin2x + U + C. 10 2 1 I6 2. dy = sin6x. cos3x. dx. Here m =- 67 z = 3, and since a is odd we apply (G). sin7x 26 sin7x / 2.Y C y O= S [cfos x. sin6. dz = C- (os2x + + C. 3. dy = sin6x. cdx. In (HT) make m = 6. cos 5 5.3 5.3..X'. sy + sin x]+si] + 3 - 1 +'. 6 +4 4.2 6.4. 2 4. dy = sinsx. cos6x. dx. In (E) make?- = 8 and n = 6. cos7X. 7 7.5 7.5.3 sTsin7x f sin~x q- 1-sin3x q- - 0 ] - 14 12 + 12 12.10 12. 10. 8 7.5.3.1 + 14.. sox. 20 306 INTEGRAL CALCULUS. and by applying (F) to the last term, we get cos7x 7 7 7 y - [six 1 sin5x + - sin3x - 6 sin x] 14 24 64 sin x 5 15 15x + [cos5x +- cos3x - co s xS ] +- C. 76 8 4 6144 sin5x 5. dy = d2 x. GOS X In (E) make = 5 and nz -2. Then — 1 4. 4.2 fsinx.dx - [sin4x + - sin2x] + 3- 1 33 cos x — 1 -= 3 [sin4x + 4 sin2x - 8 + C. 3 cos x dx 6. dy = -i. sin5x In (K) make m = 5. Then cosX 1 3 1 33.1 dx Y + + — Jinx ^ 4 Lsinx 2 sin2xJ 4.2, sin cosx _1 3 1 CO - 4 [iL + 2si - + 8 lo ta + C. L 2 sin2x 2 7. dy = dy - cos 6' In (I) make = 6. Then sinax F1 4 1 4 2 1 1 y= I +- 4 + —*5 Lcoss 3 co3s d dx sin4. cosx e Sillz4X. COS2X Introducing the factor sin2x + cos2x, we obtain p (sin2x - cos2x)dx dx sin4xz cos2x Jsini2x,. cos2x + Sin.x,f dx +, dsix?f dx cos anx S - co xi, cosx 1 21 = tanx —cot x — ~ + si si-l x o sin3x sin. TRIGONOMETRICAL FUNCTIONS. 07 74. When m -- = n, formulae (E-) and (G) cease to be applicable, but we then have P sin x f COSsnx y c= -- dx = ftanzx. dx or y - -. cx cotnZO dx. COsx -' " smxln To integrate the first of these expressions, put see2 - 1 for tan2 and in the second put cosec2 - 1 for cot2. Thus ftanxz. dx=f tan2x. tan~-2x. dx=J'sec2x. tann-2x. dx-f -/tan"^-2x.d 1 = 1~ ttann"~x J' tan"-2x. dx ~1 1 =- - 1- t~aEn8-x - fS (sec2x - 1) tanl-4z. dx n — 1 tan-l _ 1 1 1 = — tan - tan-3 _ -- tan- tal - &C, nX- I n —3 n-5 the last term being J tan x. =fx s I x -x - log cos x -- C = log see x + when n, is odd or fd = x-{x - C when n is even. cosuxdx Simlilarly, / cos X = otnX dx cotn-x. cot"-3 cotnx5x = _ -+ _ ~+ &c. - n - 3 n - 5 The last term being fcot x. dx - log sin x - C, or fdx = xt + C. 75. When the proposed form is fsinmnx. cos"xdx, in which mn and n are integers, the integration may be conveniently effected by converting the product sinmx. cosnx into a series of terms inivolving sines or cosines of multiples of x. The integration can then be performred without introducing powers of the sines or cosines. The proposed transformation can always be accomplished by the 308 INTEGRAL CALCULUS. repeated application of one or more of the three trigonometrical for1'ulae. 1 1 sin a cos b = - sin (a + b) si (a - b) ] 1 sin a sin b = cos (a-b)- cos (a - 6), cos a cos b = cos (a - ) +- cos (a + b). To illustrate this process take the following example. dy = sin3. Cos2xdc sin3x. cos2x = sin x (sin x. cos x)2 Si X (sin 2 ) 1. / 1- cos 4x\1 1 - sin - sin xI - os 4n x - sin x. cos 4x 4 \ 2 8 1 1 16 sinx J(sill z - 1 sill 3sin si — )sn y-= - sin x + Si sin 3 - ~~sin 5x 1d. -8~ cosx — cos 3x -t- cos 5x + C. 8 48 so80 76. Prop. To integrate the nl bi =dy ba:. sin-lx. clx. Put sin x. dC clv, and htaxsin-x =_ z, then v - - cos., alnd Clo -- (' - 1 )lasinn-2x. cos xdx + a. log b ~ baxsinl-lx. dx yo f. dx b:sinx. = ~- b oa"Si -1).CO b axsinns -2 cos x + (-1 ) b a. log b.fbh'sin"1-1. cos x. dx. But, by applying the formula f ctdv = uzv -S fvc to the last integral, making sin-lx. cos x. dx =: dv anld bax =, we get f b2sinl-1x. cos x. dx=- sinx ba a lg sin. ax. dx, fl nn owd, by replacing cos2x by 1 sin2x, we have /'basin-2x. cos2x. dx == fasinn-2xx ~- f bsinx x. dx. TRIGONOMETRICAL FUNCTIONS. 8309 Hence, by substitution, a lo' b f ba. sinx. dx = - baxsinn-lx. cos x + a-~ b a. sin"x _ ( log ) sixlnX. b +adx + (n - 1) f6axsin?^-2x. dx - (n - 1) f baxsinlx. dx. Iransposing, collecting like terms and reducing, we obtain bazxiBn —l2' f basinx. dx. - (a log b. sin x - n cos x) (a log b)2 + ^22 ( — 1 f baxsilln-2X.. (X. (L). (a log 6)-2 +?- 22 By repeated applications of (L) we obtain the final integral. 6az f bxdx = -l b {- C when n is even; and when n is odd, a log b f bazsin x. dx, which is given by (L) without an integration, since the last term then contains the factor n- - 1 -1- = 0, and therefore that term disappears. 77. Prop. To integrate the form dy = baxcosnx. dx, Put x =- x - -q, then cos x - sin x,, sin x = -cos x, 7,.a 6a% - 6a:.6b 9 dx —dxI. e" y=b. fa. l sin - bcx.,. 1(a log 6.sin xzl-n cosl) " (a log 6)21+'2 -k(a log - f bsinl —2xdx,, and by substitution, baxcosn —1x f baGcosx. d — log (a log 6. cos x + n sin z) ( log b) -n- b +-(a log b) + 6ax cos-2X. dx.... (.). (~ log~ 6)) +^ ^' 310 INTEGRAL CALCULUS. bax Here the final integral will be J'bazdx = - - C, when n is a log b even; and when n is odd, f baxos x. dx, to which (i) applies with. out an integration. 1. To integrate dy - eax. cos x. dx. In (iM) make b = e, n = 1, log b = log e = 1. Then eax a2 [Ca Cosx - Silln X + C. 2. dy =C e. sin3x. dx. In (L) make b = e, a = 1, n = 3, log b = 1. Then e. sin2x. 3.2 y = -i3 [sin - 3 cos x] +,2 Sf esin z clx Y — + 32 1 ( 1 = e [sin3x - 3 sin2x. cos x] - - ex(sin x-cos x)F- C. -- 10'i 1 or, y = 10 ex[si33x + 3sin cos + si Z 6cos + C 3. dy = e-axsin k. dv =- e-axsin Ix. d(lcx). In (L) make b = e, = kx, a =-, Then e-ax(a sin klx - I+ cos kx)?/= - 2 k2 - a2'8. Prop. To integrate the form dy = X. sin-lx. dx, in which X is an algebraic function of x. Put Xdx = dv, and sin-lx -= U; dcx mflen v = f Xdx = X, and du.= ~-.. y = Xsin-lx X~' and the proposed integral is thus caused to depend upon another whose form is algebraic. TRIGONOMETRICAL FUNCTIONS. 311 79. Prop. To integrate the form dy = Xcos-lx. dx, in which X is an algebraic function of x. Put XdTc dv, and cos-lx = u; then v _f Xdx X1 and d = ~ - ~ X dx y - Xcos-IX -L j ~ an algebraic form. Cor. The same process will apply to each of the forms Xttan-lxdx, Xcot-lxdx, Xsec-lxdx, &c., since the differential coefficients of tan-lx, cot-ix, sec-x, &c., are all algebraic. 1. dy -= xsin-lx. dx. Here X = x2,.'. X1 SXd. x2dx =, d Xd I d x"dx I/I f d- 2\ rd~ and f X Cf 1- - - -( 2 + )I/1 2.^ l 2 0. y s i_-l 3 / x3 I/I "2 \/ 2 _ * * y = 3 sill - _ + _)- _ 2 + C. x2dx 2. dy -1 tan-lx. Put dv = ~ = dxl. Z- and it = tan-~x. 1 -2 1 ~- x2' dx.. = - tan-x, and du= + 2.y = xtan-1x - (tan-ix)2 -/ - + tan-x. dx — d i' - XI I+ l- x 1 1 -= tar-lx - (tan -1x)2- log (l + 2) + (tan-x)2 +. = tanr'x (x - tanlLx) - log /+x2 -+t C. = ta-lxx 2 CHAPTER IX. APPROXIMATE INTEGRATION. 80. When a given differential cannot be reduced to a form exactly integrable, we may expand the differential coefficient, either by Maclaurin's theorem, by the common binomial theorem, or otherwise; then multiply by dx, and finally integrate the terms successively. If the resulting series be convergent, a limited number of terms will give an approximate value of the integral. 81. This method may also be employed with advantage, when an exact integration would lead to a function of complicated form. And the two methods can be used jointly to discover the form of the developed integral. EXAMPLES. 82. 1. To integrate dy = - dx, in a series. I -- x Expanding by actual division, we have = 1 - x - X2 -: X3 q+ x - &C 1 -- x. y-=f(1 -xX + — 2 X3 + 4- &c.)dx. 1 x2 + 1 X3 1 x + x- &C. -+ C the r 4 5 the required series. APPROXIMATE INTEGRATION. 313 Again J1 dx =log(l + x) + C 1 + x log(l + x)= x -2 x2 + X3 3 - X4 + X5 - &C.+ C, 2 4 5 where c = C - C1. But when x = 0, log(l + x) = log I =0,.. c=0, 1 1 1 1.. log(l + x) = - -X2 + x - x4 + - X5 - & 2 3 4 5 a well known formula. 2. dy = x(1 - x2) dx.. Expanding (1 -x2)x by the binomial theorem, (1-z2)i I l x2 _ l x _ 5 xx6-1 S _- &C. 1 1 1 1 5- 2 - 1 x4 - - x6 - xs &. )dC. dx 2 8 16 1 28 3. dy= Here f - -=-(l+x 2) -1-2 + 4 -. - z6+ 2, 24 4 6.2yj1 x4 — 4 x6 + &C.)d1 1 1.3 1.3.5.. z -- I7 + &C. + C. 2.3 2.4.5 2.467 +.4. +. 7 dx But 1i = _log (x+ +/)+ - 1. e1 1.fIx3 1.3.5 + C C.~ e log~+V-11~ —tI~;LX I-i 1 5 357 + &C. + C___. + 1 3.24.5 2.4.6.7 314 INTEGRAL CALCULUS. Now when x=O, log(z —+ l+x- )logl =... C- C1-O. I XI 1.3 1.3.5xT- e.' log(1-/ x2)x —: + 24.5 2.4.6.7 dx 4. To integrate dy = + both in ascending and descending powers of x. _ _ 1 1 1 1 1 = 1 _2-x2+4_6-+&c. and -2 = -- + 6- +&c. lA'-a 2A-1 42 a4 6 a. " -- --- = - tan-ix + C (17- - -x2+ - - a^ &c.)dx 1 1 5 1 -= x 3 + - X7 + &C. + C. 0 5 7 1 1 1 1 - -A —-- - - & &c.A -C'. x 3X3 5x'5 7X7 The two results become equivalent, by selecting the constants C and C, such that C, - C- = r. For, the first series = tan-lx + C. And the second " = - tan-l- + C - cot-x + C,..'. In order that the two series may be equal, we mlust have tan-lx + C - cot-lx + 6C, or tan-l + cot-lx = C-C, or ~- Cl- C. 5. dy = d-. Expanding the numerator we have (1 - e2y2) 1- e2x2 + e4x4 - e6zX6 &c. — 2.4 -2.4.-6 APPROXIMATE INTEGRATION. 315 1 1 4_ 1.3 dx' Y S (I 2 e2X2 -t ez * e x5 &C.) 2 2-.4 2.4.6 1-X2 all the terms of which are of the form f 1 2- - and have been already integrated in the chapter relating to binomial differentials. We might also expand (I - x2) 2 by the binomial theorem, then perform the multiplication indicated, and finally integrate the terms in succession. Adopting the first course we have y i= in-lx' + e2 (I x/1- - sin-i) \2 -/ 2 - 4 l- 3 + 1 i-7 —- -nx &e. 2.4 L\4 2.4 2.4 83, Prop. To obtain a series which shall express the integral of every function of the form Xdx, in terms of JX, its differential co. efficients, and x. Put X = q, dx= v: then dit = C dx, and v =x. Now substituting in the formula fudlv = uv - fVvdu we get fXdx - Xx - i'* xdx. dX Next, put = u and xdx = dv, d2X 1 then du -= d2. dx andc v -= x fd x dX *2 d2X X2 dx C 12 dX2 1' 1 2dX * 2X a2 d2X 3 d3d X;3.. fdax z3 foimilalyl. -.-... -s^ dx = --' - / ~' ^^- dx &c.&o, Similarly dx2 dx dX2 1.3 dx3 1.2.3 By substiution dX x2 d2X x3 d3X x4 c=-x-,17 2 1 o 2c3,J. 1: C, 316 INTEGRAI CALCULUS. This formula, called Bernouilli's series, shows the possibility of expressing the integral of every function of a single variable, in terms of that variable, since the severial differential coefficients dX d2X d~' d-2: &"., can always be formed. But the series is often divergent, and then of no use in giving the value of the integral approximately. CHAPTER X. INTEGRATION BETWEEN LIMITS AND SUCCESSIVE INTEGRATION. 84. The integrals determined by the nethods hitherto explained are called itadefinite integrals, because the value of the variable x, and that of the constant C, both of which appear in the integral, 1iemain uncletermined. But in applying the Calculus, the nature of the question will always require that the integral should be ta(lc'. between given limits. Thus, slppose the integral to originate. (_,r its value to reduce to zero) when x - a: this condition will -1 the value of the constant C. Then, to determine the value of the ec.'tire or definite integral, we replace x by b, the other extreme vah /I of the variable. Ex. To integrate dy = 3x2dx, between the limits x -- xz and x z. y = 3x2dx = x3 + C. But when x = x, y= 0.. = Xi3 + C and C -- x3, and by substitution in the indefinite integral y - x3 - X3. Now make t = x2, and there will result y - x23 - x3 the complete or definite integral. INTEGRATION BETWEEN LIMIITS. 317 A slight examination will show that the desired result will always be obtained by substituting in the indefinite integral for the variable x, first the inferior limit x1 and then the superior limit x2, and then subtracting the first result from the second. In these substitutions the constant C may be neglected, since it will disappear in the subtraction. 85. The integration of 3x2cdx between the limits xz and X2, when xr is the inferior limit, or that at which the integral originates, and x2 the superior limit, is indicated by the notation. rZ2 x 3x2dx. 86. The precise signification of this definite integral will, perhaps, be better understood by the aid of the following b Prop. The definite integral a Xdclx (where X is a finction of x which does nlot become infinite for any value of x between the limits x = ca and x = b.) is the limit of the sum of the values assumed by the product Xh~, as x is caused to increase by successive equal increments (each h) fiom x -- a to x.