b'\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n:*^J \n\n\n\n^:tl% \n\n\n\n\n\n\n\n\n\n\naass^3.,. J\\d O^- \n\nBook L_ V^-g) ^ \n\nCopight^I? \n\n\n\nCOPVRIGHT DEPOSnV \n\n\n\nELEMENTS \n\n\n\nOF \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nBY \n\nEDGAR W. BASS, \n\nColonel United States Artny, Retired. Professor of Mathematics in the \nU. S. Military Academy April 17, 1878, to October 7, 1898. \n\n\n\nTHIRD EDITION. \nFIRST THOUSAND. \n\n\n\nNEW YORK : \n\nJOHN WILEY & SONS. \n\nLondon: CHAPMAN & HALL, Limited. \n\n1905 \n\n\n\nLIBRARY of CONGRESSJ \n\n\nTwo Copies \n\n\nReceivea \n\n\nDEC 21 \n\n\nI9U4 \n\n\nOopyriKiii \n\n\ntrilry \' \n\n\no&^c, f2.. \n\n\ntgo^ \n\n\nCUSS CU \n\\ / O 3 7 \n1 COPY \n\n\nXXC NO! \n\n\n2LO \n\n\nB. \n\n\n\n1904, \n\n\n\nCopyright, 190 1 \n. BY \n\nEDGAR W. BASS. \n\n\n\nROBERT DRUMMOND, ELBCTROTYF2R AND PRINTER, NEW YORK. \n\n\n\nPREFACE. \n\n\n\nThis text-book has been prepared for the use of the \nCadets of the U. S. Military Academy who begin the sub- \nject with a knowledge of the elements of Algebra, Geometry, \nand Trigonometry which ranges from fair to excellent. \nThe time allotted to the subject (ten and one half weeks), \nand the requirements of the subsequent courses, especially \nMechanics, Ordnance and Gunnery, and Engineering, limit \nand determine the scope of the work. \n\nMy experience leads me to the belief that the more rig- \norous and comprehensive method of infinitesimals is suit- \nable only for a treatise, and not for a text-book intended \nfor beginners. \n\nAt the same time I believe that any presentation of the \nsubject, no matter how elementary, should in no manner \nprejudice the student against any established method. On \nthe contrary, it should, I think, endeavor to lead him to an \nunderstanding of the relations between those in general use, \nand, above all, it should aim to construct the best possible \nground work for the subsequent study of the subject treated \nin the most rigorous and extended form. \n\nThe principle of interchange of infinitesimals, which con- \nsists in replacing one infinitesimal by another when unity \n\n\n\nIV PREFA CE. \n\nis the limit of their ratio, has been used to overcome the \ndifficulty encountered by beginners in the determination \nof the limits of ratios of infinitesimals. \n\nTo the Officers of the U. S. Army who have taught the \nsubject with me, and in many cases to my pupils, I am \ngreatly indebted for much valuable assistance. \n\nTo Captain Wm. Crozier, Lieut. J. A. Lundeen, Lieut. H. \nH. Ludlow, and Lieut. F. Mclntyre I am under obligations \nfor many demonstrations and solutions. \n\nAssociate Professor W. P. Edgerton has been my collab- \norator throughout the work, and lo him much credit is due \nfor numerous demonstrations, improvements, and sugges- \ntions. \n\nI have added a list of the works of authors which I have \nfreely consulted in the preparation of this book, for the \npurpose of acknowledging my indebtedness to them, and \nfor the benefit of students who may desire to extend their \nknowledge of the subject. \n\n\n\nEdgar W. Bass. \n\n\n\nWest Point, N. Y., June 15, li \n\n\n\nPREFACE TO THE SECOND. EDITION, \n\n\n\nFor the corrections and changes made in this edition I \nam indebted to the Department of Mathematics, U. S. M. A. \nMy thanks are especially due to Professor W. P. Edgerton, \nAssociate Professor Chas. P. Echols, Lieut. George B. \nBlakely, and Lieut. F. W. Coe. \n\nEdgar W. Bass\xc2\xab \n\n524 Fifth Avenue, New York City, \nTune I, iQOi. \n\n\n\nPREFACE TO THE THIRD EDITION. \n\n\n\nI AM indebted to Professor Chas. P. Echols for the cor- \nrections and changes made in this edition. \n\nEdgar W. Bass. \nBar Harbor, Maine, \n\nAugust 28, 1904. V \n\n\n\nLIST OF AUTHORS WHOSE WORKS HAVE BEEN \nCONSULTED IN THE PREPARATION OF THIS \nBOOK. \n\n\n\n\n\nAMERICAN. \n\n\nChurch, \n\n\nNewcomb, \n\n\nRice and Johnson, \n\n\nBowser, \n\n\nByerly, \n\n\nHardy, \n\n\nTaylor, \n\n\nOsborne. \n\n\n\n\nENGLISH. \n\n\nTodhunter, \n\n\nGreenhill,* \n\n\nWilliamson, \n\n\nPrice,* \n\n\nEdwards,* \n\n\nHaddon, Examples, \n\n\n\n\nFRENCH. \n\n\nBertrand,* \n\n\nHouel,* \n\n\nJordan,* \n\n\nHaag,* \n\n\nDuhamel,* \n\n\nSerret.* \n\n\n\nHarnack\'s Introduction to the Calculus, translated from \nthe German by Cathcart. is a rigorous treatment of the \nsubject. \n\n* Treatises \n\n\n\nCONTENTS. \n\n\n\nDIFFERENTIAL CALCULUS. \nINTRODUCTION. \n\nDEFINITIONS. NOTATION AND FUNDAMENTAL PRINCIPLES. \n\nPAGE \n\nChapter \\. Constants, Variables, and Functions ... i \n\nII, Principles of Limits 25 \n\nIII. Rate of Change of a Function 44 \n\nPART I. \n\nDIFFERENTIALS AND DIFFERENTIATION. \n\nIV. The Differential and Differential Coef- \nficient 59 \n\nV. Differentiation of Functions 72 \n\nVI. Successive Differentiation... 121 \n\nVII. Implicit Functions and Differential Equa- \ntions 141 \n\nVIII. Change OF the Independent Variable 154 \n\nPART II. \n\nANALYTIC APPLICA TIONS. \n\nIX. Limits of Functions which assume Indeter- \nminate Forms 158 \n\nX. Developments 170 \n\nXL Maximum and Minimum States 206 \n\nix \n\n\n\nX CONTENTS. \n\nPART III. \n\nGEOMETRIC APPLICA TIONS. \n\nPAGE \n\nChapter XII. Tangents and Normals 243 \n\nXIII. Asymptotes \xe2\x80\x9e.. 256 \n\nXIV. Direction of Curvature. Singular Points.. 271 \nXV. Curvature of Curves 289 \n\nXVI. Involutes and Evolutes 304 \n\nXVII. Orders of Contact of Curves and Oscu- \nlating Lines 317 \n\nXVIII. Envelopes 324 \n\nXIX. Curve-tracing 335 \n\nXX. Applications to Surfaces 348 \n\n\n\nDIFFERENTIAL CALCULUS \n\n\n\nINTRODUCTION. \n\nDEFINITIONS, NOTATION AND FUNDAMENTAL \nPRINCIPLES, \n\n\n\nCHAPTER I. \nCONSTANTS, VARIABLES AND FUNCTIONS. \n\nI. In the Calculus quantities are divided into two \ngeneral classes, constants and variables. \n\nA Constant is a quantity that has, or is supposed to \nhave, an absolute or relative fixed value. \n\nA Variable is a quantity that is, or is supposed to be, \ncontinually changing in value. \n\nIn general, constants are represented by the first letters \nof the alphabet, and variables by the last ; but they should \nnot, therefore, be confused with the known and unknown \nquantities of Algebra. \n\nThe same quantity may sometimes be either a variable \nor a constant, depending upon the circumstances under \nwhich it is considered. Thus, in the equation of a curve, \nthe coordinates of its points are variables ; but in the \n\n\n\n,2 DIFFERENTIAL CALCULUS. \n\nequation of a tangent to the curve, the coordinates of the \npoint of tangency are generally treated as constants. It is, \ntherefore, necessary to determine from the circumstances, \nor object in view, which quantities are to be regarded as \nvariables, and which as constants, in each discussion. \n\nIn general, any or all of the quantities represented by \nletters in any mathematical expression or equation may \nhave definite values assigned to them, and be regarded as \nconstants ; or they maybe considered as changing in value, \nand treated as variables. Thus, in the expression ^7Tr\\ r is \na constant if we suppose it to represent the radius of a par- \nticular sphere ; but if r is considered as changing in value, \nit will be a variable. In the first case, A^nr^ is a constant, \nand measures the surface of a particular sphere ; but when \nr is variable, ^nr" is also variable, and represents the surface \nof any sphere no matter how much it may increase or \ndiminish. \n\nIt should not be understood, however, that we may in \nall cases treat quantities as constants or variables at pleas- \nure without affecting the character of the magnitude rep- \nresented by the expression or equation. For example, n \nis generally assumed to represent the ratio of the circum- \nference of any circle to its diameter, which ratio is invariable. \nIf a different value be assigned to jt, the expression ^nr^ \nwill not measure the surface of a sphere whose radius is r. \n\nIn some cases variation in a quantity changes the di- \nmensions of the magnitude represented by the expression \nor equation ; in others it changes the position only ; and \nagain it may change the character of the magnitude. Thus, \nif we suppose R to vary in the equation \n\n(oc - ay + (j, - /?)\' = ^, \n\n\n\nCONSTANTS, VARIABLES AND FUNCTIONS. 