- b; the value of A being -continua.lly) diminished, and consequently the num ber of these increments bein indefinitely increased. Thus, if Q X0 X 3... Xn-1 be the values assumed by X, Vwhen x takes successively the values a, a- A, a+-22, a+3A,.... a+ (n -1)Ah, then will / Xclx be the limit to the value (x0 +- Xl + X 2... — r,,- )h, provided nh: - b - a and Ih be diminished indefinitely. Proof. Let x and x + h be any two successive values of x, and denote by Fx the general or indefinite integral fXdx. Then by Taylor's Theorelm, d Fx hl d2x 72 1 3 1F 2.F(x + l) Fx + -'- + ~) - + &c., F(-xI zF ~'dx I dx 1.2 dx3 1.2.3 818 - INTEGRAL CALCULUS. which may be written, F(x + h) = Fz - Xh + Ph2,... (1), where P is a function of x and l. Suppose the difference b- a to be divided into n equal parts, each equal to h, so that b- a = nh. Now, putting successively a, a -+ h, a + 22h... a + (z - I)h for x in (1), and denoting the cotresponding values of P by Po, P1, &c., we get F(a + A) Fa + XY,0h + poh2 F(a + 2h) [(a + i2) + ) + ] F= (( + A,) + x7,/ + PA2 F(a + 3Ah) = [(a - 2h) + h] F= (a + 2/) + 212, + -.P2h2 C,. &e. &c. F(a + nh)= F[(a + (n2 - 1)h) +- h -- [a + ( -1 )A] + Xn-A + P1-1A2, adding these equations, and omitting the terms common to both members of the sum, there results F(a + nh) = Fa - h(XO + X1 - X... + X,-1) + h2(P, + P -+ P2.... 4- Pn-]). But, since every value of X is finite, none of the values of P will become infinite. If, therefore, we denote the greatest value of P by P, we shall have Po + P + P2....+ P-,_1 < Pn, and since F(a -F nh) =- Fb, and 1ZA b — a. F. - F - h(X0 + + X2... + X1_) < (b - a)P.h. But b - a and P are both finite, and therefore by diminishing A?, the second member can be rendered less than any assignable quantity. Hence Fb - Fa must approach indefinitely near to equality with h(X0 + X + X2... + X-_) when h is continually diminished. SUCCESSIVE INTEGRATION. 319,Successive Inteqration. d2y 87. If the second differential coefficient T y X be given instead of the first, two successive integrations will be required to determine the original function y in terms of x. Thus, multiplying by d. and integrating, we get / d12, dy 2. dx= Xdx, or -= XSdr X, + C. Jdx 2 clx Multiplying again by dx, and integrating, we get CY x = fSXLdx + f Cldx, ddx or y = X2 + CIx + C2. 88. Similarly, if there were given - X, three successive in. tegrations would give = X3 - 2 + 1 13 dIlf And if there were given d X, then = n —1 Y 1:. 3... (2- 1) + C23 + &c... + C,-1X + C,'.2. 3....(1 — 2) the number of arbitrary constants introduced being n. 89. The result obtained by performing the above integrations may be indicated thus fnXdxn = y: it is called the nth integral of Xcx". 90. Prop. To develop the nth integral f"Xdx" in a series. Employing Maclaurin's Theorem, we have 320 INTEGRAL CALCULUS. xJuXdxz L= XdxIl + [fn-'Xd j']{ + [J. n-2Xldx'] -2] + &c. )n-1 7Y Lni 1... L(-) 1.2.3. L-x.2.... 1) +-dj; l-.2.3..+2 L)+L 1 J.J-3. + [R]. [, 21. 2. 3.. (It + ) ] 1. 2. 3.O. +n' ] I c2 S..,,f The terms within the L ] are the arbitrary constants C1 C2 C3....Cn as far as'[J' Xdx] inclusive, but taken in an inverted order. 91. Prop. To deduce the clevelopmlent of j'cXdx"l fioml that ot X. By Maclaurin's Theorem-; we have X EX] + [- + -T j + X [X] - d-/x-J I + J 1. 2 + clx^ J 1J. ^i. o and this may be converted into the series [R] by multiplying each term by x', then divicding the successive terms by 1.2...., by 2. 3. 4.. (.a + 1), by 3.4. 5... (n + 2), &c., and finally annexing terms of the form 1Z31 C2 xn-2 1.2.3... (n-1)' 1.2.3... (-2) C. 1. To develop J/4 Cdr4 1 xX X 1 1 3 1 3 5x & Here X —(1- x2) 2 = 12+ - x -2.4 G & -1 +C 4 246 Also n -- 4. Therefore multiplying by x4 and dividing successively by 1. 2.3.4, by 3.4.5. 6, &c., and finally annexing the terlms containing the constants we get / l,~t'4 r~vi 0 ^-2 P ^3 ^4 6 4 a JO Ciz x' x d4_ — -J L _- + _2_ + __ ___ _ _ 4_~ -l 1 1. 1. 2. 3. 1...4' 2. 3.4. 5. 1. 3x 1. 3. 5x +2. 4 5. 8 2. 4-..7 8. 10 &c. +2.4.5.6 8 2. 4. 6.7.S. 9. 10 SUCCESSIVE INTEGRATION. 821 d2f 2. What curves are characterized by the equations d = 0, and dx2 d^y d = 0, respectively? ef d, d =f J Cldx, or y = C1x + C2, a straight line. 2d. If dx -0 then jx d2 - C dV 3 Jdx 3 ~ X2./i2 a = X Cdx or d= Cx + C2, -dx.. Clzclx -+ / Cdx or y 112- + C2 + C3, a parabola. 21 PART II. RECTIFICATION OF CURVES. QUADRATURE OF AREAS. CUBATURE OF VOLUMES. CHAPTER 1. RECTIFICATION OF CURVES. 92. To rectify a curve is 6o determine a straight line whose length shall be equivalent to that of the curve, or simply to obtain an expression for the length of the curve, in terms of the co-ordinates of its two extremities. 93. Prop. To obtain a general formula for the length of the arc of a plane curve, when referred to rectangular co-ordinates. Let AB be the proposed arc, P a point in it, OX and 0 Y the co-ordi- Y nate axes. p Put OD = x, DP = y, AP = s. Then since ds = cx /+ dx-2' E we shall have by integration /( + d2) clx.... (S), the required formula. dy 94. To apply (S) we replace dy2 by its value, in terms of x, dx2 deduced from the equation of the curve, and then integrate between the limits x = OE and x = OF. RECTIFICATION OF CURVES. 323 95. Again if y be taken as the independent variable, we shall have ds = dy V1+ d~2 and therefore s I/(+ ( ~ dX2 y... (Si), a second formula. dx2 This will be applied by substituting for d2 - its value, in terms of y, derived from the equation of the curve, and then integrating between the proper limits. EXAMPLES. 96. 1. To find the length of the para- Y bolic arc AB, included between the ordi- B nates b1 and b2. The equation of the curve is y2 = 2px., cx y o E D F F. D c dy p which substituted in (S1) gives S J si ~ j+!f3 2 dJ= (p2~ Y 2) 2dy. But by formula (B), f(p2 + y2)d = (p2 + 2)Y S(2 + 12) y. (l). To integrate the last term, put (p2 -- y2) _ z - y, 2. + y2 2y 2 + 2 - fy 2: dy P 22 + P_2 Z2 2 + )2_2 and (p2 - Y2) + P - f(e _2+j2) 2dy J - - ~I I 9;.4 C.. +/(^4 -_/z ) 324 INTEGRAL CALCULUS. And by substitution in (1), To determine the value of C,, put y _= 6 and s - 0, since the arc is supposed to comumence at the point A. Thus I 1 1 1 s ~=( 2 ) — 2plog[(p 2 + b~2)'~-b~]-} C. Ada 2 (2 + y 2) 1 l Tohen eterm t vale of C, put y b, and 0, s t If the arc be su reckone fom the vertex 0, the ordinate 2 0,= 2s ( 2 l[(2 2 +1 b ) — b(2 ]+ + 2 b 2) 2 bl loI (2. The cy2cl)oid (y 2 b 2 + r. versin -1 _ when y / b2 1_ 1, _ ((P 2 + b~)~ b I If the are be reckoned fi'onm the vertex O, the ordinate b1 - 0) e 1 = 2r-w 2r _ l V \ z2 A,~ ^ ~ 1 (p2 + b,~)~ 1 (+2 + tb p2'+ - bb b2 2 210grr + -) 2log 2. The cycloid y — V x25-~ - x2 + r-. vetsin — x Here dy= /2r-x a7Y2 2r - x -N, A c \ RECTIFICATION OF CURVES. 325 Hence by substitution in formula (S). s=y dx =2dz 2r + C. But when x O, s =O,.. C =O0, and hence s=- 2 rx or, the cycloidal arc OP = 2 chord 01 of the generating circle. When x = 2r, s = arc OPA = 2 diameter 0 C.' arc AOB of the entire cycloid = 4 diameters of the generating circle. 3. The circle y2 = r2 x. dy X dy2 +2 y2 O ~ _ l+ b-14-l+ --- 3 dx --' -~ 8 = = r. sin-l +. This result involves a circular are, the very quantity we wish to determine, and is therefore inapplicable. To obtain an approximate result, expand the differential coefficient (72 _ x2) and integrate: thus'/1 1 x2 1.3 1.3.5& X6 d 7 2 r3 2. 4 r5 2. 4 6 r7 1.3 1.3 x5 1.3.5 7& l+ - 7+27 - 7- ++27]'+ J+ 2.3 3 2.4.5 r5 2.4.36. 7 (7 J But if s = when x = 0, then C =- O and.. when x = r, 1 L 11.3 1. 3.5 8= r - 2- 3 2745 2.4.76 + &c) the value of the arc APB of the quadrant. 1 1 1.r3 1.3.5 And if r= 1, s — 1 -.- 24 - -- &C. 2 2. 3.4. 2.4.6.7 326 INTEGRAL CALCULUS. 4. The ellipse a2y2 + b2x2 a2b2. dy2 b4x2 dx2 a4y2 + dy2 _ x+ 2 a2(a2b2 - bz2) _ — b42 a2(a2 _ x2)+ b2X2 dx2.4y2 a2(a2lb - b2x2) ~ a(a2 x) 2) c2 _ b2 a2 x2 dy 2 a2 e2x2 orI + 2 = 2 2 - = 2, ~ where e is the eccendX2 a - c~ -a x tricity. (a. s~- dxi dx. d, by making - -= x. (0A - x2)2 (- X12) This expression has already been integrated approximately. 5. To determine what curves of the parabolic class are rectifiable. The equation of this class of curves is y" = axm, in which n and m are positive integers. d I m - [ - 2 -- - -ax x,i.. s = 1 + aT 1 2dx dx n ) L "2 and this can be rationalized, when - --- =', an integer, that is, m3 1 -- ^r1?\ n+ when - + (Art. 41). Hence, if one exponent, n, be even, and the other, m, greater by unity, the curve will be rectifiable; that is, an exact expression for the length of the curve can be obtained in terms of the co-ordinates of its extremities. The term rectifiable is sometimes restricted to those curves whose lengths can be expressed algebraically, or without employing transcendental quantities; and with this restriction, the value of r must be positive, otherwise s would be transcendental. Now applying the other condition of integrability, we have 1 1, n 2r- 1 ~-.+ -, =, an integer, whence = 2/m~n\ 2' ~9. n m 2, 2 RECTIFICATION OF CURVES. 327 H-ence. if one of the exponents be an even integer, and the other less by unity, the curve will be rectifiable. Combining the two results, we find it simply necessary that m and n should differ by unity. 97. Prop. To obtain a formula for the rectification of polar curves. Here we have to express s in terms of r or 3, and for this purpose we must transform the formula [S], by means of the relations dsO = c —+ J d^''..(1). - cos,... (2). y r sin.''(3), the quantity 6 being taken as the independent variable. Then (2) and (3) give dx dr dy d -r - sin T cos B-, and -r cos - sin -. ds2.d &d.2 -. - r2sin2 S - 2r sin 6 cos + cos20 1 dr2 dr d.g + r2Cos2 - + 2r sin 0 cos 6 -+ sin -20 do dQ2 +- 2+...0...(T). 1. The logarithmic spiral r - ao, between the limits r = rl, and dv~~ a0 d = log a. aO = -, where m is the modulus. d. d. - = dr =- dr, and by substitution in (T), aO r 122... * = S, 2 + /7 2 ) = ( n2 + 1)d- (2 +1) + C. 1 But s, when rr,.. C = -(m2 + )2rl. S. s = (1 -1- m2(r —r), and when r = 22, s=(l+m2)7(2-r ). 328 INTEGRAL CALCULUS. 2. The spiral of Archimedes r = a, fiom the pole to the point r rl. = a, d... =dr = a, d =o S =-Jr2 + a2);-dj-. d6 a a L This expression is entirely similar to that integrated in rectifying the parabola. 1 1 r (a2 - 1912) + a 1, (a2 + r,2).'. S = ~ ^ - a loo 2a ~ a 3. The lemniscata r2 - a2cos 26, 1 —" c' rdr a4 ( d^ 92'dr d'2 a( (a - adr _. 1 -. 3r see'' - a ~ ~ -J + 27-4 (a4 -r) a 2 2c 2. 4 a I.. 5 9,12 + 2.46 a4 &c.J dr, which, integrated from r = a to r = 0, gives for the are BIA or one-fourth of the entire length of the curve. s — 1. 13.5 [ 5 + — 2.4.9' 2.4.6.13,' 98. When the curve is characterized by a relation between the radius vector r and the perpendicular p upon the tangent. To obtain a formula for the rectification in this case, we assume the value of the perpendicular found in the Differen. Calculus, p. 154viz.: q2 dr2 622(2.2 -- p2).p =;whed c9d- 2, and r2 + dj 2 ds2 ds d d.2 dr2 4 dO2 c dl2 dO2 e 02 p2 QUADRATURE OF PLANE AREAS. 329 ds2 9.4 dj,2?94 p2 2 d2 - p2 d 2 = p2 X r.2(2 _p2) 22 _p2 * s = dr (U), the required formula. (Q,2 _ -2) Ex. The involute of the circle from p = 0 to p = 2qa. Here the equation of the curve is 22 = a2 + 22. [*. _ r d=r2 a2 +p2 X ) __ _ C C _- C. a 2a 2a ~ But when p = 0, s = 0...C= =; and when p 2 a, s = 2r2a. CHAPTER II. QUADRATURE OF PLANE AREAS. 99. The quadrature of a plane curve is the determination of a square equal in area to the space bounded in part or entirely by that curve. The problem is regarded as resolved when an expression for the area in terms of known quantities has been obtained, the number of terms being limited. 100. Propl. To obtain a general formula for the value of the plane area ABCD), included between the curve DC, the axis OX, and the two parallel ordinates AD and BC, the curve being referred to rectangular co-ordinates. Put OE=x, E7P=y, EF=:h, FP —Y1, Y and the area AiEPD=A. R Then when x receives in increment h, the area takes a corresponding increment EPP1F, intermediate in value between the E rectangle FP and the rectangle FS. 330 INTEGRAL CALCULUS. dy h d2y h2 Bu _ Xh y dx 1 dzx *1.2 But - -FP y X h y y Aly lb d2y jb2 h+?+d2 12 - &c. -1 when h- 0. d y dx2 1.2.y Hence at the limit, when h is taken indefinitely small, the area.EPP1F, which is always intermediate in value between FP and FS, must become equal to each of these rectangles, or equal to y X A.. dA = ydx, and consequently A fydx...... ( V), the required formula. 101. If the area were included between two curves DC and D1Cl,we should find by P a similar course of reasoning D A -f(Y-/)dx....... ( V), / in which Y and y denote the ordinates EP and EP1, corresponding to the'same 0 A E I. B abscissa OE. 102. To apply (V) or (TV), we eliminate y, or y and Y, by employing the equation of one or both curves, and then integrate between the limits x - OA and x = OB. EXAMPLES. 103. 1. The area ABCD, included between the parabolic arc D C, the axis of x, and two given ordinates AD and BC. Put OA=al, AD=bl, OB=a2, BC=b2, OE=x, and EP=y. Then, from the equation of the parabola, y e wVe have y2 = 29x, or y = (2p). x.. And by substitution in formula (V), - A E B X A= f 2. ( 2. C C A=(). xdx (2p) C - (2px). x+ C xy+ C. 3 o QUADRATURE OF PLANE AREAS. 331 2 But A = 0 when x = a and y bl,.'. = -- a, l ~~2~~~~o. A -= (xy - albl) = ADPE; and when x = a and J = 62 2 A =1 (a 2 - lb) - ADCB. 0 Cor. If the area ODCB of the semi-parabola were required, we should have 2 2 a1 =O, b 0, O and.'.A = 3 2b2 - circumscribing.; 0 3 and for the entire area of the parabola 4 2 2 2A = a22; a. 2b6 = circumscribing. 3 o o 2. The circle y2 = -92 - X, 0or its seg- E ment A CD. Here A = fydx = S(r-2 -__ 2)2 dx, A [ o or by employing formula (B), A = (2 X)x 1 2 d 2)1 _ cos- 6+ _A z(. - +.' =X) (.-2 X() -,.2 CoS- + Suppose the area to be reckoned from A. area A 0 when x = OA =- r. 1 1, -. (7 = 2 7 2cos-1( 1) - 72 22 22 And when x - + r, A = 2 r2- area of semicircle AEB... area of entire circle AEBD = r2. 332 INTEGRAL CALCULUS. To find the area of the segment A CD, make x - 0 - a, then A — A C 2= 2 co -a(2_ a)_ - cos2 2 —; r ~-. cos- > )1~ ca(2 - a2) =?'(A CB - CB) - a(r2- a_ 2 1 1:r.AC — Ia. CG. 2 2. segment CAD C - r. AC — a. CG. 3. The elliptic segment A C1D1. Here the equation of the curve is c on AB. b C Y — (a2 - 2)2A Bt since y is a traAsceetal / a c 2A seg'ment AC1D1 -j -. segment A CD of a circle described a on AB. b b Hlence te areai o f the entire ellipse -area circle -B - a a ca 4. The cycloid y = -(2rx - x2)~+ r. versin1 - Put OD = x, DP-=y. Then the area OP.D - f ydx. D But since y is a transcendental / function of x, it will be preferable A C \ to integrate this expression by parts. Thus A = /ydx - xy - Sxdy. QUADRATURE OF PLANE AREAS. 383 But fioml the equation of the curve we have dy /2r- x 2 dx ~ \ ~ or dy = dx. dzx x x.A = xy -f /rx -x2. dx. Now f /2zrx- x2dx - fyldx where y1 is the ordinate DP1 of the generating circle, corresponding to the abscissa OD = x, or f V/rx - xdx -- area OP1D... area OPD = xy - area OP1D, and when x = OC = 2r. area semi-cycloid OA - = OC X CA - area semicircle OPI C 1I 3 = 2r.'r ~- - n'2 = 3 sr,.2 2 ar.. rea entire cycloid = 3-r2 = 3 area generating circle. 104. Prop. To determine a general formula for the quadrature of polar curves, their equation having the form r Fd. Let QX be the fixed axis, QP the B radlius vector, forming with QX an

Q Ph[ but < QP 0O. Also the ratio 1 dr t d2r t2 x rLt 2 (r 2r d 3 +d21;, QP__K 1.2,2 r X rt 0 t d2r 1 2 dr2 t2 - -' dd2 1. 2. (1' d2' bI when t = 0. 334 INTEGRAL CALCULUS. Hence at the limit, when t is replaced by dO, and QPP1 becomes dA, the value of QPP1 will be equal to QKP or QP1O. Thus we shall have dA - -72. d6...A -,,fr2dO.... (V2), the required formula. 1. The spiral of Archimedes r = aO. 1 1 1 1 r3 A = fj.2d=- a2fd82d = ca3 + C = -+. 1 3 If A = O when r = 2r, then C= - Ga r /1i3 r 3\ 3 ^ 3 A 1; and when = 1-2, - (2 s\ a 6\ a For the area of one convolution estimated from the pole, we have the limits r1 = 0 and r2 = 2ara. 4..A =- ^ 3a2. 2. The logarithmic spiral from r = r1 to r -= r2. o0 o 1 d Here r a... dr = log a. a. dl and d6 = r log a r 1 1 1 2 -. A = 4 d == ~-^~_...2 + C. 2 2 log a 4 log a -m(r22- r12), between the limits r1 and r2: the quantity m denoting the modulus 3. The hyperbolic spiral from r = r1 to r 1- 2. a adO 02 a Here r = —, dr -- =d a- d =. _ 7r. A _ =-1= ad ar + - a(r- - 2) betweenli ldie im22'2 its r1 and 1r2. QUADRATURE OF PLANE AREAS. 335 4. The lemniscata r -= a2cos 2 8. 1 1 12 A = -fr2d0 =- a2 f cos 26Bd - a2sin 20 + C. Put A =- 0 -whe hen 0; t C 0, and A - sa2sin 20, 4 1 1 which gives, when r = 0, or 0- j A - -a2, E. Elntire area = a2 - square described on semi-axis. 105. Prop. To find a formula for the quadrature of a plane curve, wher. its equation is given by a relation between the radius vector, and the perpendicular upon the tangent. pdr 1 1 fI prdr Since d6 = = A 2 = /d. (,.2 _ 2)2 02 -p ) 1. The involute of the circle A 1 f pirdr \ 1 =% f ]2 f(r12 ) a2'rdr (?2 2 - (72 - 2)= - + C; and this, between the limits p = 0, and p = 2era, within which the entire circumference is unwound, gives A -= qr3a. Co?. The area included between the involute ABS, the circle, and the tangent AS, is equivalent to that swept over by the radius vector, and therefore equal to -* 6a c2(r2- where c- a+2b, a and b 2. The epicycloid p2 -= c -,2 w e a2) being the radii of the fixed and generating circles. 336 INTEGRAL CALCULUS. A = f p2rdr I c 2 (- ) rdr. 2 ) 2- 2 a ( Put (r2 _a2)2 Z Z. ~.2 Z2 _a2, rdr =zdz, (C2 _ 2)2 - ( - a2 - 2), c I1 (2 _ a2 _ z2)- z2dz Z(c2 _- a2 2-)2C (C2 - a2).r dz e(c2 am -)c (c - (C2 ) C- _ z2) Z(C2 - at _ Z2) fC C2 _- 2)C 81~2 -] a C. ~ 4-~-~ - si~+ n- -- -- - + S 4a 4a Lc2 - a This, between the limits — a and r=c, v gives 4a 2 (a2 3a+b + Tb2) = OIPVO. But OIL =- t rab. 1 b2q.IP VL =- epicycloid -- (3a + 2b) f2 2a and, ]VI1L_ - (3a + 2b), the entire epicycloid. a If b = a, then epicycloid 4 rb2 -_ ra2 = area fixed circle. If b - a, then epicycloid = 5rb2 == 5aa- = 5 area. fixed circle. CHAPTER III. QUADRATURE OF CURVED SURFACES. 106. Prop. To obtain a general formula for the quadrature of a surface of revolution. Let AB be the arc of a plane curve T which revolves about the axis OX, P and P P, points taken on the curve so near to each other that the aic- PP1 may present A its concavity to OX at every point. 4 x D D, Put OD x, DP y, D)P,= -, DD1Pl -, P - P =s. The surface generated by the arc PP1, is intermediate in magnitude between those generated by the chord PP-, and the broken line PTP1. Denoting these surfaces by C and B, we have B (PD -+ TD1) 2P T + (TD12 - P1D12) r f! -- -jPD + P,D,)PP,. 2 (2PD + VT)PT + (2P1D, + P T)P1T (2P) + VP1)PP1 F dy( dI 2d d(lv h +2 X[2f+t I dA 2 -&.] [A2+ C2 dX2 1.2+ ) ]c Dividing numerator and denominator by h, and then passing to 22 338 INTEGRAL CALCULUS. the limit, we obtain - = 1. And hence the limit to the value of the surface C, generated by the chord, will be a proper expression for the elementary surface generated by the arc PP1, when that arc becomes indefinitely small. But at the limit, when h = dx, C = -2ry + d-Yd/x. Hence we have for the differential of the surface, dA = 2 ~Y( + 1d X, and.. I~ f + di).. ( W) or, A = 2yds.....( T). 107. To apply (W), we elimin:lte, by means of the equation i., the generating curve, y. and, ad then integrate between the given limits. Similarly, we apply ( WI) by expressing y in terms of s, or ds in terms of y and dy. EXAMPLES. 108. 1. The surface of the sphere. y Here the generating curve is a circle whose equation is y2 _ -2 2 -_ X2r dy; 2dy y + x 2.2.'. A f2rJ = rydx rf dx - = 27r7x -+-. Put A - 0, when x - -r; then C = 2qrT2... A = 2tr(r + x), which, when x + -r, gives for the surface of the entire sphere A = 4rur2 = 4 great circles. For the zone whose height is h- =x -- x1, we have A = 2tr(x2. - xj) = 2rrh. QUADRATURE OF CURVED SURFACES. 339 1. The paraboloid of revolution, y2= px, d=y + y2 l X d- = ~ + — = 2 - _, 0 x2 X = 2 (y2 + p2) dx = 2rfS(2px + p2) cl (2x X+1p2)7+ C [(2px2+p2) 2 (2pX1+p2) 2]. If the surface be reckoned from the vertex, we shall have x1 - 0. - o. A p [(2px,2 - p2) - 29]. 2. The surface generated by the revolution of the Catenary about its axis. The equation of the curve is s2 - x2 + 2az.x (x -+- a)dx acd aclxx *. ds ( a a - and cly-v= d-~dx-: 2=-x. x +- 2ax -2 + 2ax 8 Now, applying formula ( WT), and integrating by A i r B parts, we have,\ D/ A = 2f 8yds = 2r(ys - fsdy) = 2<{(ys - afdx) A = 2r (ys - ax) + C. But when. =0 0, y = and s = 0,.. C = 0.. = x2 2 (y - } +2ax - ax). 3. The surface generated by the revolution of a semi-cycloid about its axis. Here dy = - d and s = 2 -rx~= - 8. x.A - 2Srf yds -= 2(ys - fsdy) = 2<{(ys - f2 rx 2~ Adx). O =B 2wyt -. Srx+(2 - _ 2] + r C. 02 But when x =- 0 A 0,.. = - — ~ r. 3d 340 INTEGRAL CALCULUS. 0 I A* o * A jlYx 3 ) r2]; 32 and when x = 2rl, A - Sq2r2 2- ~rs,27 the entire surface. 4. The surface generated by the revolution of the cycloid about its base. In the formula A -= 2,fyds, the quantity y denotes the distance of a point in the revolving curve froml the axis of revolution, and must therefore be replaced in the present instance, by 2r- x... A (2?r - )ds = 2~(2?' - )/ dx = 25r 1 (4r7- (4) + 9. But A=-, when x= 0..'. C=0. A. =2 2r(4rx2~ — -x), g O and when x = 2'. 32 64. A --. ~2; and the entire surface 2A -= c~,2. 0 109. Prop. To obtain a general formula for the quadrature of any curved surface, whose equation is referred to rectangulatr co-ordinates. c Let CA LB ibe a portion of the surface incladcd between. the planies /~ ~~~ F of xz anld y,'z a-ind the planes BP 1, G AP1 drtaiwn Ia:rallel thereto, / / Put.- OA= xz OB1 A P1= -,, PP =- z, A CBP = A, and let,/ D, z =- (x,y)...(1) be the equation y G' of the surafe. Then, since the value of A will be determined by the assumed values of the independent variables x and y, we shall have A = (xy). Now when i: receives an increment Al1 -= h, the area A takes the increment lAD, becoming (dA h d2A h2 d3A 13 ~A -d-x ^ -1 s^lx2j 15 ci 1t " QUADRATURE OF CURVED SURFACES. 341 Similarly, when y alone takes an increment Blbl = k, A takes the increment BG, becoming dA k d2A k2 d3_A Ic3 A, = ~(x,Y + k) = + +' +. + &c y y2 1.2 y3 123 But, when x and y increase simultaneously, A takes an increment AD) + -BG - PI, becoming dA 6 dA k d2A 12 d2A hk d2A c2 3~- + ~'+ I dy 1 I' dx2 1.2 dxdy * 1 dy2 1.2 d3A ] 3 cd3A hI2 k d3A /jk2 d3A 3 qE- t- 4- J- f- &e. x3 1.2. 3 dx2cly 1.2 + dxdy2 1.2 dl3 1.2.3 PI= A A - (A - -A) - 2 -) d 2A /h d3A 1 2/c d1A hAk2 + +-H + &C. dxcly 4 cd 1. 2 dty2 1.2 P- I 2A d 3A A d3A k hk -cl xdy dcl2dy 1. 2 + dx.d dl 1. +. which, at the limit when AI 0 and k = 0, reduces to PI d2A PI, c~ xdy.... ( ). Now this quotient, which results from dividing the elementary surface PI by its projection _P11 on the plane of xy, is equal to 1, where v denotes the angle formed by the tangent plane at the cos v v point P with the plane of zy. But from the theory of surfaces (Diff. Cal., Art. 177), we have I cos v _ o d z2 dy2 dcIA c1 + d2 + I+ co- c + dxdy -V dx dy2 cI2A Now, since the second differential coefficient is obtained by dxdy 342 INTEGRAL CALCULUS. differentiating the function A of x and y, first as though x were alone variable, and then as though y only varied, we shall obtain d2A the value of A by multiplying the value of ~- by dxdy, and then performing two successive integrations with respect to x and y, the order of these integrations being immaterial, since that of the differentiations is arbitrary. This double integration is indicated by the symbol ff, and the result is called a double integral. Thus /V z CIa2 dZ 1 A 1 + -v 2 +,)C dxdy... (V Z), the required formula. The limits of these integrations, in the case represented in the diagram, are y = 0 and y -= OB1 - b,x 0 = and x. = OA1 a. But if the surface were terminated laterally by a cylinder (instead of by planes parallel to xz and yz), the elements of this cylinder being parallel to the axis of z, and its base in the pla.ne of xi, represented by the equation y, =-f, then the superior limit of the first integration would be y - yl =fx, the inferior limit being still zero. This will be rendered plain by an example. 