3 \n\nwe shall have a series of circles differing in size ; but by \nchanging a or (3 and not R the position only will be \naffected. \n\nBy changing b\'^ within positive limits, the equation \n\xc2\xaby + b\'^^\'^ = ^^b"^ represents different ellipses, but negative \nvalues for b"^ cause the equation to represent hyperbolas. In \ngeneral, however, constants are supposed to have fixed \nvalues in the same expression, unless for a particular dis- \ncussion it is otherwise stated. \n\nFUNCTIONS. \n\n2.. A quantity is a function of another quantity when \nits value depends upon that of the second quantity. Thus, \n\\ax is a function of 4, <2, and x. In general, any mathemat- \nical expression which contains a quantity is a function of \nthat quantity. If, however, a quantity disappears from an \nexpression by reduction or simplification the expression is \nnot a function of that quantity. Thus, oc" -|- [c ^ x){c \xe2\x80\x94 x), \nax/bx^ and tan x cot x^ are not functions of x. \n\n3* A function of a single variable is one whose value \ndepends upon that of a single variable and varies with it. \nThus, \n\n\n\n4^y(i \xe2\x80\x94 jic"), yr^x^ + 2/jc, log (a ~\\- jc), sec x^ \n\nin which x is the only variable, are functions of a single \nvariable. \n\nAny function of a single variable is also a variable, and \nvaries simultaneously with the variable. \n\n4. The relation between a function of a variable and its \nvariable is one of mutual dependence. Any change in the \nvalue of one causes a dependent variation in that of tlie \nother. Either may, therefore, be regarded as a function of \n\n\n\n4 DIFFERENTIAL CALCULUS. \n\nthe other ; and they are called inverse functions. Thus, if \nX passes from the value 2 to 3, the function 20^ will vary \nfrom 8 to 18 ; and conversely, x will increase from 2 to 3, if \n2X^ changes from 8 to 18. \\i x be again increased the same \namount, that is from 3 to 4, the function will increase from \n18 to 32. Similarly, with other functions we shall find that, \nin general, equal changes in the variables do not give equal \nchanges in the corresponding functions. \n\nIt is therefore necessary, in referring to a change in a \nfunction correspondiiig to a change in the variable, to con- \nsider the states from which and to which the function and \nvariable change, as well as the amount of change in each. \nWith that understanding, corresponding changes in a func- \ntion and its variable are mutually dependent. \n\nIn the equation of a curve^ the ordinate of any point is a \nfunction of the abscissa, and the abscissa is the inverse \nfunction of the ordinate. \n\nThe function is considered as dependent, and the vari- \nable as independent ; for which reason, the latter is called \nthe independent variable^ or simply the variable. \n\nRepresenting a function of x^ as ^^byjj;, we havejv = x^\\ \nsolving with respect to x^ we have x^=^ yy\\ a form express- \ning directly Jt: as a function of j\'. \n\nThe difference in form in the following important exam- \nples of direct and inverse functions should be observed. \n\nHaving, _y = x^\\ then x = \\/y. \n" y \xe2\x80\x94 a-\\- x\\ " X = y \xe2\x80\x94 a. \n" y = ax; " X = y/a. \n\n\n\nCONSTANTS, VARIABLES AND FUNCTIONS. 5 \n\n5. A state of a function corresponding to a value or ex- \npression for the variable is a result obtained by substituting \nthe value or expression for the variable in the function. \nThus, \n\n\xe2\x80\x94 00 , \xe2\x80\x94 i6rt;, \xe2\x80\x94 2^, o, 2^, 16^. 00, \n\nare the states of the function 2ax^ corresponding, respec- \ntively, to the values or expressions for x^ \n\n\xe2\x80\x94 00, \xe2\x80\x942, \xe2\x80\x94I, o, I, 2, 00, \n\nand \n\no, 1/2, S/TfV, I, o, \xe2\x80\x94I, o, \n\nare the states of the function sin corresponding, respec- \ntively, to the expressions or values of 0, \n\no, ^/6, 7r/4, 7r/2, ;r, 37r/2, 27t. \n\nA function of a variable has an unlimited number of \nstates. It may have equal states corresponding to different \nvalues of the variable ; and it may have two or more states \ncorresponding to the same value of the variable. Thus, \n\n5 and I, 7\xc2\xb1l/i2, 13 and 5, i3\xc2\xb1V24, 25 and 13, \n\nare the states of the function 2X -\\- \\\xc2\xb1. V4X, corresponding, \nrespectively, to the values of x, \n\nI, 3\xc2\xbb 4, 6, 9. \n\nTrigonometric functions have equal states for all angles \ndiffering by any entire multiple of 27t. \n\nIn connection with any state of a function corresponding \nto any value of the variable, it is frequently necessary to \nconsider another state of the function, which results from \n\n\n\n6 DIFFERENTIAL CALCULUS. \n\nincreasing the value of the variable corresponding to the \nfirst state by some convenient arbitrary amount. In order \nto distinguish between these two states of the function, the \nfirst is designated as a primitive state, and the other as its \nnew or second state. \n\nAny arbitrary amount by which the variable is increased \nfrom any assumed value is called an increment of the vari- \nable. It is generally represented by the letter h, or k, or by \nA written before the variable ; as, Ajc, read " increment of \n^"* \n\nLet x\' represent any particular value of x, and h, or t\\x\' , \nits increment ; then will 2ax\'\'^ and 2a{x\' + Hf., or \n2a(x\' + A x\'Y, represent, respectively, the primitive and \nnew states of the function 2ax^, corresponding to x\' and its \nincrement h, or l\\x\'. The general expression 2a{x -\\- Ji)\'^ \nrepresents the second state of any primitive state of the \nfunction 2ax^, and from it we obtain the second state corre- \nsponding to any particular primitive state by substituting \nthe proper value of x. \n\n6. A function of two or more variables is one which de- \npends upon two or more variables and varies with each. \nThus, \n\nX sin y, xy, x^, y log x, x^ -j" y xy \xe2\x80\x94 3_);, \n\nare functions of x and_>\' ; and \n\n^\xe2\x96\xa0\\- y -\\- ^y y + tan x/z, z ^vciipc^y), \\^x^ -\\- y^ + log -2^, \n\nare functions of x, y and z. Each variable is independent \nof the others. Particular values or expressions may be \n\n* Increment as here used is in an algebraic sense, and includes a \ndecrement, which is a negative increment. In general, an assumed \nincrement of a variable is regarded as positive. \n\n\n\nCONSTANTS, VARIABLES AND FUNCTIONS. 7 \n\nassigned to one or more of the variables, and the result dis- \ncussed as a function of the remaining variables. A func- \ntion of two or more variables possesses all of its properties \nas a function of each variable. By substituting in the \nfunction 2x\'\' -\\-y, any assumed value ior y, as 5, the result \n2X + 5 is a function of a single variable. \n\n7. A quantity is a function of the sum of two variables \nwhen every operation indicated upon either variable includes \nthe sum of the two. Thus, \n\n\n\n3^ ^ X \xc2\xb1y, sin {x \xc2\xb1 y), log {x \xc2\xb1y), a^^y, \n\nand all algebraic expressions which may be written in the \nform \n\nA{x\xc2\xb1yY-\\-B{x \xc2\xb1yY-^ -\\- -f H, \n\nin which A^ By etc., are constants, are functions of the sum \nof the two variables x and \xc2\xb1y. \n\nMx(x-{-yY, Vx \xe2\x80\x94y \xe2\x80\x94 2y, Vx-]-y, ^+^, xsin{x\xe2\x80\x94y)y \n\nare not functions of the sum of x and j. \n\n\n\nsin(^-^\xc2\xb1/), A{x\'\xc2\xb1yr, 3^og{x^\xc2\xb1/), V2{x\'\xc2\xb1/)-{-ja, \n\nare functions of the sum of the two variables x\'\' and \xc2\xb1y% \nbut not of the sum of x and \xc2\xb1y. \n\n2{dVx-{- ay""), cos\'(<^V^ + ^/), 2 A/log {bVx-^ ^/ \xe2\x80\x94 3^), \n\nare functions of the sum of the two variables d V^and ay"^. \n\nIn any function of the sum of two variables, a single \n\nvariable may be substituted for the sum, and the original \n\nfunction expressed as a function of the new variable. \n\n\n\nO DIFFERENTIAL CALCULUS. \n\nThus, z may be substituted for [x -^ y) in the function \n(^{.