110. 1. Required the surface of the tri-rectangular trialngle ABCo From the equation of the surface x2 + y2 -+ 2 =.r2, we obtain d x and dz y p dx z dy z dz2 cda2 r 1A+f dxdl[^ d o ~ a A — -dy= 2 —/2 B =' f sin — ( )2.) The lim.its of this first integration with respect to y are y 0 andy = r2 —2 DE. QUADRATURE OF CURVED SURFACES. 343 But when y = 0, = 0 -02 _ 0, and when y = / x — 2 = 1, and sin-(1) - r. 1. 1 1'. A= - srf dx = r- rx + C - r2 between the limits x = 0, and x = r. 2. The axes of two equal circular semi-cylinders in- tersect at right angles, form- / ing the figure called the groin. Required the entire / surface intercepted upon the / two cylinders. A Assuming the axes of the cylinders as those of x and y respectively, the equation of the cylinder whose axis coincides with x will be y2 + -2 = 2, and that of the cylinder whose axis coincides with y will be x2 -- z2 r2. The entire surface to be estimated is projected upon xy in the rectangle ilB CF, and the triangle 0 GF is the projection of one-eighth of this surface. To compute this portion to which the equation,2 +- 2 = r2 applies, we have A l d-z2 - d72] dxdy, in which the limits of integration are y = 0 and y = x, x = 0 and x = r. Z=tdz _ x dz But from the equation x2 + 2 = 2, we get d=-'- =0. dxct~ z, dy0' rclxdyyx _ xdx y- r = - - _ between the given limits. - - J _x2 d [ x2 or A = r~r - x2 + C = 2 between the limits x = 0 and x = r.. SA - Sr2, the entire surface of the groin. CHAPTER IV. CUBATURE OF VOLUMES. 111. Prop To obtain a general formula for the volume generated by the revolution of a plane figure about a fixed axis. Let OX. the axis of x, be the axis of revolution, ABCF the generating area. Put A E OD-x, DP_-y, DD —=h, D1P1-Yl, and let y = Fx be the equation of the o F D D bounding curve AB. The volume generated by the revolution of the small quadrilateral BPPlDl is intermediate in magnitude between the cylinders generated by the rectangles PD1 and ED1. But dy tl dly h2 cylinder ED _y,12h _ y12 clx dX 1.2 &) cylinder PD1 -, ry2h y2 1,,+ 2 dy7 + P2 * z * 2 + dx y d 1 2. y dx- y = when h = 0. Therefore at the limit the volume generated by DPPD1 -- cyl inder PD1, or dY =- ry2dx, and consequently V -= qrfy2dx... (X) the required formula. To apply (X), we substitute for y2 its value in terms of x derived from the equation of the bounding curve AB, and then integrate be tween the given limits. CUBATURE OF VOLUMES. 345 112. 1. The sphere. Here the equation of the circle which bounds the generating area is x2 + y2 = r2.' V- fry2s=x =,rf(2 _ x 2)C = (.2x- 3 3) + C. Put V= when x -r, /1 \- 3 / 3. then C-= -- r3 _ 3_. \. /." A- r 0 +T B 1 2 4 V.. V = - ('2 -- X3) -+ 2-'3 and when x = r, V =4'3. 3 3 i, 2. The ellipsoid of revolution, generated by the revolution of the semi-eliipse about its greater axis 2a. Here 2=2 ( - _2), (a2_2)c 6 2.)- which gives between the limits x = - a and x = + a. 4 2 2 V = arb2a =- (2a. r6b2) -= circumscribing cylinder. 3 o0 d 3. The paraboloid of revolution y2 - 2px. V= 2?p f xdx — = rp2 +- C. If V = O when x 0; then C- O and V-= -ipx2; which becom'es, when x - xi and y = y, 1 1 V -= pxl =2 - x1. y12 =2 - circumscribing cylinder. 4. The parabolic spindle generated by the revolution of the para, bolic area A QB about the double ordinate AB. Put OQ = a - b, O D = b, DP =,y. Then QC - a y. x2= 2p(a-y) and V=( a — -) dx- f(4a2p2- 4a2x2 —c)dx. / 5(6x3 5 + -P-P 346 INTEGRAL CALCULUS. But if V = when - = 0, then - = 0; and when x = OA b. ah3 b5\ Q V _ a2b- -p + 20p2)P b2 or since a. / 21) B o D VF a2b (1 - + )- = l- ra2b = volume A 0O. 16..volume A QB = - ra2b. 15 5. The volume generated by the revolution of the cycloid about its base. Put O V=2r = -x, DPD =, V =- z = 2r - y. Then from the equation of the cycloid, d =( -)Iz and since dz =- dy. dz ) —f- v' ^ ^d x 2 o _ dy ydy — E -A 5 A2 0 D B V r:jy2dx='-fy2( )2d dy -fj(2r-y)7dy. But by formula (A), 1 01 0 3 2-2 _ ^Y ^-) 25- - +,+ >,(2r-y) dy f2(2r -y) 2dy - - (2 -y) + 2 I (2-y() - 3 ~ 3_ fJy 2(Q-y) cly= y (2r —y) -'f j (2r-) dy 1 1 1 1 1~-~ p d-(- y)+'-(2. Also jy 2(2r- -y) dy - J= - = versin- V - - +)2(: ++y+,) v- —. versinill - G. 0. r CUBATURE OF VOLUMES. 347 Put V - 0 and y- 2r. Then C C-= - r2 and when y = 0. 5 5 = - r,32 = volume B VO. 5 5. volume BVA _- 5r3r2 = 2r-r -2' - circumscribing cylinder. 2 S 6. The volume generated by the revolution of the cycloid about its axis. (See last Fig.). Put VI - X, IP — = y VO = 2r, and to facilitate the integration, introduce the variable angle VCE = 6. Then x = r(I - cos 6), y r= r(sin 6 + 6), dx = r. sin 6d6. V' = Sfy2clz' = fr3 f(sin3) + 26. sin26 +- 82. sin 6)d, 1 2 4 But fsin3J. d6- = sin2C. cos - -cos -, fiom 6 = 0 to = r. 0 3 0 2f. sin2. d = -6. d6 -fc. cos 206d 62 - sin 26 + fsin 26. d by integrating by parts. 2 ~ 262 1 1 1 s I, 1 = 82 9 sin 2 - -cos 20 -= ffrom 6 = 0 to 6 =%, 2 M 2 4 and J62. sin 6. d - 6 2 COS 9 + 2 f6. cos Od3 =- 62 COS +- 26 sin 0 - 2 fsin Od - 62cos 6+ 26 sin 6 - 2 cos d 2 _ 4, fiom l = to 8 = -.. Entire volume =,3 q2 - ~)-~ 113. Prop. To obtain a general formula for the volume of all solids which are symmetrical with respect to an axis. Such solids may be generated by the inotion of a plane figure, as ABCD, of variable dimensions, and of any form, whose centre G remains upon the axis OX, its plane being always perpendicular to OX, and its variable area X being a function of x, its distance from the origin. 348 INTEGRAL CALCULUS. By a method entirely similar to that a.ppliedl to solids of revolution, we may show that dcV = Xdclx, and.*. Xdx... (X1), 0 the required formula. To apply (X2) we must express the C value of the area X in terms of x, and then integrate between the proper limits. Cor. The same formula is applicable to any solid generated by the motion of a section of variable dimensions parallel to a given plane, when the area of the section can be expressed in functions of its distance from the fixed plane. 114. 1. The ellipsoid with three unequal axes. x2 y2 z2 -ere rwe have + + a2 b2 C2 or b2C2Z2 + a2c2y2 -- c262~2 = ct2C2. Make CC1 = x, and put successively B y 0 and z -- O. Then when ~ __,~ 2 A c 2 I and when b'bc Z =0, y = -oa2 x 2 = )D C1. 1F area BD,FEl = X = (a2 - 2); and this value substituted in (1i) gives bc12 _ Z) d z = (a2 - 2 23) +C c 2 3 Put V = when x = a; then C = 2 - a andl when x=-a a 2 4 2 V -= 7 bca = entire eliipsoid -- circumscribing cylinder. 3 CUB TURE OF VOLUMES. 349 2. The elliptical paraboloid cz2 + by2 = a2x. Put successively y = 0 and z = 0; then C = a -) and CD = a. Then X a=. And,7ra,2,, 1 ra2a2 2 A V -. dz - = ~2 +.._ / i/bc 2 tb-c If V=0 when X- 0, then C= 0, and 1 qca2x2 2. bc,~ I 7 a2X 2 1 1. When x OA x -- x1 V - X circumscribing cylinder. 3. The groin or solid formed by the intersection of two cylinders whose axes are perpendicular to each other. 1st. Let the bases of the cylinders be 0 equal, semi-circles. Then the generating section ABLCD / \ will be a square./ \ Put OG= G = EA=, O-, = -- — x, - CT, =- y = ElA,. l Then ALB1 CD1 = 4y2, and from the / equation of the circle EO, F 2 _ %x _ -. 2.' V-= fIXdx (SZ —42) = (Srx-4)dx 4rx2 - 3 + C. But V= 0 when = 0,.'. = 0, and when x r, 8 2 2. V= - r`3= r..2r. 2r-= circumscribing parallelopipedon. 2d. Let the bases be unequal parabolas. Then the generating section will be a rectangle. Put OG —c a, GE-b =b, 1 = b0 x, G1E = y, EA1 = y 850 INTEGRAL CALCULUS. Then y2 = 92x, y2 = 2p1lx. X.. 2y. 2y1 = Sx'. V fXdx -Xc 8 pp/~ f-xxc 4x2 /pp1 = 2x. yy1, and when x = a. 1 1 V = 2abb = - a. 2b. 2b = circumscribing parallelopipedon. 4. The Conoid, with a circular base. A F Put B -A = a, DE- 2r, D = x, I- = y. Then the generating triangle IFH7 -- X- ay\ =a x - x2. \. = Xclx =a C - X2. dx \ a. segment D G1. D G ~E and when x ='2r, V a (semi-circle DIHE). or volume conoid =- volume circumscribing cylinder. 2 Cor. A similar result will be obtained if we suppose the base to have any other form, the generating triangle being still perpendicular to the base. 115. Prop. To obtain a general formula for the volume of a solid bounded by any curved surface, whose equation is referred to rectangular co-ordinates. First suppose the volume bounded by the co-ordinate planes of xy, xz, and yz, by planes parallel to xz and yz, respectively, and by the curved surface Cafb, whose equation is z1 = (x,y). z Put OA, = z, OB1-AP1 = P y, P1p1 =i, P1P — z 1, / > A1la - clx, P1G c-= i, pi2p =. ( 3' _ Let the volume be iltersected by O G planes AG, and ali, parallel to yz, / and including between tlhem the lamina or slice A1l: let this la- L,/_______1 /mia be ct by G, I, mina be cut by planes bi^ BD1, &c., CUBATURE OF VOLUMIES. 351 dividing it into prisms such as PI1, &c.; and, finally, let each prism be subdivided into elementary parallelopipedons, such as rzd by planes parallel to xy, the successive planes being at distances from each other denoted by dx, dy, and dz, respectively. Then the volume of one of these elementary parallelopipedons will be ex1pri ss:.ct by cdxclclz; and. if this be integrated wvith respect to z, regarding x and y as constant between the limits z 0 and z = z~ = P P = F(x.,y), the result obtained will represent the sum of all the parallelopipedons contained in the prism PJi. A second integoration, with respect to y, between the limits y=0 and y —A G,, will give the sum of the prisms contained in the lamina A1; and a third integration, with respect to x, between the limits x= 0 and x = Oa1, will give the sum of the laminae, which constitute the entire volume. Hence the required formula is V = Sfdxdydz........ (1). The symbol.J fff denotes three successive integrations, with respect to the variables x, y, and z, and the result is called the triple intlerac of dxclydz. Cor. If the volume were bounded on every side by the curved surface, the same formula (1) would apply; but the limits of integration would be different, those of the first integration being z = zi anld z = z2 where zr and aZ are the two extreme values of z corresponding to the same values of x and y, and derived from the equation of the surface; those of the second integration being y = Y1 and y = y, the extreme values of y corresponding to the same value of x, and derived from the equation of the section perpendicular to OX; and, finally, those of the third integration being x1 alnd x2, the extreme values of z. 116. 1. The tri-rectangular spherical sector. Here the limits of the integration are z= 0 and P1P-= -/r2 —s2 —y2, y=- and Y-D1y=LE:=, x —0, and x= OA=r. 352 INTEGRAL CALCULUS.. = f ffdxdydz = ffJzdxdy ff= J 2 x_ 2 __ y2. dzdy. But / f r22 ~x-yc2 _ y=2 dy O - yy )2 / y \ __ -(/ 2_ /_ _ 2_2_y2 / 2) - 0 X +2 w 0ose a a 1.i2in 17,02 _x) between the limits given 4 1B. t(r2x.2) s ml be between the limits. 2. The volume cut from a paraboloid of revolution, the equation of whose enerating curve is y2 by a right cylinder with a circular base, its axis passing through the focus, and the diameter of its base being equal to p9. The equation of the paraboloid being y2 + z 2: 2px, and that of fth e ( - X x2) 9l- -- -T the limits. - 2nd the to the liner x, the limits of interation in the presen case will be z. -I- - I/pV ~ y2 and 2 = 2pxy ~, yc + P 2 ~ )x ~ x and y:- = px-~ x2, / x- 0 and x- =p. 0 - V. =-fjfdxdydz =ffrclxdy =J:'(2ix -- 2)2 dlxdy. o A _I )-2+. px dy But V/^=- {-yy d~ —-. = y(p( - y2) + pX. si~n= xl/~2 — xT2 + 229. sin-' \ + ~~ between the limits. CUBATURE OF VOLUMES. 853 ~2 v-X P~ 2dX.'