^ + j)") giving the function in the form az^. In a similar \nmanner we may write \n\ntan {x \xe2\x80\x94 y) = tan z, a^+^ = a^^ 2aV\\og{x\xe2\x80\x94y) = 2a\\^\\og z; \n\nbut it must be remembered that z in the new form is a \nfunction of the two variables x and y. \n\n8. A state of a function of two or more variables, corre- \nsponding to a set of values or expressions for the variables, \nis the result obtained by substituting those values or ex- \npressions for the corresponding variables. Thus, \n\n\n\n\xe2\x80\x94 20, \xe2\x80\x94 6 \n\n\n\nare states of the function ^x -\\- T^y -\\- 2 corresponding, re- \nspectively, to the values or expressions for x andji^, \n\n(-4,-2), (-2,0), (-8, + 10), (0,1), (2,5); \n\nand \n\no, 4/1/3, I, ^3y c\xc2\xbb, \n\nare states of the function tan {x -\\- y) corresponding, re- \nspectively to the values or expressions for x andji^, \n\n(0,0), {7r/lZ,7t/g), {7t/l2,7r/6), {27t/g,7t/^), (o, TT^). \n\nAny function, in which all of the variables are indepen- \ndent, is a variable, and has an unlimited number of states. \n\n9. A function of several variables may be equal to some \nconstant value or expression ; in which case one of the \nvariables is dependent upon the others. Thus, the first \nmember of the equation 2x -\\- 3;\' = 7 is a function of the \ntwo variables, x and 7 ; but x and j\' are mutually dependent. \n\n\n\nCONSTANTS, VARIABLES AND FUNCTIONS. Q \n\nAny equation containing n variables expresses a depen- \ndence of each variable upon the others; and there are only \nn \xe2\x80\x94 \\ independent variables in such an equation. In other \nwords, the number of independent variables in any equation \nis one less than the total number of variables. \n\nIn any group of equations, the number of independent \nvariables is equal to the total number of variables less the \nnumber of independent equations. \n\n10. An Algebraic function is one that can be expressed \ndefinitely by the ordinary operations of Algebra ; that is, by \naddition, subtraction, multiplication, division, formation of \npowers with constant commensurable exponents, and ex- \ntraction of roots with constant commensurable indices. \n\nAlgebraic functions have particular names based upon \npeculiarities of form. \n\nA ratioiial function of a variable is one in which the vari- \nable is not affected by a fractional exponent. \n\nAn integral function of a variable is one in which the \nvariable does not enter the denominator of a fraction, or in \nother words, is not affected by a negative exponent. \n\n^m _|_ ^jt;;\xc2\xab - I _j_ ^j^;;\xc2\xab - 2 _j_ Gx -{- H, \n\nin which m is a positive integer, and A, B, etc., do not con- \ntain X, is a rational and integral function of x. The coeffi- \ncients A^ B, etc., may be irrational or fractional. \n\nA rational integral function of a variable is also called an \nentire function of that variable. \n\nA linear function of two or more variables is one in \nwhich each term is of the first degree with respect to the \nvariables. \n\nThus, 2X -f 3j -f 72; is a linear function of x^ y and z. \n\nA function is homogeneous with respect to its variables \n\n\n\nlO DIFFERENTIAL CALCULUS. \n\nwhen all of its terms are of the same degree with respect to \nthem. \n\nA linear function is a homogeneous function of the first \ndegree. \n\nII. A Transcendental function is one that cannot be \nexpressed definitely by the ordinary operations of Algebra. \n\nIn general, a transcendental function may be expressed \nalgebraically by an infinite series. \n\nTranscendental functions include expo7iential^ logarithmicy \ntrigonometric^ inverse trigonometric^ hyperbolic and inverse \nhyperbolic functions. \n\nAn Exponential function is one with a variable, or an in- \ncommensurable constant, exponent; as, \n\nA Logarithmic function is one that contains a logarithm \nof a variable; as, \n\nlog X, log {a-\\-y)^ 2ax\' \xe2\x80\x94 ::i;/log x.* \n\nA Trigonometric function is one that involves the sine, \nor cosine, or tangent, etc., of a variable angle ; as, \n\ncot X, sec 2X^, (^\xe2\x80\x94 sin x)/x^^ versin\' x. \n\nAn Inverse Trigonometric function is one that contains \nan angle regarded as a function of a variable sine, or cosine, \nor tangent, etc. \n\nSin~^jj;, tan""\'j>/, read " the angle whose sine is j " ; " whose \ntangent isj\'" ; are symbols used to denote such functions. \nHaving given y = versin x, then x \xe2\x96\xa0= versin\'^j/ ; and if \nu = cos y^ then y = cos~^u, etc. \n\n^ Napierian logarithms will always be considered unless some other \nbase, as a, is indicated by loga. \n\n\n\nCONSTANTS, VARIABLES AND FUNCTIONS. I I \n\n12. Hyperbolic Functions.\xe2\x80\x94 From Trigonometry, we \n\nhave \n\nX\' ^ x\' ^\' \n\nsm ^ = ^ r h -T , f- etc.; \n\n13. li ll \n\nUv *A/ i/v \n\ncos X = 1 -f- \xe2\x80\x94 \xe2\x80\x94 r\xe2\x80\x94 4" etc. \n\n2 4 6 \n\n\n\nPlacing V \xe2\x80\x94 I = z, and substituting xi for x^ we have \n\nx^ x^ oc\' \nsin xi = i{x^\xe2\x80\x94 -^ \xe2\x80\x94 - -f -\xe2\x80\x94 + etc.); \n\n|3 li 11 \n\nCOS ^z =: I -| [- [- -\xe2\x80\x94 -f etc. \n\n2 |4 |6 \n\nFrom Algebra, we have \n\nx^ x^ X* \n\ne" =1 + ^-1 ^ _|. + etc.; \n\n^ 13 li \n^"^= \\ \xe2\x80\x94 X -\\- r [- -T etc. \n\n^ ii l\xc2\xb1 \n\nHence, \n\n^^ + ^-^ , Jt:\' Jt:\' ^\' \n\n\xe2\x80\x94 T-=\' + T+-l^+|^+^*<=\xe2\x80\xa2\' \nTherefore, \n\nsin xi = i(e\'\' \xe2\x80\x94 ^~^)/2; cos-^/=: (^^ + e\'\'^)/2. \n\ncos ^7 is real, and is called the hyperbolic cosine of x. It \nis generally written cosh x. The real factor in the sine of \nxi is called the hyperbolic sine of x, and is written sinh x. \nThus, \n\nsinh X\xe2\x80\x94 {e\'\' \xe2\x80\x94 ^~*)/2 ; cosh x\xe2\x80\x94 (^^ + ^"\'\')/2. \n\n\n\n12 DIFFERENTIAL CALCULUS. \n\nFrom which, \n\ncosh\'^ X \xe2\x80\x94 <SA.X\\\\v ^ = I. \n\nComparing with x^ \xe2\x80\x94 y^ \xe2\x80\x94 i, we see that cosh x and sinh \nX may be represented by the coordinates of points of an \nequilateral hyperbola, referred to rectangular coordinate \naxes, with the original its centre. Hence the name hyper- \nbolic functions. The functions sinh x and cosh x are not \nperiodic functions for real values of x^ but increase with x \nindefinitely. \n\nBy analogy with trigonometric functions, other hyper- \nbolic functions are defined, and written, as \n\nsinh X \n\ntanh X = \xe2\x80\x94 - \xe2\x80\x94 \n\ncosh J^: \n\ncoth X = \xe2\x80\x94 - \xe2\x80\x94 \ntanh X \n\nsech X \xe2\x80\x94 \n\ncosh X e"^ -\\- e~\'^* \n\ncosech X \xe2\x80\x94 -r-, \xe2\x80\x94 = \xe2\x80\x94 ^. \n\nSinn X e^ \xe2\x80\x94 e~ \n\nIt follows that \n\ntanh\'\' X + sech^ x \xe2\x80\x94 \\ \xe2\x80\x94 coth\'\' x \xe2\x80\x94 cosech\'\' x \n\n13. Inverse Hyperbolic Functions \xe2\x80\x94 Writing J^^ = sinh x, \n\nwe have x = sinh"^ y. \n\n\n\ne^ \n\n\n\xe2\x80\x94 e \n\n\n\n\ne"" \n\n\n-\\-e- \n\n\nx> \n\n\ne^ \n\n\n+ e- \n\n\n-X \n\n\ne^ \n\n\n\xe2\x80\x94 e~ \n\n\nx\'> \n\n\n\n\n2 \n\n\n\n\n\nFrom 7 = sinh x \xe2\x80\x94 (^^ \xe2\x80\x94 e \'*\')/2, we find e"^ = y\xc2\xb1Vi +^1 \nhence, sinh"^ y z=i x= log (j\xc2\xb1Vy^ + i)* Similarly, \ny = cosh X = (e"" -\\- <?~^)/2 gives cosh~^ y=\\og(j \xc2\xb1V/ \xe2\x80\x94 1); \n\nc^ \xe2\x80\x94 c~^ i-l-y \ny = tanh x \xe2\x80\x94 -\xe2\x80\x94 gives tanh"\' y -- I log ; \n\n\n\nCONSTANTS, VARIABLES AND FUNCTIONS. \n\n\n\ncoth X = ~ \xe2\x80\x94 gives coth ^ y \xe2\x96\xa0= \\ log \n\n\n\ne \xe2\x80\x94 e y \xe2\x80\x94 I \ny = sech X = ^ ^ ^_^ gives sech-^j/ == log ^ ; \n\n\n\nr=cosecn:r=:\xe2\x80\x94 \xe2\x80\x94 gives cosech y = log -^-^. \n\n,?\'*\' \xe2\x80\x94 ^^ ;; \n\n14. Explicit and Implicit Functions. \xe2\x80\x94 When a function \nis expressed directly in terms of its variable or variables, it \nis an explicit function ; otherwise it is an implicit function. \n\nThus, in the equations \n\nJ = 2^\' + 3^, y = tan\'x, y = f, y = log 2ax\\ y = f{x, z), \n\ny is an explicit function of the variables in the second \nmembers, and in the equations \n\n^y + 3V=:^\'3^ f^\\ogx\\ f = r\'-x\\ f{y,x)=o, \n\ny is an implicit function of x. \n\nThe relation between an implicit function and its vari- \nables is sometimes expressed by two equations. Thus, \ny = 32/j u^ \xe2\x80\x94 Vx, in which y is an implicit function of x. \n\ny =/W> u= (p{x)\', and y \xe2\x80\x94 f{u), x = (p{u), \n\nare forms expressing jf as an implicit function of x. \n\n15. Increasing and Decreasing Functions \xe2\x80\x94 A function \nthat increases when a variable increases, and decreases \nwhen that variable decreases, is an increasing function of \nthat variable. Thus, 2X^ y.-r^, 2^, ax^/b, tan x, are increas- \ning functions of x. \n\nA function that decreases when a variable increases, and \nincreases when that variable decreases, is a decreasing \n\n\n\n14 DIFFERENTIAL CALCULUS. \n\nfunction of that variable. Thus, i/.r, {c \xe2\x80\x94 xf^ b/ax^^ cot.r, \nare decreasing functions of x. \n\nFunctions are sometimes increasing for certain values of \nthe variable, and decreasing for others. Thus, {c \xe2\x80\x94 a)\' is \nan increasing function for all values of x greater than c\\ \nbut decreasing for all values of x less than c. 2ax\'^ is an \nincreasing function when x is positive, and decreasing when \nX is negative. The positive value of j = \xc2\xb1 S^ r\' \xe2\x80\x94 x^ is an \nincreasing function for values of x from \xe2\x80\x94 r to o, but de- \ncreasing for values of x from o to + r. The negative \nvalue of jv is a decreasing function for negative values of x^ \nand increasing for positive values of x. sin x^ cos x^ sec x^ \nvers X, are increasing functions for some values of the vari- \nable and decreasing for others. \n\ni6. Continuous and Discontinuous Functions. \xe2\x80\x94 A func- \ntion is continuous between states corresponding to any two \nvalues of a variable when it has a real state for every inter- \nmediate value of the variable, and as the difference between \nany two intermediate values of the variable approaclies \nzero, the difference between the corresponding slates also \napproaches zero. Otherwise a function is discontinuous \nbetween the states considered. \n\nThe varying height of a growing plant is a continuous \nfunction of time. \n\nIf for any value of the variable a function is unlimited, \nimaginary or changes from one state to another without \npassins: through all intermediate states, the continuity of \nthe function is broken at the corresponding state. \n\n\xc2\xb1.Sf 2px is continuous between states corresponding to \njc = o and \xe2\x96\xa0%\xe2\x96\xa0 = oo . \n\n\n\n-^b/a y a^ \xe2\x80\x94 x"^ is continuous between states correFpond- \ning to X = \xe2\x80\x94a and x ^= -\\- a. \n\n\n\nCONSTANTS, VARIABLES AND FUNCTIONS. 1 5 \n\n\n\n\xc2\xb1.bla\\ x\' \xe2\x80\x94 a\' is continuous between states correspond- \ning to \n\n^ = \xe2\x80\x94 00 and jc = \xe2\x80\x94 ^, ^ = (2 and ji; = oo , \n\nbut is discontinuous between states corresponding to \nX =^ \xe2\x80\x94 a and x \xe2\x96\xa0= a. \n\nSin X, cos :r, e"", and all entire algebraic functions are \nalways continuous. \n\nA continuous function in passing from any assumed state \nto another must pass through all states intermediate to \nthose assumed ; but it may have intermediate states greater \nor less than the states assumed. Thus, the function \n\n\n\n^/r\'^ \xe2\x80\x94 x^ is continuous between the states o and r/2 V^, \nwhich correspond to x = \xe2\x80\x94 r and x = r/2 ; but it is greater \nwhen X ^^ o than either of the states considered. \n\nA function always continuous changes its sign only by \npassing through zero ; but a discontinuous function may \nchange its sign without passing through zero. \n\nUnless otherwise stated, functions will be regarded as \ncontinuous in the vicinity of states under consideration. \n\n17. Functional Notation. \xe2\x80\x94 A function of any quantity, \nas .T, is generally represented thus, /(x), read "function of \nx" Other forms are also used; as, /\'{^), \xe2\x96\xa0F{x), F^x), \n0Gt), 0\'W, 4-{x), il:^(x). \n\nThus, ax/(i-^x) maybe represented by /(.t). The/, \nor exterior symbol, is called the functional symbol, or sym- \nbol of operation. It represents the operations involved in \nany particular function. Thus, having f{x) = ax/{i -\\- x), \nf indicates that x is to be multiplied by ^, and that the \nproduct is to be divided by i -f- x. Its significance re- \nmains unchanged throughout the same discussion or subject, \nand placed before the parenthesis enclosing any other quan- \n\n\n\n1 6 DIFFERENTIAL CALCULUS. \n\ntity it indicates that the quantity enclosed is to be sub- \njected to the same operations that x is in the expression \naxl[\\ -j- x). Thus, \n\n. /(^) = r^., /(sin 0) = "^ ^\'" "^ \n\n\n\n1+2 I -|- Sin \n\nIn order to represent different functions of the same \nquantity the functional symbol only is changed. Thus, if \nF{x) is selected to represent 2 \\^bx, then some other forn^ \nas FX^), or 0(^), etc., should be taken to denote ^cx^ + 2X \n\nDifferent functions of different quantities are represented \nby different symbols within and without the parentheses. \nThus, Vx\'\' \xe2\x80\x94 a^ and 4>\'V(i \xe2\x80\x94 y^) may be denoted by f{pc) \nand <p{y), respectively. \n\nA function of x^ is written fix\'\') or F{x^), etc., and the \nsquare of a function of x is designated by/(:^) or <p{x) , \netc. \n\nWhen the quantity is represented by a simple symbol, \nthe notation fx, (px, Fx\'^, tj^x^^ etc., is frequently used. \n\nT^cVmy\'^ may be expressed as a function of my\' by some \nform, 2L\'S, f{my\'^) ox f\'{iny\'\\ etc. \n\nHaving represented az\' by /(2), and 3^ \\^az\' by F{az\'\'), \nwe may write y Vaz\'\' = F{az^) = F[/{z)]. \n\nHaving a"" = (p{^), and b Va-^ = ^\'(^*), we write \n\n2db ya^ -\\- h ^ \' \n\nAn expression containing several different functions of a \nvariable, as 2ax:\' \xe2\x80\x94 log x -j- 3 sin jjc, may be considered as \n\n\n\nCONSTANTS, VARIABLES AND FUNCTIONS. 1/ \n\na function of the several functions of the variable, and \nrepresented thus: \n\n4>V^{^y)^ ^Wj/(-^)] represents a function of three differ- \nent functions of different variables. \n\nP\'unctions of two variables are denoted thus: /(-^j >\'), \nf\\x,y), F{y, z), (p{x, y), tp{x, z), ip^{x, z), etc.; and func- \ntions of three variables by F{x,y, z), ^p{r, s, t), etc. \n\nFunctions of any number of variables are indicated simi- \nlarly by writing all the variables within the parentheses. \n\nIn all cases a functional symbol indicates the same oper- \nations in any one subject. \n\nThus, if f{x, y) = ax -{- by, then f{s, t) = as -\\- bt\\ /{i, 3) \n= 2a -\\- ^b] /(o, m) = bm. \n\nHaving 0(^, z,y) = 2x \xe2\x80\x94 cz-^-y"^, then (p{r, s, t) = \n2r-cs-\\- t\\ \n\nFunctions are frequently represented by single letters; \nthus, \xc2\xb1 y Bi^ \xe2\x80\x94 x^ may be represented by y, giving \n_y = \xc2\xb1 y/ R^ \xe2\x80\x94 x^\\ and/(^, y) by z, giving z = f(x,y). \n\nF{x -\\-y)y f{x + Ji), ^{/ + r^) are forms denoting func- \ntions of the sum of two variables. \xc2\xa77. \n\nILLUSTRATIONS. \n\nHaving /(^) -x^-\\- Px^-\' + Qx^-^ + . . .-\\-U,m which F, Q, \netc., do not contain x, then \n\nf{3bc) = {3bcy^ + F{3bc)^-^ + . . . + ^7. \n\nf{a - x) = {a- xy^ -I- P{a - x)\xc2\xbb\'-^ + . . . -j- i/ \n\nf{o) = o^-\\- Fo\xc2\xbb\'-^ + .:..-{- W \n\n/(.\')=0 + i\'(.^)\xe2\x80\x94 + .. .. + 1/. \n\n\n\n1 8 DIFFERENTIAL CALCULUS. \n\nHaving then \n\n(P{a) = Ao\' + ca, <p{x + r) = 4(x -^ yf + c{x -\\-y). \n\nip{ax^) = 4{ax^)\'^ -f c{ax^), ^(sin G) ^ 4 sin^S + ^ sin 6. \n\nF(x) = a^ , ^{x -\\- y) = a^^y = a^ y^ay = F{x) X F{v). \n\nF{x)) z= a\'^y, F(x)^ = Fjf = (a^)^ ={ay = F(xy). \n\n/x = \\ogx, /(limit ^) = log (limit ;i;). \n\nfnx=a^, fn (limit x) = a"\xc2\xab\xc2\xbb" ^. \n\nn y- c\\ w \xe2\x80\x94 {jvy \n\nIf (p(i) = \\z, i>{x) = sax, and J^(w) = ^^ _ ^j, > \n\nthen ^(^[^(.)]) = \'^^\'\'^^~^-(s\'\'Viy ^^^^^^ \n\n\n\nIf 0(jf, jf) \xe2\x80\x94 2j; + sin_j/, and i}){z) = 2>V !^, then ?^f0(jr, jj/)] = 3y2^-|- sin \nIf /(-^j J\' ^) = 7<^^r^/2, and /"(j)^) = \\^y \n\n\n\nthen \n\n\n\n-^(^[/(-O\'.-)])] = 2a^<\'-V^)\'. \n\n\n\n18. Lines are classed as algebraic or transcendental \naccording as their equations involve algebraic functions \nonly or contain transcendental functions. \n\nAny portion of any line may be considered as generated \nby the continuous motion of a point. The law of its motion \ndetermines the nature and class of the line generated. \n\nLet s represent the length of a varying portion of any \nline in the coordinate plane XY, of which the equation in \nX and y is given. .9 depends upon the coordinates of its \nvariable extremities, and varies with each; but the equation \nof the line establishes a dependence between these coordi- \nnates. Hence, j" is a function of one independent variable \nonly. \n\nIf the line is in space, its two equations establish a de- \n\n\n\nCONSTANTS, VARIABLES AND FUNCTIONS. 