.V-= 2 L[ x V^^+ 2px. sin-i2 1 ]/ d V3 2cm2p.. <> (2_2) _ -272,3 Sill- / 2X + -= (p2 - xL2) + 2- px sin- li- 12 23 in = + -) between the limits x -= 0 and x =p. 117. Prop. To obtain a general formula for the volume of a solid bounded by a surface whose equation is referred to polar co-ordinates. Let the volume be divided into elementary wedges such c as GiD] CO by planes drawn through the axis OC. Let / \ each wedge be subdivided / \ into elemenitary pyram/ids, such as FGCDEO, by coni- / cal surfaces generated by / the revolution, about the axis OCC, of lines OD, OE, ~ &c., inclined to OC in constant angles. Finally, let each pyranmid be subdivided into element'ary parallelopipedons, such as fdl1 by concentric spherical surfaces with their centres at the origin 0. The co-ordinates of a point d are Ol =r, dOODl —, and A OD =- v; and the three edges of the elementary parallelopipedon fd, are d:I -_ d=, (l de = =rdi, and dcZ = r cos. cdv; the last expression being obtained by observing that when the line OD revolves aroundl the axis OC, the point d describe:; a small are dt whose 2 3 354 INTEGRAL CALCULUS. centre lies upon the axis OC, and whose radius is the perpendicular distance of d from that axis, and therefore expressed by r. sin ZOd =- r cos 0. Hence the volume of the parallelopipedon will be expressed by r2cos dv. dIdr, and.. V f/Jr2cos dcv. dc.dr....(1) will be the required formula for the entire volume. The first integration, if perfoirmedl with respect to r, while v and I remain constant, will give the sum of the parallelopipedons contained in the pyramid 9D'TFGO, the limits of the integration being r = 0 and r -= 01)= F'(Fv, ). A second integration with respect to 6, while v remains constant, will give the sum of the pyramids contained in the wedge GCD, CO, and the third integration with respect to v will give the sum of the wedges which constitute the entire volumne. 118. 1. The hemisphere with radius equal to a. Here the limits of the itegrations are = 0 and r =- a, 6.- 0 and = -,, v- = and v 2 —r. 2 V. v= ffJ''2 cS. dv.ddrG = f Jr3 coo8 s 0 vd - af cos. dv. d 3 33 0 0 0 0 0 2. The volume cut from a sphere who,,se r lldils is a, by a cylinder with a circular base whose radius -- b, the centre of the sphere being on the axis of the cylinder. / Here we shall have for half the required volume or AB CDE, A 0 B| 2 V =fffJr2 cos. dvddr, the limits of integration being CUBATURE OF VOLUMES. 355 1st. r- 0 and r — O = b se 6, 6-0 and d-=cos —l- v = 0 and v- =2'. a b 1 2d. r=0 and r-a, -=cos — and 0 -, v-=0 and v= 2-. The first set of limits give fffi cos ddv.ddr = - ffr3 cos 6dv.dd - ff b3 sec3 cos dv. Sd 1 3 = b3Sffsec2.dvd = -b3tan. dv 3 0 = b3tan (cos- -)dv 1/ a'-b2 2 - 3 tan tan n-l )v = - b2 -- - and the second set of limits give o 3 0 a3f sill dv 3 = - rC -a3 s in fav a3 sin \S — 1 I3 2 3 ( t / ) e V = 2[3-_(a.-b2)v a-b62]\ and V3 (a 2 o3 PART III. INTEGRATION OF FUNCTIONS OF TWO OR MORE VARIIABLES' CHAPTER I. INTEGRATION OF EXPRESSIONS CONTAINING SEVERAL INDEPENDENT VARIABLES, 119, When a differential expression, containing two or more indepeodent variables, can be obtained directly by diffierentiating.soime function of those variables, it is said to be an exact diferential. Thus xdy + /ydx is an exact differential, being equal to d(xy); so also is 3x2dy 3ydx — + (ydcx -- 3xcdy, being equal to d (32y —3x); but x2Cdy -Sclx is not an exact differential, there being no expression whiech, when cdiff1ereitntiated, will produce that proposed. 120. If Ca differential be exact, its integral can be determined in all cases e b methods which will be explained, but we shall first establish whereby to distinguish exact differentials. 121 Prop.1 To determine the conditions which indicate that any proposed differential is exact. Let the proposed expression be Pd -+ Qdy, in which P and Q may be functions of one or both variables. If this expression be the exact differential of some function u of ar and y, we shall have du =, Pdx + Qy......... (1), FUNCTIONS OF SEVERAL VARIABLES. 357 But by the general process for differentiating a function of two independent variables, we have due = d + d d.(2) d x dy And since (1) and (2) must, from the nature of the supposition, be identical, the following conditions will exist, viz.: P d e (3) Q (4). Now differentiate (3) with respect to y, and (4) with respect to x, and there will result dP d 2t dQ d2tu an d - dy dxdy dydx But it has been shown that the result of differentiating u, with respect to x and y, successively, is the same, without reference to the order of the differentiations, or that d2L d2u dP dQ dxJdy dydx dy dx Hence, when the proposed differential Pdx + Qdy is exact, the condition (5) will be fulfilled. The converse is equally true, as will appear fully when we attempt to integrate such expressions, and hence the condition (5) is called the test of integrability. 122. Now let the proposed expression be Pdx + Qdy + Rdz, invoiving three independent variables. If this be an exact differential of some function uz of x, y, and z, then du dux du du - x + -- dy + -- dz = Pdx Qdy + Rdz dr dy cdz du du du whence P _ - Q T= R - ~, or by differentiation, dx dy dd " dP d2u dP d2u dQ d2x dQ d2u dy dxdy' dz dxdz' dx dydx' dz dy dz' dR d2u dR d2xu dx d-dx' dy dzdy 358 INTEGRAL CALCULUS. d2u d2U d2,u d2U d2u d2u But dxdy dydx' dxdz dzdx' dydz dzdy Hence we have three following conditions of integrability, dP dQ dP dR dQ dR dy dx' dz dx' dz dy Similarly, if the expression were Pdx + Qdy +- Rdz + Sds + &c., 1 involvihg n independent variables, there would be - n (n-l), con. ditions of the forms dP _dQ dP _ dP d S d _Q dR d Q dS dy dx-' dz- dx ds' cdz dy ds -dy 123. 1. Is a2ydx + x3dx + b3dCy + a2xdy an exact differential? Here P = a2y a+ x3 and Q -= b3 -- C2x. dP dQ 2 dP dQ d = a d = a", d*. d and the expression i? dy' dx dy dx integrable. dx dy xd al 2. Is ~ ^+ + -~' an exact dilierential? 1 I (.2 + 2;, y yx +,2) Here P = (, _ y2)+ and Q - y-l[l - (2 + y2) 2X], 3. Isxdex d iffetial 3. Is 3xdy - 4y2dx an exact differential? 4y2 Q=3 dP dQ dy — 4 =3X, -- 8y, (X + 3) and since 8y and 3 are not equal, the expression is not integrable. 124. Prop. To oblain a general formula for the integration of the form du = d Pdx - Qdy, when the condition of integrability is satisfied. EXA CT DIFFERENTIALS. 359 Since the term Pcdx has resulted from the differentiation of the function u, with respect to x only, y being regarded as invariable, it follows that u will be obtained by integrating Pdx with reference to x alone; but as ut may have contained terms involving y alone, which terms necessarily disappear in a differenltiation with reference to z, we must complete the integration of Pdx, not as usual by adding a constant C, but by adding a quantity Y, which is some unknown function of y and constant, and we thus provide for the reappearance of such terms as may have disappeared in the first differentiation. Thus we get u = Pdx + Y,... (1), in which the value of Y remains to be determined. Differentiating (1) with respect to y, there results du d/J'Pdx dYr du dyl dyd q d, But -- Q. cly dy dy dy dY Q df Pdx dY ( dJfPdx\) ~ - ~-~9 ~ ~dy =)~ 1 dy dy dy dy) and by integration Y =- - d fPdxj dy. This value reduces (1) to the form df Pd( u JPdx + / Q -dy (2), which is the required formula. 1250 It is necessary to prove, however, that the coefficient d f Pdx Q - of cly, does not contain x, since otherwise, the second integration would be attended with the same difficulty as the first. Differentiating that coefficient with respect to x, we obtain dQ d2S Pdx Cd Q d2SPd.x dQ dP dx dydx dx dxdy dx dy 360 INTEGRAL CALCULUS. and this is equal to zero by the condition of integrability, which is supposed to be satisfied. Hence the coefficient of dy in (2) cannot contain x. 126. This proof also establishes the truth of the converse of the d/P dQi first proposition, viz.: that when the condition is satisfied, the integration is possible. 127. By a sinilar process we obtain a second formula u -f Qdy+ - [ d ] d......(3). in which the coefficient of dx does not contain y. Cor. If there were given dt = Pdx + Qdy + -Rdz, we would write u =- fPcl + V in which V is a function of y and z. Then differentiating with respect to y, we obtain dy dy dy dy dVf Pdy dJ Pdx dy d'' dydy and by integrating with respect to y and adding a function Z of z, we get V= f[Q d+ Q j dy 3+ z. Now differentiating with reference to z, we obtain dZ du df Pdx dfQdy d F dSfPdx dl dz dz dz dz dzLJ dy Jy -f[s d dj'Pdx dfQdy __d __f AI in cz s ide dt o+ d y. in which the coefficient of dz is independent of x and y. EXACT DIFFERENTIALS. 361 U = f Pdx +fQ - df]Pdxi/.. ~-I~+/[Q^^ -I ~S[ ~.S~cl~ f Pdx f d dy Pdx 8 + -[R d_ dfd + d(I df dy)]dz. 128. In practice it will be found usually more convenient, and always more instructive, to apply the method, rather than the form. ula explained above; especially where there are three variables. EXAMPLES. 129. 1. Integrate dt = (3x2 + 2axy)cdx + (ax2 + 3y2)dy. Here P 3x2 + 2axy, Q = ax2 + 32. dP d_ _x- dQ d=- 2ax = cd' and the expression is integrable. dy dx1 But f Pdx = f(3x2 + 2axy)dx = 3 + ax2y,. fdfd ax2 and Q - d ax2 3y2 - ax2 3y2. These values reduce (2) to the form, u = x3 + ax2y + f 3y2dy = 3 + - ax2y + y3 + C. 2. Integrate dc -- (3x2 — x2)dX - (l+6y2 —3 2y)dy. dP dQ P = 3xy2 - x2 Q =-(1+6y2-3x2y), d = d SPdz - S (3y2 - X2)d = _x2y2 - 1 3 4 2 3 d Y du ^ 3x2y = - (1 J- 6y2 - 3x2y)- 3x2y = - - 6,y2' dy ay dY'. d dy -dy - y2dy, and Y - y- 2?/ - C'. *y 3 1 u = -X2y2- y 2y3 + C'. 2i 362 INTEGRAL CALCULUS. 3. du = (sill y + y cos x)dx -t (sin x + x cos y)dy. dP dQ = cos y + cos x - d' = J'(siny + y cos x) dx = x sin y y sin x + Y, dY du -. - x cos y - sill = 0,..' Y=, dy dy and u = sin y + y sin + C. ydx v x dy _ x ydz 4. du = y- + I- x a-z a - (a - 2)2 dP dQ d P y dR dQ x dR dy a- clx' dz (a z )2- dx' dz (a - Z)2 dy yd,v xy dV du x u-f d = +,.'. ~ =0 d V=- a-z a-z dy dy a —z Then dZ, du y - =. dz dz (a - z)2.u=- Y + C. a —+ 130. In practice the preceding process may be abridged by first integrating Pdcx, then integrating the termns in Qdy, which do not contain x, and finally integrating those terms in Rdz which do not contain either x or y, and adding the results. That the complete integral will be given by this process, appears immediately, from the consideration that the integration of Pdcx necessarily gives all the terms in the integral sought except such as contain y and z without x. Hence in integrating Qdy we must not consider any term which contains x. as otherwise we would introduce into the integral new terms containing x. Similarly the integration of the selected teri's in Qdy gives all the remaining terms except such as contain z only, and therefore in integrating Rdz we must neglect all terms involving both x and y. HOMOGENEOUS EXACT DIFFERENTIALS. 363. xcdx + ycly + zc zdx - xdz Ex. du - ~ __ - +rdxp2d / l. d - x +- _- + dz2 + y2dy (2 + y2 + 2) This satisfies the conditions of integrability, and by taking the terms in Pdx we get J'Pdx (= 2 - +f id( = (2 + y2 + ~2)2+ tan-1 (x2+2+z2)!2 Now taking the terms in Qdy which do not contain x, we get fQdy = Syf y -= 1 3 and finally taking the terms in R which do not contain x nor y, - 1 2 fSiRd =J-f — z 2. 4..u = (2 + 2 + 2) n + tan-i +1 3 - + 2 + + C. Z 2 Emffo20 geneoTus Exact Dffer entices. 131. Although the methods of integration just explained apply to all exact differentials, yet another and simpler process can be used when the expression belongs to the class called homogeneous. A differential expression is said to be homogeneous when the suIn of the exponents of the variables is the same in the coefficient of every term. Thus ax2c{:9 - by2d?/ xdy + ydx, x2zcIx + xz2dx- xyzdy, and cx' (x - 3) are homogeneous differentials. The degree of the terms is estimated by this sum of the exponents; thus in the first expression it is 1, in 123 97 the second it is 3, and in the third it is 2 - 1 -- lo lo 364 INTEGRAL CALCULUS. 132. Prop. If an exact differential be homogeneous, and the terms of any degree except - 1, its integral may be obtained by simply replacing dcl:. d/, and dz, &c., by x, y; z, &c., respectively, and dividing the i e-ult by n + 1, when n denotes the degree of the terms. Proof. L t cld = Pclx + Qdy + Rcz + &c., be homogeneous and exact, in which P, Q, R, &c., are algebraic functions of x, y, z, &t., of the degree n. This must have resulted from the differentiation of a homogeneous algebraic function u = P1x + Qy + + &c.....(1), of the degree 2 + 1, since differentiation diminishes by unity one of the exponents in the term differentiated at every step. Put y = y1x, z = zl, &c., and substitute in P1, Q1, R1, &c., which quantities contain x, y, z, &c., involved to the nztc degree. Replace also y by ylx, z by zlx, &c., in (1); then each term in the value of u will contain the factor xn+, and.'. = P2xn+l..... (2), in which P~ is a function of y,, zx, &c., but does not contain x. Differentiating (2) with respect to x we get dX - = ({a + 1)P2.... (3). A similar substitution in the value of cdu gives du = Pdx + ~Qd(ylx) + Rd(zlx) + &c..... (4); and therefore the partial differential coefficient derived from (4) cldx by differentiating the products ylx, z'x, &c., with respect to x only, is P = P + QY 1 + Rzl, &c..... (5). Multiplying (3) and (5) by x and equlating the results, we get (n + 1)P2xn+l = Px + Qy1x + Rzxl = Px + Qy + Rz + &c. HOMOGENEOUS EXACT DIFFERENTIALS. 365.Z. = P2Xtn+1 = PX + Qy + Rz + &c. 2^ ~ n + 1 as stated in the enunciation. 133. When n - 1, this formula would make u = oe. In this case it is easily seen that the formrula ought not to be applicable, because it is not then true that the desired integral is an algebraic function of the degree n -q- 1; but on the contrary it is transcendental. 1. To integrate du (2y2x + 3y3)dx + (2x2y + 9xy2 + 8y3)dy. dP dQ dP = 4yx q- 9y2 = d-Q dy4 9dX the differential is exact; it is also homogeneous, and since n = 3 or 1- = 4, u- = y 2 x + + S 2y4 + oder 0, we hve - -- 1...dxPz (- 2y)d + ( y2 - Xy)dz - 2. duz ~- - - dP t integr d y il dQ 2y -tu. I_ d dy r dx' dz z dx dr z9 dy c, if he differential is exact; and being also homogeneous and of the order 0, we have n + 1 - 1. Px4 - y + RZ Yx xy -y2 2 y y ~xyz 134. When there are three or more variables, the application of the test of integrability will often be very troublesome. In such cases, if the differential be homogeneous, it will be found more convenient to apply the preceding process as though the differential were known to be exact, and then to ascertain, by actual trial, whether the given expression can be reproduced by differentiation. 366 INTEGRAL CALCULUS. Ex. du (2 q-z)dx d/ 2(xSy-zx)cls /xdz d 2s2vx 282 /y t S26 This being homogeneous and of the degree --, its integral, if possible, must be 2(2~ v )- _ y2xy_ 4- 2 (x/y-zlv. K + 2z + C 2~ 2 2 22 2 2 2 O -2Z+ z'iy ~+4y 4x+ 22~+ 2 ~ X2~ S2 S This, differentiated, gives )-z x -2 lX, ( Iy Y\ 2 )d xy dy 2(xyIV-zx)ds dz s2 282 83 S2 which is identical with the proposed expression (1). It must be distinctly understood that in the differential expressions, here considered, the variables x, y, z, &c., are wholly independent of each other. If, then, the conditions of integrability be not fulfilled, the integration must be impossible, since there is no relation between the variables, by the aid of which we might hope to transform the given differential into another of an integrable form. It would be otherwise if a relation between the variables were given in the form of a differential equation, such Pcdlx + Qdy = 0. Here the form of the first member may be greatly modified by the introduction of a variable factor (or by other methods), and thus the integration may be facilitated. CHAPTER 11. DIFFERENTIAL EQUATIONS. 135. A differential equation between two variables x and y is a relation involving one or more of the differential coefficients such as d I ( 3y 2f /V /dWy 3/2^ 2 -d d.y dy &c., or their powers L, t c. dx dx:s2 dxaC powerxs ) ()I (djl Suich equations are arranged in classes dependent upon the order and degree of the differential coefficient. Thus, when the equation dc 1y d(n involves only the first powers of it is said to be dx' dx',tis id of the nt t order and 1sl degree. When it contains only the powers of the 1st differential coefficient, viz.: d-,' (,d)''-'.(, it is of the 1st order and nth degree. And when it contains the nzt powers of one or more differential coefficients, and a coefficient of the'.tht order, the equation is of the ntl' degree and cl thl order. 136. The resolution of a differential equation consists in finding a relation between x and y and constants. This relation, called the primitive, must be such, that the given differential equation can be deduced fiom it, either by the direct process of differentiation, or by the elimination of a constant between the primitive and the direct differential equation. Hence the same primitive may have several differential equations of the samne order. Thus the equation ay + bx + c = 0.... (1) dgi by differentiatio 368 INTEGRAL CALCULUS. and by elimination between (1) and (2) we get the indirect differen. tial equations dy dy bx ~y + c - by = 0... (3), when a is eliminated; aydc-axdand ay +- c ax - - 0... (4), when b is eliminated. In each of the equations (2), (3), and (4), the variables are con nected by the same relation as in (1), which latter is their common primitive. 131. As the integration of differential equations can be effected in comparatively few; cases, it is found convenient to arrange them in the order of the difficulties presented, commencing with the simplest ease. D',/ferentcal Equations of the First Order and c Degree. 133. These are of the general form P+ Q - =0 or Pdx-t- Qcdy -0, in which P and Q may be finctions of both x a(nd y. The integration will obviously be possible, by the method applied to differential expressionas, whenever Pdx + Qdy is an exact differential, and the required solution will be of the form 7(X, y) = C, where Cis an arbitrary constant. 139. Again, the integration can be effected whenever the separation of the variables is possible, that is, when the equation can bo reduced to the form XdCx + Ydy = 0, where X is a function of x only, and Y a function of y only. The form of the solution will then be Fwh + rqy = e, which requires only the illtegration of functions of a single.variable. DIFFERENTIAL EQUATIONS. 869 The separation of the variables is possible in several cases. 140. Case 1st. Let the form be Ydx + Xdy/ = 0, in which the coefficient of dx contains only y, and that of dy contains only x. Divide by XY, the product of the two coefficients, and there will result dx dy X +- - = 0, in which the variables are separated. 141. -Ex. Given (1 + y2)dx - xdy = 0, to find the primitive relation between x and y. Divide by (1 + y2)x), then dix dy'y i -1 - 2 = 0,.. 2x2- tan-ly = C, I l+y2-. which is the required relation. 142. Case 2d. Let the form be XYdx + X, Yjdy - 0, in which each coefficient is the product of a function of x by another function of y. Divide by X1 Y, X. +, = 0, and the variables are separated. 143. Ex. Determine the primitive of (1 - x)2ydx - (1 + y)x2(y 0. Divide by x2y, (1-)2 I f y dx 2dx dy ( lx +dy - O0, or, - — dy O, xf xY x Y.'~. -— 2 log + - logy-y - C. 24 370 INTEGRAL CALCULUS. 144. Case 3d. Let the proposed equation be homogeneous, or of the form (Xnf M + axl+lyl-1 + b6,x+2ym —2... + f x?+cym-c)dx n (eymL -+,^n+lym-l 1_ g[Zn+2ym-2.. +... S -S)6l _ O + (exny- + fxi+iy- -.x.+iy..qx- y... ) dy = 0. Put y - xz, thel dy - xdz + zdx, and by substitution xn+7 "(zm + az-m -r + b.zl-2... + pz,-c)dx +.fl+m(erm A- f"'_ + gm.... + qzm-s)(xdz + zdxt) = 0. dx (.. l +J fz.-l.+ 9z7-2 + qzm-S~)dz *x z"7+az~,-1+bZm-2 -t — +ezm+ +f: m gzm-1. +qzm — +s 5? + OtiUl-1 + +Pm-2 ~j~pzfLlC + e 4ft~l +ft l.+ qz 0 and the variables are separated. 145. Ex. To find the primitive of x2dy - y2dx - xydx = 0. Put y = xz, then dy = xdz ~- zdx. or, by restoring the value of = -, 2. xdy - ydx = dxc/x ~- y2. y = xz, then x(xd + dd )- zdx) xzdx = (1 - ) dx, dx dz G _z *=d~,. log. = sill1z + C. (1 -z2) 146. The same method of transforlmation may be extended to CYl tl l such differential equations as involve any function of- unmixed with:? DIFFERENTIAL EQUATIONS. 371 the variables, provided the equation would be otherwise homo. geneous. Erx. xydy - y2dx = (X + y) 2e'dx. Put y xz, or, =Z x 2z(xdz + zix) - x22d = X2(1 + -)2e-zdx dx ezzdcz e * -- (T+- t)2) d l = - -+ + g Y.'. log. + -- + C. x x y 147. Case 4th. Let the form be (a d- bx + c/)dx + (a 1 + bx + cy)y = 0. Put a - bx - cy/_v, and a1d +- blx -+ cly U=. Then dv = bdx + cdy, and du = bldx - c1dy, and by elimination c~dv - cdu.. bdu- bldv dx c,d - cd and similarly dy 6- = bl bcl - bc bc b - c. By substitution v(cdtv - cdu) + u(bdu -- b6dv) = 0 which is a homogeneous equation. 148. This method fails when bcl - bc = 0, because the attempt to eliminate either dx or dy causes the other to disappear bc also. But since we then have c1 -b- the proposed equation reduces to (a + bx + cy)ddx + (a, + Bx + -~y)dy =0 o..e (.) z -- Cy = - cd = Put bx- cy- =z, then x -- and d -. — Yd" 6 b 372 INTEGRAL CALCULUS. and, by substitution in (1), (a dz - c- y -i i =O (a + z) ~ + 1 + b ) - 0, (. + z)cd ca - c a -I- alb - blz an equation in which the variables are separated. 149o Ex. Find the primitive of (1 + x + y)dx + (1 + 2) d- 3y)dy = 0. Put 1+ x + y =-v, 1+ 2+x + 3y = u; then dx + dy = dv, and 2dx +- 3dy = du,. dx = 3dv - du, and dy du - 2dv.. ~ v(3dv - du) + uc(du 2dv) = 0. Now put u = rv, then du c rdv-F-vd', and, consequently, v(3dv - -dv - vdr) -- rv(rdv - vr - 2dv) = 0, l(2r - 3)cl,. - dr -dvI (r -1 )cl. 01 2 2-' o o log j-4 2 loo r — +-j -— ) t —[ -/ -- 1 or 1 or log [u - 3v2u -- 322] + tan-' 2 or log (1+ -x-t-x+2 — y+ 3y2) -- tan' (l — x] 150. (Case 5/it. Let the form be dy -- Xydx = ldx..... (1), in wlich X 1(and X1 are functions of x, The peculiarity of this form is that no power of y except the first SEPARATION OF THE VARIABLES. 373 enters into it, and for this reason it is usually called a linear equation. Its solution is always possible. Put y = X2z where X2 is an arbitrary function of, which may be so assumed as to facilitate the integration; and z a new and undetermined variable. Then dy= Xi2dz - z+dX2, and (1) can be reduced to the form X2 dz z dX2 + X2zcldx = Xldx....... (2). Now lett f2 be determined by the condition zdX2,- Xcldx....... (3), and (2) will become X2dz + X2zdx = 0. z - - Xdx and logz =:~ -Xcx,. ~zX = e This value, substituted in (3), gives e-fXxIcldX2 = Xldx or dX2- e fxdxXd dx. n' f efX jezxXdx, and y = zX2= e-fsXdxJe fsX^Xdx... (4), which is the required relation between x and y. 151. Let there be now taken the more general form dy + Xydx = Xlym.dx...... (5). This is easily reduced to the linear form (1). For put z = n -- 1 and z- y-n. Then dz = - -2 ny —ldy or dy = - yd Substituting this value of dy in the equation (5), and reducing, we get dz Xdx — + - = Xldx, or dz - nXzdx = - nXtdx.... (6), which becomes identical with (1) when we replace z by y, - nX by X and - nX by X1. 374 INTEGRAL CALCULUS. Z= - - ner1fxdZ'fXle-?^'xddx,...... (7). Cor. In forming the integral j'Xdx, it will be unnecessary to add a constant C. For if we replace fXdx by X3 +- C, the formula (4) will become y - e-3 -cfes + Xldx = e-X3e-cSX3 e. Xlx =d e-X3 J-e Xdx, in which the constant C has disappeared. 152. 1. To determine the primitive of (1 + x2)dy - yxclx = ad.,. yxdx a Here dy - -x -y- = a1 dx, which, compared with the linear I + X2 + x2 form, gives X a X —= 1 + ~ 2 and X1 =. l~-' (1 + x+ x2 f 1 + X2 log 2. dy + ydx x = y3dx. xdx Here X= 1, Xi =, YM = Y3, or gn = 3, and n- = 2 - 2ef. fx2. * - -e + - + -: ex2 _x ^o [04+ e-] (l + C].-. ~ y-' qX2 dx - - l-C2 ( + x~)" ( l+.~;~ r 1- [(I 2x + x ~ y2.. d! +- a- ydx = bxdx. Here X -1, X —x, y -y or m = 3, and n - 2, and, by substitution in (7),. ~. ~ — = _ 2evd~. fxe-2/~. dx- =: e2[xe-~ + e-~ + C]. 3~ dy ~ ~-~ydx -= bdx. SEPARATION OF THE VARIABLES. 375 Here X =' X = b.. fX jX x == log (1- x)a, e/fdx- elog(l-)( =( (1 -_ x)a fX1 efX dx. ^^ dx = (l -d = - - ( X)a+l + C. C b Y-(1-x)" a- x). 