19 \n\npendence between the three coordinates of its extremities, \nso that one only is independent. \n\nThe same result will follow if a system of polar coordi- \nnates is used. \n\n19. Convexity and Concavity. \xe2\x80\x94 The side of an arc of \nany curve upon which adjacent tangents, in general, inter- \n\n\n\n\nsect is called the convex, and the other, or that upon which \nadjacent normals intersect, the concave side. A curve, at \nany point, is said to be convex towards the convex side \nand concave in the opposite direction. \n\n20. Graphic Representative of a Function of a Single \nVariable. \xe2\x80\x94 The relation between any function and its vari- \nable may be expressed by the equation formed by placing \nthe function equal to a symbol. Thus, placing /jc equal to \nJ, we have J = fx, which expresses the relations between j \nand X, and therefore between the function fx and its \nvariable x. y \xe2\x80\x94 fx is also the equation of a locus, the \ncoordinates of whose points bear the same relations to each \nother as those existing between the corresponding states of \nthe function and variable. Therefore, by constructing, as \nin Analytic Geometry, any point of this locus, its oj^dinate \nwill represent graphically the state of the function corre- \nsponding to the state of the variable similarly represented \nby its abscissa. The locus thus determined is called the \ngraph of the function. It is important to notice that it is \n\n\n\n20 \n\n\n\nD IFFEREN TIA L CA LCUL US. \n\n\n\n\nthe ordinate of the graph, not the graph itself, that repre- \nsents the function. \n\nTo illustrate, let the line AB be the \ngraph oi fx. Then the ordinate PA is the \ngraphic representative oi fx, corresponding \nto a value of x represented by OF. Sim- \nilarly, /^\'^ .represents /x when x \xe2\x80\x94 OP\' . \nThe ordinates PM and P M\' of the \ngraph MQM\' represent two different \nstates of the function corresponding to \nthe same value of the variable, \xc2\xa7 5. \n\nThe ordinates PM, RN, and SO of \nthe graph MNO represent equal states \nY of the function corresponding \n\nto different values of the vari- \nable, \xc2\xa7 5. \n\nThe graph of a function \nwhich is of the first degree with \nrespect to its variable is a \nright line, otherwise not. \n\nThe graph of a continuous function is a continuous line. \n\n21. Surfaces. \xe2\x80\x94 Any portion of any surface may be con- \nsidered as generated by the continuous motion of a line. \nThe form of the line and the law of its motion determine \nthe nature and class of the surface generated. \n\n22. Let u represent the area of a varying portion of the \nsurface generated by the continuous ^ M \nmotion of the ordinate of any given \nline in the plane XY. \n\nu depends upon the coordinates of \ntlie variable extremities of that portion \nof the mven line which limits it, and varies with each ; but \n\n\n\n\nCONSTANTS, V^ARIABLES AND FUNCTIONS. 21 \n\n\n\n\nthe equation of the given line establishes a dependence be- \ntween these coordinates. Hence, z^ is a function of but one \nindependent variable. \n\n23. Let r=f{v) be the polar equation of any plane \ncurve, as DM, referred to the \npole F, and the right line FS. \nLet u represent the area of a \nvarying portion of the surface, \ngenerated by the radius vector \nrevolving about the pole, u will \nchange with v and r; but v and r \nare mutually dependent. Hence, \nu is 2i function of but one independent variable. \n\n24. Let any line in the plane XY, as AM, revolve about \nthe axis of X. It will generate a sur- \nface of revolution. \n\nThe same surface may be generated \nby the circumference of a circle, whose \ncentre moves along the axis X, with its \nplane perpendicular to it ; and whose radius changes with \nthe abscissa of the centre of the circle, so as to always \nequal the corresponding ordinate of the curve AM. The \nradius of the generating circumference is, therefore, a func- \ntion of the abscissa of its centre. Hence, the generating \ncircumference, and any varying zone of the surface gener- \nated as described, is a function of but one independent \nvariable. \n\n25. The area of any surface with two independent vari- \nable dimensions is a function of two independent variables. \nFor example, the area of any rectangle with variable sides, \nparallel respectively to the coordinate axes X and Y, is a \nfunction of the two independent variables x andj>. \n\n\n\n\n22 DIFFERENTIAL CALCULUS. \n\n26. Having any surface, as ATL, let A BCD = ?/ be a \nportion included between the coordinate planes XZ, YZy \nand the planes DQR and BPS, parallel to them respec- \ntively. Let OP \xe2\x80\x94 X and OQ^^yht independent varia- \n\n\n\n\nbles, u will depend upon x^y, and z\\ but the equation of \nthe surface makes z dependent upon x and j. Hence, u is \na function of but two i7ide pendent variables. Similarly, it \nmay be shown that any varying portion of the surface \nincluded between any four planes, parallel two and two, to \nthe coordinate planes XZ and YZ^ is a function of but two \ni7idependcnt variables. \n\n27. Graphic Representative of a Function of Two \nVariables. \xe2\x80\x94 Placing any function of two variables, as/(x jf), \nequal to z, we have z =/(x, j) which expresses the rela- \ntions between the function and its variables. \n\nThe locus whose equation is z \xe2\x80\x94 f[x,y), is called the \ngrai)hic surface of fi^x^)\'), for the reason that the oj^dinaie \n\n\n\n\xe2\x80\xa2 CONSTANTS, VARIABLES AND FUNCTIONS. 23 \n\nof any of its points will represent graphically the state of \nthe function corresponding to the states of the variables \nsimilarly represented by their respective abscissas. \n\nIt is important to notice that it is the ordinate of the \ngraphic surface that represents the function, and not a \nportion of the surface as in the case described in \xc2\xa7 26. \n\nThe graphic surface of a function which is of the first \ndegree with respect to each of two variables is a plane, \notherwise not. \n\n28. Volumes. \xe2\x80\x94 Any portion of any volume may be con- \nsidered as generated by the continuous motion of a surface. \nThe form of the surface and the law of its motion deter- \nmine the nature and class of the volume. \n\n29. Let any plane surface included between any line in \nthe plane XY^ as AM, and the axis of X be revolved about \n\nX. It will generate a volume of revo- y M \n\nlution. The same volume may be gen- \nerated by the circle, whose centre \nmoves along the axis X, with its plane \nperpendicular to it ; and whose radius \nchanges with the abscissa of the centre of the circle, so as \nto always equal the corresponding ordinate of the curve \nAM. The radius of the generating circle is, therefore, a \nfunction of the abscissa of its centre. Hence, the generat- \ning circle, and any varying segment of the volume generated \nas described, is a function of but one indepe7ident variable. \n\nIt is important to notice in this case, that the generating \nsurface is limited by the ordinates PA and P\' M, corre- \nsponding to the extremities of the limiting curve, which \nordinates are perpendicular to the axis of revolution. \n\n30. Having any volume, as ATL, bounded by a surface \nwiiose equation is given, and the coordinate planes, let \n\n\n\n\n24 \n\n\n\nD IFFERENTIA L CAL CUL US. \n\n\n\nABCD-ON ^=- F be a portion included between the coor- \ndinate planes XZ, YZ, and let the planes DQR and BFS \nbe parallel to them respectively. \n\nLet OP ^ X and OQ=y be independent variables. \nV will depend upon x, y and z ; but the equation of the \nsurface makes z dependent upon x andjj;. Hence, F is a \nfunction of but two independent variables. \n\n\n\n\nIn a similar manner it may be shown that any varying \nportion of the volume included between any four planes, \nparallel two and two, to the coordinate planes XZ and FZ, \nis a function of but two independent variables. \n\n3I\xc2\xbb Any volume with three independent variable dimen- \nsions is a function of three independent variables. For \nexample, the volume of any parallelopipedon with variable \nedges parallel, respectively, to the coordinate axes X, Y \nand Z, is a function of x^y and z ; all of which are inde- \npendent. \n\n\n\nPRINCIPLES OF LIMITS, 2^ \n\n\n\nCHAPTER II. \nPRINCIPLES OF LIMITS. \n\n32. The Limit of a variable * is a fixed finite quantity or \nexpression which the variable, in accordance with a law of \nchange, continually approaches, and from which it may be \nmade to differ by a quantity less numerically than any \nassumed quantity however