4. Find an expression for the sum of the series x x3 x3 5x X X X5 X7 -+1.3 + 5 + 57+ Differentiating, we obtain d y X2 X4 X6 x3 x5 d ++.- + _. +&C1+x(x+ 1 3- + +&e.) dX 1 1.3 1.3. 5 1 1.3,. = 1 xy.. dy - xydcx = dx, a linear equation. Also fdx - =- fxdx = -- 2..X2 V'2' y - e- (fe dx), the desired expression. 153. Case 6th. Let the form be dy -+ by2-dl = amdx..... (1). This form, which is called Riccati's equation, involves only the second power of y. Its integration has been effected for certain values of the exponent mi, but a solution applicable to all values of m1 has not been discovered. 154. It is obvious that when m = 0 the equation (1) will be integrable, for then dy + by2dx = adx. dy -= dx, ad the variables are separated.b and the variables are separated. 376 INTEGRAL CALCULUS. 1 1 l_ x b2 dy ___?/_ Thus we have 2a2 b~ dx = 1 1- 1 1 by + a b2 y -a2.2a2 bx=log 1 1+ C. y - Next suppose mn to have some value other than 0. 1 z Assume y -= + 2-, where z is an undetermined variable, but bx' obviously a function of x. Then dx 2zclx dz 1 2z z2\ dy =-b2- 3- + 2 and by2 b2 +- 3 + o^" 3 X'2 2 4 Substituting these values in (1), it becomes dz bz2dx bz2 - + ~ =- axmdx or dz + ~ dx = axm+2dx.... (2). x2 x4 - - Now this equation (2) will be homogeneous, and therefore integrable when m= - 2, and thus a new integrable case is found. Again, if m = — 4, the variables x and z can be separated, for then z2 dz dx dz -r b dx - ax-2~dx.. x x2 a- bzb xwhich is a third integrable case. 155. To obtain others, put - = y and xnt+3 =x. z Then dy d Then dz_ —d- - _ = y1 2 Y +3 xm+4 dx 1 + +3 dx. dX I xan - - = m -+ 3 x1 Hence by substitution in (2), r+4 dyli+ ( b m4 = + adx 3) 2- ( x+3). - 3dx t 2 3 ) y12 x i. 3) SEPARATION OF THE VARIABLES. 377 b a?n -+ 4 Now put -^ = b6, and -- in m-+"- - - in +3 l+3 and (3) will become, after reduction, dy1/ + bly,2dx = al.xlml.... (4), which is identical in form with (1). Hence (4) must be integrable when the exponent nm has either of the three values 0, - 2, or - 4. Moreover when a relation has been obtained by integration between x1 and Yl, a simple substitution will give the desired relation between x and y. We have therefore to examine whether by assigning either of the values 0, -2, or - 4 to mz1, any new values of mi will arise. - + 4 3m,1+ 4 But mi = -.. mlzmn+3n-l —1 -m-4, and min- ~ - m -t 3 mj-I-1.when m1- = 0, mi - 4, a case before considered; and when m - 2, n 2, but when M^ =-~4, qn =~3, a new case. Hence Riccati's equation is integrable when mm =- - also. 156. In a maanner entirely similar to that by which (1) was transformed into (4), may we transform (4) into a new equation dy2 + b2y22cx2 = c2X2 clx2.... (5), _, -+ 4 32 +- 4 in which mn2=. - ~, and therefore min =- ~ + ~ m211 -+- 3' m122 +1 And by repeating the process, a series of such equations may be formed; so that it will be possible to find a relation between x and y when any one of the following quantities or exponents shall be - 4; viz.: mn + 4 mqi + 4 _ 2 +- 4 mn, or ml-=, or nm ~2 =- or n3 &c._ o+3 m 1 + 3` q- "' 3T8 INTEGRAL CALCULUS. But m = when in = 4 12 4 m 5, whe-l m2 -- 4 16 ft2 -- & mn = 77 m3 - 4 Bence by successive substitutions, m- 5~ when ms- -~4 16 m -: -- 4 Thus Riccati's equation is integrable for all values of m included in the series. 8 12 16 20 24 - 4, -;,, - 5, - 7, - 9, -1l, sc. 5" T' 57 9' 11' 4n 157. The general formula for these numbers is 2- in which n is any positive integer. To prove this, suppose one of the numbers, as in8, to be of the 4n form required - 1; then will the adjacent terms m7 and m9 be of the same form, with the number n increased or diminished by unity. For we shall have, 4n 4 3 4n 3m8 + 4 -2n-1 4-4 + 4 4(n+ 1) Mn 4n 2n +1 2(12+1)- - 2n - 1 and 4'a 8+ 4 2 - 1 4n 4 4(n -1) 9 wnm.8 + 3 - 4. - 2n - 3 2( n- 1)-1 2n - 1 both of which forms are similar to that given above as the general form. SEPARATION OF THE VARIABLES. 379 But we have seen that one integrable case is that in which 8 4.2 m = - 2.. - 1 which being of the required form, the 12 adjacent numbers - 4 and - - are also of that form; and thence the same reasoning can be extended to the other numbers in the series. 158. Second Transourmation of Riccati's Equalion. —In the given equation dy + by2dx = axdc.......(1). 77. put y, and xm-~ = diY I _ l dx and x1 9 x +i d then dy = - lxa x dx -- a d /10 9' + l' (m + 1) Also, y2 = 2' and therefore by substitution in (1), Y1V - (m +_1y -— j- -1 — bxd d' _ a dy1 + (In +D^ = _+ i d d a b - or, dy1 + y + 1 y2dxl x= 1Z fldx5. (8). mn + I w +I Now make a =b bl, = a-, m, m + 1 em - 1 -- 1 + and the equation (8) will reduce to dy1 + byidj12dx = ajxllmdxl... (9). which is of the same form with (1). The equation (8) will evidently be integrable whenever m, has any one of the values included in the series before found, that is) m 4n when mn1 or its equal has the form - ~1 But if m + 1 2n- 1 9m 4n m +1 =-2 -1' 4,then 2mnn - mn = 4mmn + 4n, mb + I 2n - I 4n2 and' m. - 2n1 - 1 380 INTEGRAL CALCULUS. Hence wse have a new series of integrable cases corresponding to 4n all values of m, included in the formula - 1~ Thus Riccati's 2n + I equation is'iotegrable whenevelr the exponent m can be expressed in 4n the form —, the quantity n being a positive integer. It appears also, that whenever the given value of nz is found in 4n the second series, the terms of which have the form -2 + 1' an 2 q + 1, or n' > n - 1, the solution of the problem will be impossible. Similar remarks apply to the superior limit; and we conclude that when the new function V' contains any coefficient of an order higher than a - 1, the function IU will not admit of a maximum or minimum. 78. Prop. To find the form of the function y and the values of the limits xo and x, which shall render U = X VTdx a maximum a0 or minimum, where r dy dny X 6 ^ di llYA vl.-f/\,,d 1I.i idx U,0 bcos in ths ce (. The general equation D) U= 0 becomes in this case (p. -il41) 484 CALCULUS OF VARIATIONS. L l\+f,,^o 1+ ltd,) +p,))+ (dl) + & ] i [+ &X) d)+fi d] + &x C\dx- dP2 + P 3 + &\ - f + (dq) F(P dP + ql L fx o [N d'o d2 &.] Sy. dx - This beingf written in the form l-a0 a- ydx shows that b is the same as before, and therefore the form of the function y is not changed by supposing V to contain explicitly the ~linmi ting -values of x,, d o, &c. Also the terms i:n a~ - 0 are of the same natude as if V A Id xP + BJ,&Y + C1 ( ) + &c. + A6 dx0 + Bxy0 ) ah,1&c beiig wcostants. For in the exo pressions I mdx, / mclrdx, &c., 1a - ao, + bfydx —O hos the sae sppstion is made as in the rf ems - t \ 1F &c.) limiting values of x,, I, Also the terms in al - "0 -- 0 are of the same nature as if V' did not contain the limits, forming a series A. ]31x, C,, &e., AO, Bo, Co, &c., being constants. For in the expressions dld, Modx, &e.. 0 O the same supposition is made as in the terms /~ cP,.. \ [,,iz,~ MAXIMA AND MINIMA OF ONE VARIABLE. 485 and the other coefficients of the several increments in the equation a~- 0 = 0, where V did not contain the limits; viz.: that the value of y, derived from the equation b = 0, has been substituted in mi1, ino, &c. This substitution being effected, and the definite integrals f'1 mndx,' m0od, &c., being formed, the quantities A1, B1, Ao, B,, &c., will become entirely constant. Thus the mode of treating the- equation D U 0 is in all respects the same as in the case previously considered. The following examples will illustrate the cases considered in the last two propositions. 79. Ex. Having given the area c of the figure BAAB1,, bounded by the axis of x, by two ordinates passing through the given points B and AB, and by a curve A Cz, to find the nature of the curve and the values of the extreme ordinates BA and BA1l, when the perimeter of the figure is a minimum. Put OBx0,, OB,-xz, BA-yo, B1A —y1 A, Then, since BB1 - x, -x, y is constant, we haveo B, X~ BA + B1A1 + ACCA1 = Yo + Y1 +/( +1 / 2clx = V + /=1 Vdx =a minimum. Also fX1 Vf = Jlydx c. UT V!"+/ ( 1(V+ X V') dx- a minimum..o Here U contains a term VT", exterior to the sign of integration, involving the limiting values of y, and, therefore, by the method 486 CALCULUS OF VARIATIONS. applicable to such cases, combined with that of relative maxima and minima, we have DUD= D 0 + ~ + I (I + c2i)+ Xy d = 0. Now V + XV' =f (y ) dy) and therefore by formula (b) V XV', +' + P d d( F+?,P) dy" dy But p + () dx (" - y dz= d (0 - Y) d? odx',ci Ct = (a _ )2 a2 _ (P _ y)2] W or, (X - c2)2 + (y - 3)2 = a2, the equation of a circle. Hence, the curve A CA1, is a circular arc. To determine the values of the ordinates yo and y,, and that of a, the radius of the circle, we recur to the equation a1 - a - 0, which becomes, in the present case ( ViT+ V')c lX- ( +X V')o dXo+ (P,)I 6y1- (PI)o 0 o + N,'y, +1x'Cy0o-0, (1), since V-+ X V' does not contain P2, P3 &c., and V" contains only Yo and Yl. Also, since the points B and B, are given, dze _: 0, and dx = 0. Thus, (1) is equivalent to the two conditions MAXIMA AND MINIMA OF ONE VARIABLE. 487 (P1),+ V"= o, (P1)0- OT= O. c V" d V" But IN' -I and N" -- =1. dY0o dy Hence, by substituting the values of N', 1N" and P,, we obtain +4-/ + 1-= and (1 + -1 0 Ldx ^^/ J] L dx\ dx2/ J o dy\ /o\ _:/oy + Go and k And therefore the arc ACA1 is a semicircle, the tangents at A and A1 being perpendicular to OX. Also, radius ac - (x~1 x), and y= yo. 1 But area BAA1B — 2 e Y0o +- -"2r = c, and. yo becomes known, thus making the solution complete. 80. Ex. To find the curve of swiftest descent from one given curve to another, the motion being supposed to commence at the upper curve. Let AB and A B1 be the given curves, and B o \-E F CC, the curve required. 0 ) r Put OD=- x, DC -yo, OE=x, C I, EP = y, OF=x- FC1 y1, CP='s. Then, by the principles of Mechanics (before Y cited), the velocity acquired by the body in A descending from C to P along the curve CPC1, is expressed by ~ls dx f1 dy2,g x IP = vg (y -:o); and also by - t= - +; dt -- Cd 1ie (=It+ G/d l dx2d.dt - [2g (yo) [l +M ]d,,. 488 CALCULUS OF VARIATIONS. Jo U=l Y-) d( " + ]dx 1 Vdx = a minimum. ~ XQ' - YO d ) JX Here V contains the limit Yo explicitly; and therefore D U will contain the additional terms [/x l (d) x] o+ [/,o d which terms appear in the equation a1 — ao = 0, but not in the equation b = 0. Also, since Y=f / i(, we have, by formula (b), dy [2C)(YYo)] This is the differential equation of a cycloid having the axis parallel to y, the cusp or extremity of the base at the upper point x0, yo, and the diameter of the generating circle = 2C. The equation a1O- =a 0 gives, in this case, VYdx1 - Vcdx0 + (P1)1 &y1 - (P1)>Jy + Qfxa (dx)odx)do But V dV- dP- N dP' d/2~+ ( -1 d 1,'. (v - dxo r (~ + / -- d dy - - Nc - (y - yo)]d This is the differential equation of a cycloid having the axis parallel to y, the cusp or extremity of the base at the upper point x0, y0, and the diameter of the generating circle -- 2C. The equation a, - a, - 0 gives, in this case, But alP1 since n, A — -- - dP1 dyo f dy- d dx' dx dxP MAXIMA AND MINIMA OF ONE VARIABLE. 489' Xno9Zdx = (P,) - (P,)1, and Je), dx d ( t[(P1)o - (Pl)l Again, if the differential equations of the two given curves be _yo dy1 dyo = t,, and -d tl, d0, dx1, we shall have the following conditions connecting the values of dx,, oy,, dx1, and Sy1, viz.: o+ (-)dx 0 todxo and Jy i+(-dx) cl1= c Now substituting the values of Syo, Jyj,f lnodx, and f d ( i 70dIx in (1), and placing the coefficients of dXo and dxl, separately, equal to zero, we get V' + (P)1 [- (J) J- and F0 -+ (Pi) 0 to- )2 [(P1)0 - (PI); (7) - [(P1)0 - (P) o] - (d)0o or - Y_)2J ((1i/ ) [(=0.. ) (2) and [(I + dy2 +)4 (Y - Y)] -(d42 )[(i c~ x (y -- )2 + t (7z ) [(o ~ t<2 ((y - Y)] 1 ~. (3). 490 CALCULUS OF VARIATIONS. ad = -O; and theref ore the cycloid From (2) we obtain 1 +t 1 () = 0; and therefore the cycloid intersects the second curve at right angles. Also, from (3) we get 1 + to d) 0;.. t to and the tangents to the two curves, at the points of intersection with the cycloid, are parallel. The co-ordinates of those points are readily found. 81. Prop. To determine the forms of the functions y and z, and the values of the limits x1 and x%, which shall render U =: f1 TZdx a maximum or minimum, where JX V —/[ dy cl2y dny dz d2z dm" ~Lx' dx dx' dx,' dx' dx.'"x2 dxL' The equation ) U = 0 becomes in this case Vdx, - Vodx, + [ - d + &c.] y_1 - [ dP + &c.] oy + [-LR- - (d o f) -[P2-&. ]~ ( \ * +F d(d-) y].~y (-1) dP16] nydr ~{ — [- df +~ ~-[' dP' + &c.] L Jr + [FP2' - c.]1 (+- _ ^- ]o (dz & Lo 0 dx dx dx.' —& lm1p J. (z -0.m' 7. dx P,,-1 MAXIMA AND MINIMA OF ONE VARIABLE. 491 If the functions y and z be independent of each other, their variations 6y and &z will also be independent; and, by reasoning as in previous propositions, it will appear that we shall have the conditions -. l — d2 ( 1) (0 ) c~, lF2 d2P' __..- "... (-,1 dx - &e.... -~ (- )"' dx'c' dX d dz2 Cm-0.1 ). And for the equation of the limits Vldx~- Vod-o o+ [P- dc- + &c.] yi - P dP + &c. ] o + [P2 - -& cl-.' [P2 - &.] (d -) &c. &c. dx 0 + [1 da+~&c]; - [ - cl d1 + [P' - &c.] ( )- [P2' - &c.]o -— ) &c. &c.. (2). The mode of treating these equations is exactly the same as that employed when V contained but one function, and by reasoning, as in that case, it may be readily shown that the number of equations applicable to the solution of the problem will not, in general, be affected by any equations of condition restricting the limits. For every such equation of condition will diminish by unity the number of terms in (2), either by reducing to zero the variation which appears in such term; or, by uniting two terms in one, and thereby dimrinishing by unity the number of equations deducible from (2). But the given equation of condition will just supply the place of that which has disappeared. Thus it will suffice to prove that (1) and (2) furnish the requisite number of equations in a single case, as when the limits of x are alone fixed. 492 CALCULUS OF VARIATIONS. Now the first of equations (1) is of the order 2n in y, and mn -+ n in z, and the second of equations (1) is of the order m -]- n in y, and 2m in z. They are therefore of the forms d cl12ny cdz cdm+z" Fl~~~ L c; l* dx2n) 2' d2X d* m+n]= ~ (3). [ y/ d+lly dz d2mZ~ 1 F2 [, d 7t. * m+n' Z clx2 0....(4). If, then, we differentiate (3) 2m times, and (4) mz + n times, we shall have 3m1 +'L + 2 equations with which to eliminate the 3m-+n dz d3in+nz quantities, - d3sm+.n and the resulting equation will be of the order 2m + 2nr in y. The integral of this equation will contain 2nm -+-2 constants. But the number of equations given by (2) is exactly 2n + 2m, viz.: the 2n equations, 1 dP2 +&c.1 =0 [P1 lP2 +&c.] 0, [P2- &c'] 10, a.; and the 2mr equations, P -+&C. — 2 P [p&1c -c 0, [P2'- &c.] = 0, &c. Hence the problem is in general determiniate, but there are exceptions entirely similar to those considered in the case of a single dependent function y. 82. If the functions y and z be connected by an equation L = 0, and if it be possible to resolve that equation with respect to y or z, so as to obtain a result of the form z =-f(x, y, dj &c., the values of d d2 &c., can be formed by differentiation, and substituted in dx c that of V, which will then contain x, y, and the differential coefficients of y with respect to x, thus presenting a case already considered. MAXIMA AND MINIMA OF ONE VARIABLE. 493 83. But since the proposed equation L = 0 is often a differential equation difficult to be integrated, we are often compelled to adopt the method already noticed, (Page 444) in which by the introduction of a new indeterminate quantity X, and a suitable determination of its value, we are enabled to obtain nn expression for J U which shall contain but one of the variations dy and 6z under the sign of integration. Thus, if we denote by S, the sum of the terms exterior to the sign of integration in the value of U, (Page 445) there will result 8 U = + /1 [1VX -t- (P ~ / &c.] Sydx 0 ('h'dx + I[^Y+ Xa-l.( 1~ + &c.] JZdx; and if we so assume the quantity X as to fulfil the condition N/' + l x + d(l'f + X'1) + & 0 dx it will appear by reasoning, similar to that employed when y was the only function, that the condition U 0= cannot be satisfied (so long as the form of 6y is arbitrary) unless we have the two conditions O=0 and.+Xa- d(P + x)- &c. 0. dx Hence, we have for the solution of the problem, the three general equations aL=0, +x d d(P +' + +&c. 0, dx and 2Y' q- Xc~' d (Pm' +-?/') whichd b~i' + just dx ~&c. 0. which are just sufficient to determine the three unknown quantities, X, y and z. 84. We will now give, in conclusion, examples to illustrate the cases and methods above explained. 494 CALCULUS OF VARIATIONS. Ex. To find the nature of the line which is the shortest distance between two given points in space there being no restriction by wvhi(h the line is required to be confined to one plane. The general value of the length of the arc of a curve of double curvature is t/ l \- cl"dxcllx2 f ( + ~ d +- d2j)2 x taken between the proper limits. Hence in the present case we shall have U = l + d2 +T - 2 dcx =- a minimum. Here 1 + dv2 1 cdx, 7 V djy -0 N' -_c 0 dy dz dV dx p d V dx dy -cy2 zc2' 1 dd 2 dx ~dX2+12 dx2 P2 0, P2' =, &c. Hence the equations dP dP1. 1V-_ —~+ q&c.-=O and 1' -.. +.&c. -0 become ~ O and 0 dx dy ddz or P P1 --- = c and Pi = d z = dx2 d2 2 dx2 Eliminating first dx and then d/ between these two equations, we reaily obtai reslts of the forms readily obtain results of the forms MAAXIAA AND MINIMA OF ONE VARIABLE. 495 - a - i which m andn a ae constants, dx ~ dx.. y = mx -+ p, and z = nx -- q. These are the equations of a straight line, which therefore is the shortest distance required. To find the values of the constants m, p, p, and q, we introduce the given limits X., yo, Zx, x1, y1, z1, and thus get Yo ='^Xo - 2, zo - Px, Y =-o maxo + 1,, Z= nx -- q,, which stiffice to determine m, n,,p and q. 85. If the limiting values of x only were given, those of y and z remaining indeterminate, the terms exterior to the sign of integration would give (Pl)l = 0, (P,)o - 0, (P1'), = 0, (P1')o - 0, which are equivalent to the two equations m = 0 and n = 0, thus leaving the other two constants p and q indeterminate, and presenting one of the cases of exception already noticed. 86. Ex. To find the shortest distance between two given surfaces. Let the equation of the first surface be fo0(xo,,,0 ) =... (1) and that of the second surface Ji(x1, Yl, Z1) = 0.... (2) As in the last example V + d2 + cx22 ( + dx2 alx) and we immediately deduce as before y =i n2x + p..... (3), z = x + q...... (4) which show that the shortest path is still a straight line. To fix the co-ordinates of the extremities of this line we form the complete increment of (1) and (2) thus: 496 CALCULUS OF VARIATTONS. ~+ 1l~~($ ^ ctl/ (21~ ~+ +.:::. 0~..(5) L'~+ 6o d^\ch/ dz (,\ci x]A + cQ. =O +\~ +d (dXXt l+.(d +^.ld ^ 1 zj 0z..O (6) L^- dY1 \dP, d0 \dy)3 0 ~ d Put for brevity dfo df d/fo dfi dy o y dzfo 1d 10- c~ ) 1- (J~) O dj'I I >df1 dx( dxl d.X dx1 ~nd substitute for ~(,') (d"), () ~)x0 their values derived from equations (3) and (4). We shall thus obtain (1 ri + n1o) C+ m + 1) 0 /o + oZo0 = 0 (1 -- mm ^-+ nil) dxl + my 6y i 4- 911 = 0. Now eliminating, by the aid of these equations, dxo and dx,, from the equations V0dx0 + (P1)0)oo q- (P1')0Zo- 0 V1dxl - (P1,)J6y + (Pj1')8z=- 0, and placing equal to zero the coefficients of 6Yo, SO, y,, 1z, we obtain oTo -o (P1) (1 + t.mo A- -noO) 0..... (7) m1, V1 - (P), (1 + - n21 -+ n7) 0........ (S) o20 o -(P1,)o (1 + n2mo - nno) 0...... (9),21 1 - (P1')1 ( + m, +,1?,) - 0...... (10). If now we replace Vo and (P,)o &o. in (7), (8), (9) and (10), by their values 1(1 i?2 + + m 2) ( m -,9 & MAXIMA AND MINIMA OF ONE VARIABLE. 497 we readily find from (7) and (9) mn = 0, n = A0...... (11) and from (8) and (10), n = mn and n = nQ.... (12). Now eliminating Xo, yo, Zo xi, y, zt, which quantities occur in the values of m, n0, mn1, and n1, by means of the six equations, Yo = MezXo + P, Y1 = /IZXo + P, o = lxo + q, 1 = nx+ q, fo0(X, Yo, o) = fio(, ( Y1, 1) = 0, there will remain the four equations (11) and (12) with which to compute the values of mn, n p, and q; thus the line of shortest distance will be fixed in position; and, by combining its equations with those of the given surfaces, we can find the values of X1 IY iZ Yo Zo.o 87. The equations (11) and (12) show that the line of shortest distance is normal to both surfaces. For the assumed values of mZ and no indicate that they represent the tangents of the angles formed by the projections of the normal to the first surface on the planes of xy and xz with the axis of x; while m and n denote the tangents of the corresponding angles formed by the projections of the line of shortest distance. A similar remark applies to the quantities nu1 and nl1, and the normal to the second surface. 88. ZE. To find the shortest distance traced on the surface of a given sphere between two given points in the surface. Here the quantity to be rendered a minimum is the same as in the last two) examples, viz.: XQ x1 dx^Wy2 dx2/ but since the path is restricted to the surfalce of a given sphere, the 32 498 CALCULUS OF VARIATIONS. co-ordinates x, y, and z, of any point in the required path, will be connected by the relation dy dz 0 2). 22 + y2 + z= r2 or = + y +... (2). Hence the variations of y and z will not be independent of each other. dz Now we might form from (2) the value of which, substituted dx' in (1), would reduce V to a form in which it would no longer contain the function z, or its differential coefficient, or we may adopt the method of Lagrange, which is usually the easier. Taking the second method, we have v dy2 (d+ 2 2?dX2 dx2) dy dz dx dx -P PI'= C2 /~d^~d' - /~dy dZ2 V dx2 dxz 2 dx2 dx2 dV dV N- =0, ZV' —= dy d' dL dy dL, dz dy dx' dy Y dx' dx Hence the equations NV+oc d /- + &c. = 0, dx and N'+XM' d(&~' + x2 C dx become, in this case, dy dP, dy dX dx dx dx dx dz dP,' dz dX X -- X X -- z = O0 dx dx dx dx MAXIMA AND MINiMA OF ONE VARIABLE. 499 dy or + d. ( X dX \ dy2 dx2 Y~ T dz d d + -0....(4). dx +d 1 dy2 d_2) Eliminating -- between (3) and (4), we get CX dy dz d dx d. dx \ V dx2 dx- /VI dX2 dX2 and by integration dy dz dx dx c=....(5); +1 dy2 dz22 d7+ dXor, by changing the independent variable from x to s, (5) becomes dy dz' Z-Y yw-=c... 7-(6). By similar reasoning we may obtain dx dy an dz dx Y dx- c~.... (7), and xZ -Z.C2*(s). sds s dk XdZds (8) Multiplying (6) by x, (7) by x, and (8) by y, and adding, we get C C2 ex + cz + c =, or z- +- + -y= 0... (9), Cl Cl the equation of a plane passing through the origin. Thus the required line of shortest distance on the surface of the sphere, is confined to a plane passing through the centre, and is, consequently, a great circle. 500 CALCULUS OF VARIATIONS. The equation a, - aO = 0 in this case disappears, since dxo =0, dxl= 0, yo- = 0, 6y = 0, Jz= 0, and z = 0. C C2 The constants - and - are found by substituting C1 Cl xo, o, Zo, an xl, 1,Y1, for x, y, and z in (9). 89. If the limiting values of x only were given, or the problem that in which it is required to find on the surface of the sphere, the shortest path between two parallel sections, the variations Syo, Vy, Jzo, 6z4, would not reduce to zero, and the equation a - aO = 0 would give the four conditions (P1+ X)o = 0, (P + xh)1=O, (P1'+ Xh')o=O, (Pi'+ X/d')= 0, or, i+ oyo - 0....(10) / d2 c 2 \ and - X z = X (11 ); an __//^ d 2 2. ) which apply to the inferior limit, with two similar equations for the superior limit. Eliminating X0 between (10) and (II), there results dx dy d 0 / cy drz df 2 + d-2 Hence, the constant c 0 in (5); and that equation becomes. z yd 0-O; or, dy = dx dx' y MAXIMA AND MINIMA OF ONE VARIABLE. 501 l. r= ly - l og y z + lo log mz _, z; and y = mzz. This is the equation of a plane passing through the axis of x, and forming an arbitrary angle (tan-l') with the plane of xz. Hence, the required path is the arc of any great circle perpendicular to the planes of the parallel sections. TURSEND