small. \n\nThus, any constant, as C, is the limit of any variable, as \n/(^), when, under a law, /(;t:) approaches C to within less \nthan any assumed value however small it may be. \n\nVarious symbols are used to indicate a limit under a law. \nThus, assuming that f{x) approaches a limit C as x \napproaches a, we write \n\nlimit /(^) = Lt. f{x) = \\imf{x) = limit /(j*:) = C. \n\nEach form is read, " the limit oi f{pc) as x approaches a."" \n\nAny variable which under a law approaches zero as a \nlimit is called an infinitesimal. Thus, \n\n1^"^^^ [i - cos ^1 = o. \n\nAny variable which under a law can exceed all assumed \nvalues, however great, is called an infinite. It is not a defi- \nnite quantity. \n\n* In this chapter the term variable is used in its general sense (\xc2\xa7 i). \nand includes all functions of variables. \n\n\n\n20 DIFFERENTIAL CALCULUS, \n\nAn infinite cannot be a limit. Thus, \n\nis a form indicating that as x approaches zero, i/x is un- \nlimited. \n\nA tangent to any curve is a limiting position of a secant \nthrough the point of tangency, under the law that one or \nmore of its points of intersection with the curve approach \ncoincidence with the point of tangency. \n\nIn some cases, due to the form of the function or to the \nlaw of change, the variable can never become equal to its \nlimit. Thus, \n\nlimit \n\n\n\nlimit ^+i^ii^_r+_i1 \n\nX "4~ I \nBut % I, for all values of x.^ \n\n\n\nThe circumference of a circle is the limit of the perim- \neter of an inscribed regular polygon as the number of its \nsides is continually increased. The radius is the limit of \nthe apothem, and the circle that of the polygon, under the \nsame law. \n\nAn incommensurable number is the limit of its successive \ncommensurable approximating values. Thus, the terms of \nthe series 1.7, 1.73, 1.732, etc., taken in order, are approach- \ning 1/3 as a limit. \n\nIn all cases, whether a variable becomes equal or not to \nits limit, the important property is that their difference is \nan infinitesimal. \n\nAn infinitesimal is not necessarily a small quantity in any \nsense. Its essence lies in its power of decreasing numeri- \n\n\n\nPRINCIPLES OF LIMITS. 27 \n\ncally ; in other words, in having zero as a limit, and not in \nany small value that it may have. It is frequently defined \nas " ail infiiiitely small quantity "y that is not, however, its \nsignificance as here used. \n\nIn representing infinitesimals by geometric figures they \nshould be drawn of convenient size ; and it is useless to \nstrain the imagination in vain efforts to conceive of the \nappearance of the figure when the infinitesimals decrease \nbeyond our perceptive faculties. Usually one or two auxil- \niary figures representing the magnitudes at one or two of \ntheir states under the law give all the assistance that can \nbe derived from figures. \n\nIn all cases, when referring to the limit of a variable, it is \nnecessary to give the law ; for the limit depends not only \nupon the variable, but also upon the law by which it \nchanges. Under a law, a determinate variable has but one \nlimit ; but it may have different limits under different laws. \n\nAn important consequence of the definition of a limit is \nthat if two variables, in approaching limits under a law, \nhave their corresponding values always equal, their limits \nwill be equal. Thus, for all values of x^ we have \n\n(c^ \xe2\x80\x94 x^^/(a \xe2\x80\x94 x) =^ a -{- Xf \nhence \n\nJ\xe2\x84\xa2\'i (\xc2\xab\'- ^\')/(\'^ - *) = lim [a + x]= za. \n\n33 \xe2\x80\xa2 A variable which^ in approaching a limit ^ ultimately has \nand retains a constant sign cannot have a limit with a contrary \nsign. \n\nFor suppose /(j^) becomes and remains positive, and that \nlimit f{x) \xe2\x80\x94 \xe2\x80\x94 C. From the definition of a limit, f(pc) \nmay be made to differ from \xe2\x80\x94 C by a value numerically \n\n\n\n28 DIFFERENTIAL CALCULUS. \n\nless than C. It would therefore become negative, which is \ncontrary to the hypothesis. In a similar manner, it may be \nshown that a variable always negative cannot have a posi- \ntive limit. \n\n34* If the difference between the cori^esponding values of \nany two variables, approaching limits^ is an infinitesimal^ the \nva: iables have the same limit.^ \n\nLet U and V represent any two variables giving \n\nU- V= S, or U= V+d, \n\nin which S is an infinitesimal. \n\nLet Cbe the limit of C/^, then U =\xe2\x96\xa0 C \xe2\x80\x94 e, in which e is \nan infinitesimal. \n\nSubstituting we have \n\nC:-e=F+d, or C - V = d ^ e, \n\nthe second member of which is an infinitesimal. Hence, C \nis the limit of V. \n\n35. The limit of the sum of any finite number of variables \nis the sum of their limits. \n\nLet U^ \xe2\x80\x94 V, JV, etc., represent any variables, and A, \n\xe2\x80\x94 B, C, etc., their respective limits ; then \n\nl7=A-\xe2\x82\xac, - F= -B+S, IV=C- Go.\'etc, \n\n\'n which e, d, go, etc., are infinitesimals. \n\nAdding the corresponding members we have \n\nC/~ F+lV-{- etc.= A- B -rC-^etc- e + (^- &? + etc. \n\n* In order to avoid the frequent repetition of the expression \'" under \nthe law," it will be assumed, unless otherwise stated, that the chancres \nin all the variables considered together, or in the same discussion, \nare due to one and the same law; that all variables and their functions \nare continuous between all states considered, and that they have \nlimits under the law. \n\n\n\nFEIiVCIFLES OF LIMITS. 2g \n\nHence, \n\nlimit [l/-V+JV-^etc.] ^ A -^ + C+etc. \n\n= \\\\mU \xe2\x80\x94 lim F+ lim W-\\- etc. \n\n36. In general, the limit of the product of any two variables \nis the product of their limits. \n\nLet U and V represent any two variables having A and \nB^ respectively, as limits. \n\nThen U \xe2\x80\x94 A \xe2\x80\x94 e and V= B \xe2\x80\x94 S, in which e and d \nare infinitesimals. Multiplying member by member, we \nhave \n\nC/V=AB-Be-AS-^ed, \nand \n\nlimit [CrV] = AB = limit Cr limit F, \n\nIt follows that, in general, the limit of any power or root \nof any variable is the corresponding power or root of its limit. \n\nThus, limit U" = (limit UY, and limit U~^ \xe2\x80\x94 (limit t^)\xc2\xab, \n\nHaving a"" = N, x and N approach corresponding limits \n\ntogether; hence ^""^-^ = lim JV = lim a"", and lim x = \n\nlog lim JV. Also, since x = log JV, we have lim x = lim \n\nlog JV. Therefore log lim JV = lim log JV. \n\n37. In general, the limit of the quotient of any variables is \nthe quotient of their limits. \n\nWith the same notation as in \xc2\xa7 36, we have \n\nlimit ^ = lim lUV-\'\\ = lim ^7[lim F]- = \'^^^ = |. \n\nWhen B = o, and A ^ o, U/V\'is unlimited. \nWhen ^ = o = ^, the principle fails to determine the \nlimit which by definition is determinate, \n\n38. It follows from \xc2\xa7\xc2\xa7 35, Z^, 37, that, in general, the \n\n\n\n30 \n\n\n\nD IFFEREN TIA L CALC UL US. \n\n\n\nlimit of any function of any variables is the same functio7t of \ntheir respective limits. \nThus, in general, \n\nlimit /(^ V, ) =/(Iim U, lim F, ), \n\nand to obtain the limit of any function of variables we, in \ngeneral, substitute for each variable its limit. \' \n\n39. Exceptions to the above general rule arise, and are \nindicated by the occurrence of some indeterminate form, as \n\n0/0, CO /oO , OCO , 00 \xe2\x80\x94 00 , 0\xc2\xb0, 00 \xc2\xb0, I*. \n\nTo illustrate, having /(:<;) = {pc"\xe2\x80\x94 i)/{x \xe2\x80\x94 i), the general \nrule gives limit /(^) = 0/0, whereas we find \n\n/(2) = 3> /(i-5) = 2.5, /(i.i) = 2.i, /(i.oi) = 2.01, \n/(i.ooi) = 2.001, etc., \n\nand the nearer we take x to i, the nearer will /(:^) approach \nto 2. By taking jc sufficiently near to i,/(^) may be made \nto differ from 2 by a number less numerically than any as- \nsumed number however small. Hence, 2 is (\xc2\xa7 2^2) the limit \noi fi^x) as x-Wf-^Y. It should also be ^^bserved that 2, con- \nsidered with the states of /(^) which immediately precede \nand follow it, conforms to the law of continuity. \n\n\n\nY \n\n\n\n\n^ \n\n\nM\' \n\n\n\n\n\n\nw \n\n\n^ \n\n\n^;=^=^ \n\n\nA?/ \n\n\n\n\n\n\n,y \n\n\n\n\n\n\n\n\n/b/ \n\n\n/ \n\n\ny \n\n\n\n\ny \n\n\n\n\n^^ \n\n\n\n\np \n\n\nAX \n\n\np\' \n\n\nX \n\n\n\nH \n\nTo illustrate a failing case geometrically, let the curve \nBMM\' be the graph of a function. Take any state, as \n\n\n\nPRINCIPLES OF LIMITS, 3 1 \n\nPJ^ corresponding to ^ = OP, and increase x by PP\' rep- \nresented by L.X. Draw the ordinate P\' M\' and the secant \nMM\'. Through M draw M Q\' parallel to X. Q\'M\', de- \nnoted by Ay, will represent the increment of the function \ncorresponding to Ajt:. \n\nQ\'M\'/PP\' = Ay/ Ax = tan Q\'MM\' will be the ratio \nof the increment of the function to the corresponding in- \ncrement of the variable. \n\nAt J/ draw MX tangent to the curve. Then, under the \nlaw that Ax approaches zero, the secant MM\' will ap- \nproach coincidence with the tangent MT, and the angle \nQ\'MM\' will approach the angle Q\'MT, or its equal XHT, \nas a limit. \n\nHence \n\nlimit {Ay/ Ax) \xe2\x80\x94 lim. tan Q\'MM\' = tan XHT, \n\nwhereas the general rule gives o/o as a result. \n\nWe observe from the above illustration that tAe limit of \nthe ratio of any increment of any function of a single variable \nto the corresponding increment of the variable, wider the law \nthat the increinent of the variable approaches zero, is equal to \nthe tafigent of the angle made with the axis of abscissas by a \ntangenty to the graph of the function, at the point correspond- \ning to the state considered. \n\nWhen J/\' coincides with J/ the secant may have any one \nof an infinite number of positions other than that of the \ntangent line MT, for the only condition then imposed is \nthat it shall pass through M. \n\nTherefore, while limit (Aj/Ajc) is definite, and equal to \n\nthe tangent of the angle that the tangent line at J/ makes \n\n\n\n32 DIFFERENTIAL CALCULUS, \n\nwith X, limit Aj/limit A^ = o/o indicates that the tangent \nof the angle which the secant makes with X becomes inde- \nterminate when M\' coincides with M. \n\nLimit (Aj/Aj^) is, therefore, one of the many values that \nlimit AJ^^/limit l\\x may have under the law. \n\nThe exceptional cases, in general, require transformation \nin order that factors common to the numerator and de- \nnominator may be cancelled, or from which the limit may \notherwise be determined. They are of the highest impor- \ntance, for the Differential Calculus, as it will be seen, is \nbased upon the limit of the ratio of the increment of the \nfunction to the corresponding increment of the variable \nunder the law that the increment of the variable vanishes. \nThe remainder of this chapter will, therefore, be devoted to \ncertain important exceptional cases and methods. \n\n40. ^^^^^ :sr--j^ ^ my\xe2\x80\x94 ^ \n\nx\'m-^y X \xe2\x80\x94 y \n\nThis formula is deduced in Algebra for all commensu- \nrable values of m. Since (\xc2\xa7 32) any incommensurable \nnumber is the limit of its successive commensurable ap- \nproximating values, the formula holds true when m is \nincommensurable. \n\nLimit ^_Q^ a^ j aP^_a_ \n\nAs X \xc2\xbb^^ CO , it reaches a value k > a, thereafter \n\n\n\n< 1; also 1 \xe2\x80\x94 \n\ni \\x \n\n\n\n< \n\n\n\n\xc2\xab" la\\ \n\nkW \n\n\n\nbut \n\n\n\nlimit ^ll\xc2\xb1Y~"^Q . limit ^ ^ q. \n\n\n\nPRINCIPLES OF LIMITS. \n\n\n\n33 \n\n\n\n42. Limit(i + yy\'y \xe2\x80\x94 e. \n\n\n\n2 \n\n(i-;;)(i-2j;). . . [i-(^-iM ^^^ \n\nAs j^\' B-^ o each term approaches, as a limit, the corre- \nsponding term of the series \n\ni + i + ^4-,-^ + ...+ l^ + etc., \n\nthe sum of which is shown in Algebra to be ^ = 2.71828 . . . \n43. Limit a^ \xe2\x80\x94 I \n\n\n\nh-^-^o \n\n\n\nlog a. \n\n\n\nPlace a!" = 1 + J, whence h = loga(i + ^), and j/b-^/^m^o ; \ngiving \n\n\n\nlimit \n\n\n\nlimit \n\n\n\ny \n\n\n\nh^o h ~-^^\xc2\xb0log.(i+j) \n\nT \n\n= lim \n\n\n\nlog^. \n\n\n\nlog.(i + yY\'y \\ogae \n44. If unity is the limit of the ratio of any two -variables, \n\nthe limit of any function of one will be equal to the limit of \n\nthe same function of the other. \n\nLet C/" and ^represent any two variables, giving X\\v^\\iU/V \n\n= I. Then \n\n\n\nlimit/(C^) = lim/[^]=/( \n\n\n\nlim \n= /(lim F) = lim/(r). \n\n\n\nV \n\n\n\nlim V\\ \n\n\n\n34 DIFFERENTIAL CALCULUS, \n\nThus, \n\nlim[C:+ VX = lim [C -f ^] ; lim {CU^, = lim [CV} \n\nlim C ^ = lim C^ ; lim [ U/ W] = lim [ V/ IV]. \n\nTherefore, in searchi?ig for the limit of any function under \na law, we may replace any variable entering it by another vari- \nable, provided that, under the satJie law, unity is the limit of \nthe ratio of the two variables interchanged. The great ad- \nvantage in so doing arises when it enables us to determine \nthe required limit more readily. Thus, in the last example \nabove we may be able to determine the limit of F/^more \nreadily than that of U/W. \n\nIn making the above substitution it is important to notice \nthat the limits only are equal, and that corresponding values \nof the quantities interchanged, in general, are not equal to \neach other. \n\nThe privilege of replacing one variable by another under \nthe conditions described, so facilitates the determination \nof limits in certain exceptional cases, that it is important to \ndetermine under what circumstances the limit of the ratio \nof two variables is equal to unity. \n\n45* In general when lim \xc2\xa3/"\xe2\x80\x94 lim F, \n\nlim lU/VX^Xim U/Xim V= i. \xc2\xa737. \n\nThat is, in general, unity is the limit of the ratio of any \ntwo variables when, under the same law, they have the same \nlimit, or, what is equivalent, when the difference between \ntheir corresponding values is an infinitesimal. \n\nIf, however, lim 17=^ lim F = o, it does not follow that \nlim [CI/ V] = I {\xc2\xa7 39). Such cases require special investi- \ngation, and the following are selected on account of their \nsubsequent impoitance. \n\n\n\nPRINCIPLES OF LIMITS. \n\n\n\n35 \n\n\n\n\\ \n\n\nM \n\n\nQ\' \n\n\nQ \n\n\nV \n\n\nm\' \n\n\nP \n\n\nAX \n\n\nP\' ^ \n\n\n\nQ ^ \n\n\n\' \n\n\n; \n\n\nL.^^^"^ \n\n\nP\' X \n\n\nP Aa^ \n\n\n\n46. Take any plane surface, as PMM\'P\\ included be- \ntween any arc, as MM\', the ordinates of its extremities, \nand the axis of X, \n\nThrough J/ and J/\', Y \nrespectively, draw MQ\' \nand M\'Q parallel to \nX, and complete the \nrectangle MQM\'Q\'. \nLet y = PM, and y\' = P\' M\' . \n\nThen as PP\'= Ax, b-> o, we ultimately have \n\nPQM\'P\'\'%PMM\'P%PMQ\'P\\ \n\nand y\'M-^y. \n\nTherefore ^^ IML^l = n^ IA^ = , \nand limit \\PMM\' P\' / PMQ\' P\'\\= i. \n\nHence (\xc2\xa7 44) \n\nlimit PMM\'P \' ^ j.^ PMQ^P \' ^ ^^^y_Ax _ \nAx:^->o A^ Ax Ax \n\nIf the coordinate axes make an angle with each other, \nthen \n\n\n\n=y- \n\n\n\nlimit \n\n\n\nPMM\'P\' ,. ysin Bax . ,. \n\n= lim =y sm c/. \n\n\n\nAx \n\n\n\nAx \n\n\n\n47\xc2\xab Let MPM\' be the surface generated by the radius \nvector PM = r, revolving about \nP, as a pole, from any assumed \nposition, as PM, to any other, as \nPM\'. Let Av represent the \ncorresponding angle MPM\' . \nWith /^ as a centre, and the radii \nPM and P M\' , describe the arcs \nMQ\' and M\' R respectively. \n\n\n\n\n36 DIFFERENTIAL CALCULUS. \n\n\xe2\x80\xa2Then, as Az^ ^W)-^ o, we ultimately have, in any case, \narea RPM\' ^ area MPM\' % area MPQ% \nand limit [area RPM\'Jzx^d. MPQ\'] = i. \n\nHence, I\'^^o [area MPM\'/sirGSi MPQ\'] = i. \nTherefore (\xc2\xa7 44) \n\n\n\nlimit z= lim ^^ = hm \xe2\x80\x94 \n\n\n\nA57B^-^0 \n\n\n\nl\\V \n\n\n\n/\\V \n\n\n\nAv/2 __ r^ \n/\\v 2 \' \n\n\n\n48. Let FMM\' P\' be any plane figure as described in \xc2\xa746, \n\nand MQM\'Q\' the cor- \nB responding rectangle. \n\nQ\' Revolve the entire \n\nM^ figure about X. \n\n\xe2\x80\x9e/ Then as Aj\\;b->o, we \n\nr \xe2\x80\x94 ^x \n\n\n\nM \n\n\n\nP Ace \n\n\n\n.M \n\n\n\nPax \n\n\n\nultimately have \n\nVol. gen. by > Vol oren. by > Vol. gen. by \nPQM\'P\' < PMM\'P\' < PMQ\'P\\ \n\n\n\nBut \n\n\n\nlimit [ Vol. gen by /VoL gen. by~| \nAx:^o L PQM\'P\' I PMQ\'P\' J \n\n\n\nlim \n\n\n\nny^ Ax __ \nTZy^ Ax \n\n\n\nHence, \n\n\n\nlimit FVol. gen. by /Vol. gen. by"1_ \n\\x^^o L PMM\'P\' I PMQ\'P\' _\\- ^\' \n\n\n\nTherefore (\xc2\xa7 44) \n\n\n\n.. . /Vol. gen. by\\ /Vol. gen \\ \n\nlimit / pMM\'P\' . I PMQ\'P\' ) \n\n\\ AX / ^ Ax ^ \n\n\n\nlim \\ny^ Ax/ Ax\\ = Tty^, \n\n\n\nPRINCIPLES OF LIMITS. \n\n\n\n37 \n\n\n\n49. Let MJVM\'N\' be a portion of any surface included \nbetween the coordinate planes ZX\\ ZY and the two planes \nN\'SE and NP D parallel to them respectively. \n\nDenote the corresponding volume MNM\'N\\ OFFS \nby V. Construct the parallelopipedons OFFS-OM and \n\n\n\n\nOFFS-FM\', and represent their volumes by F and F\' \nrespectively. Let OF = h, OS = k, OF :=^ /, OM = z, \nand FM\' = z\' . \n\nThen as h^m^k-w^ o, or what is equivalent, as /\xc2\xbb^-\xc2\xbb o, \nwhence 2\' ^^2r, V will, in any case, ultimately be, and \nremain, between F and F\' . \n\n\n\nBut \n\n\n\nHence, \n\n\n\nlimit .p/pr \nlimit \n\n\n\nlimit \n\n\n\n\\.VlF-\\ \n\n\n\nIzhk/z\'hk] = I. \nand (\xc2\xa7 44) \n\n\n\nJ^o [^Aect. (9/^7^^] = lim [/^/M] = liiv \'zhk/hk\\ = z. \n\n\n\n^8 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\n50. Let MNM\'N\', denoted by S, be a portion of any \nsurface included between the coordinate planes ZX, ZY \nand the two planes N\' SE and NPD parallel to them \nrespectively. \n\nLet OP = h, and OS = k. \n\nAt M draw the tangents MB and MB\' to the curves \nMN and MN\' respectively, complete the parallelogram \nMBQB\', and denote it by T. T is the portion of the \ntangent plane to the surface at M included between the \n\n\n\n\nplanes which limit S. Let /5 equal the angle which T \nmakes with XY, giving T cos (3 = OPFS. Inscribed \nin S^ conceive an auxiliary surface, composed of 11 plane \ntriangles the sum of which, as ;/ ^-^ co , will have 6" as a \nlimit and such that the sum of their projections upon XY \nwill equal OPFS, The two triangles MN\' M\' and MNM\' \nin the figure illustrate a set fulfilling the conditions. Let /, \n/\', etc., represent the areas of the triangles and ^, ^\', etc., \nthe angles which their planes respectively make with XY. \n\n\n\nPRINCIPLES OF LIMITS, 39 \n\nThen OPFS = ^/ cos ^ = r cos /?, \n\nand :Et cos 0/T cos P = 1 (i) \n\nAs ;z ^-> 00 , S remaining constant, each triangle is an \ninfinitesimal, and we have \n\nThe same effect and result follows if -5" is made infini- \ntesimal and n remains constant. Hence, under the law \nthat A and k vanish, or, what is equivalent, that OJ^y repre- \nsented by /, is infinitesimal, we have \n\n^[s/2^] = ^ W \n\nUnder the same law, /3 is the common limit of 6, 6\\ etc. \nHence (i) \n\njl^\'^ l^t cos e/T COS /?] = lim [^f/T] = i. . (3) \n\nTherefore (2), (3), ^^\\[S/ T] = i, and (\xc2\xa7 44) \n\n^ Ji\'^jLo [^M^]= li\xc2\xab^ i^M^I = lim [{M/cos (5)/hk\\ \n\n\xe2\x80\x94 I /cos ft. \n\n51 \xe2\x80\xa2 Unity is the limit of the ratio of an angle to its si?ie, of \nan angle to its tangent^ and of the tangent of an angle to its \nsine, as the angle approaches zero. \n\nLet OCM = given in radians \nbe any angle less than 71/2 ; then \ntan > > sin 0, and as ;^-> o, \nwe have always C \n\ntan ^ \n\nsm sin \n\nB\xe2\x80\x9ej limit ta,L0 ^ ,i\xe2\x80\x9e^ -J- = I. \n\n<*s^o sm <?> cos <p \n\n\n\n\n40 DIFFERENTIAL CALCULUS \n\nHence, since 0/sin <p is always between tan 0/sin and \nunity, we have \n\n!^"^^^ [0/sin ] = I. \n\nSimilarly, since i > 0/tan > sin 0/tan 0, we have \n\n52. U\'m\'fy IS the limit of the ratio of any arc of a7iy curve \nto its chords as the arc approaches zero. \n\nLet s denote the length of any arc of any curve, and \nconceive it to be divided into n equal parts, the consecutive \npoints of division being joined by chords forming an in- \nscribed broken line whose length is designated by/. \n\nThen }:t\\^IP\\ = ^. \n\nUnder the above law the equal arcs of s vary inversely \nwith n\\ hence the same effect and result may be caused by \nretaining any fixed value for \xc2\xab, and making s approach \nzero. Hence, \n\nand if \xc2\xab= I, J^^^Jarc/chord] = i. \n\n53. Unity is the limit of the ratio of the surface generated \nby any arc of any curve under a law to that generated by its \nchord as the arc approaches zero. \n\nLet s^p and n denote, respectively, the same quantities \nas in the last article, and conceive s and;> to move together, \nunder a law, so as to generate two surfaces represented by \nS and P respectively. \n\nAs ^/\xc2\xab^^oo, any state of s without regard to form is the \nlimit of the corresponding state of/, and 6" is the limit oi P. \n\n\n\nPRINCIPLES OF LIMITS. \n\n\n\n41 \n\n\n\nThat is, \n\n\n\nlimit \n\n\n\nls/p-\\ = I. \n\n\n\nAs ;2B-^oo, the equal arcs of s approach zero; hence the \nsame effect and result may be caused by retaining any fixed \nvalue for 7i and making ^:^^o. Therefore \n\n\n\nlimit \n\n\n\nVs/P^ \n\n\n\n\\in \n\n\n\nlimit \n\n\n\nSur. gen by an arc \n\n\n\n\'^^^ LSur. gen. by its ch \n\n\n\nre ~j \n\n\n\nThe results determined in \xc2\xa7\xc2\xa7 51, 52 and 53, with the \nprinciple \xc2\xa7 44, are of great value in finding the limits in the \nfollowing exceptional cases. \n\n54. \\.^\\MM\'~s be any arc of any plane curve, T^il/ \nand F\'M\' the ordinates of its extremities. \n\n\n\n\n\n\n\n\n\n1 \n\n\nr \n\n\n\n\nY \n\n\n,>C \n\n\nL\' \n\n\n\n\n\n\nM \n\n\n^ \n\n\n^^=^^ \n\n\n., \n\n\n\n\n\n\nY \n\n\n\n\n\n\n\n\n^^/ \n\n\nV \n\n\nV \n\n\n\n\n\'n\' \n\n\n\n\ny^ \n\n\n\n\np \n\n\nAx \n\n\np\' \n\n\nX \n\n\n\nH \n\nThrough J/" draw the chord MM\' = c, the tangent MT \n= 3, and MQ\'= FP\'\xe2\x80\x94 Ax, parallel to X. \n\n1-1 .^.^/^ 1 ^ s\'mMM\'T \n\nFrom the triangle MM\' T, we have \xe2\x80\x94 = -: \xe2\x80\x94 TT-r^TTi\' \n\n^ \' ^ sm M7M\' \n\nAs Ajic approaches zero, the arc s and the angle M\'MT \n\nvanish, but the angle T remains constant. Hence, the \n\nangle J/J/T approaches [180"\xe2\x80\x94 T], and \n\n\'b \n\n\n\nlimit \n\n\n\n^ limit fsin^M^I ^ sin_( \nj\xc2\xab^o |_ sin T J ! \n\n\n\n80\xc2\xb0- r) \n-\xe2\x96\xa0 \xe2\x80\x94 ^= \xe2\x80\x94 ^ \xe2\x80\x94 I. \n\nsm 1 \n\n\n\n42 \n\n\n\nD IFFEREN 1 \'I A L CALC UL US. \n\n\n\nHence, since (\xc2\xa7 52) \n\nlr.\':^A] = i, we have (\xc2\xa744) iH [V^] - l \nTherefore (\xc2\xa7 44) \n\nb ,. FaVccs Q\'MT\'^ I \n\n\n\niri-ii=\xc2\xab-^=i.\xc2\xbb.L- \n\n\n\n] \n\n\n\nt^x J COS Q\'MT \n\n55* Let r ^ f{v) be the polar equation of any plane \ncurve, as AMM\\ referred to the ri^ht line PD, and \npole P. \n\n\n\n\nLet AM = ^ be any portion of the curve, and PM = r \nthe radius vector corresponding to M. \n\nRegarding ^ as a function of v (\xc2\xa7 18) let v be increased \nby MPM\'\xe2\x80\x94ixv. The arc MM\' will be the correspond- \ning increment of s. Draw MQ\' perpendicular to PM\\ \nand denote PM\' by r\' . Then (\xc2\xa7\xc2\xa7 44, 52) we have \n\n\n\nlimit arc MM \' _ ch. M M\' \n\n\n\nAv \n\n\n\nAv \n\n\n\nlim \n\n\n\n/mQ\'" 4- Q\'jW \n\n\n\n_ . i /(^ sin A e\')"^ +(/\xe2\x80\x94;- cos AvY \n\n\n\nPRINCIPLES OF LIMITS. \n\n\n\n43 \n\n\n\nHence (\xc2\xa7 51) \n\n\n\n\n\n\nWe also have \n\n\n\ntan FMD = }\'\'^\'^ tan Q\'M\'M=\\im ^-^, \n\n\n\nt\\Vl \n\n\n\n= lim -7 = lim \xe2\x80\x94. \n\nr \xe2\x80\x94 r cos Av r \xe2\x80\x94 t \n\n\n\nHence, \n\n\n\nlimit r - r \n\n\n\nwhich, substituted in (i), gives \n\n\n\nlimit arc M M\' ^ ^ ^, _^ \n\n\n\nAz\'B^O AZ^ \n\n\n\ntan\' FMB \n\n\n\nIf the radius vector PM coincides with the normal to \nthe curve at M, we have \n\ntan FMI? = 00 , and }^\'^^ [arc MM\'/ Av] = r. \n\n56. Let any plane figure, as j \n\nPMM\'P\\ included between any \nplane arc, as MM\'\\ the ordi- ^ \nnates of its extremities and the \naxis of X, be revolved about X. \nThen (\xc2\xa7\xc2\xa7 44, 53) \nlimit Sur. gen. by arc MM\' / \n\n= lim ^^""^ ^^\'^\' ^^ \'\'^\' ^^\' ^ lim "^^-^ ^^\'^\'^ \nAx Ax \n\n__ \xe2\x80\xa2 \'^{y + y ) ^ Vcos QMM* _ 2 ny \n\n"" A^^ cos (2\'JO^* \n\n\n\n\n44 DIFFERENTIAL CALCULUS. \n\n\n\nCHAPTER III. \n\nRATE OF CHANGE OF A FUNCTION. \n\n57* A function changes uniformly with respect to a vari- \nable when from each state all increments of the variable \nare directly proportional to the corresponding increments \nof the function. \n\nIt follows that from all states equal increments of the \nfunction correspond to equal increments of the variable; \nalso that the ratio of any increment of the function to the \ncorresponding increment of the variable is constant. \n\nThus, having 2ax^ increase x by any amount denoted by \nh, then 2a{x -\\- K) \xe2\x80\x94 2ax \xe2\x80\x94 2ah will be the corresponding \nincrement of 2ax, It varies directly with h, it is the same \nfor all states of the function, and 2ah/h = 2^ is a constant. \nHence, 2ax changes uniformly with respect to x. \n\n\'Let fx be any uniformly varying function, and /i any in- \ncrement of X. /{x -\\- /i) \xe2\x80\x94 fx will be the corresponding \nincrement of the function, and \n\n[/(^ + ^) \xe2\x80\x94 fx\\/h = constant = A. \n\nHence /(x -\\-A)=A/i-{- f{x), \n\nin which .r = o, gives f{h) = Ah -\\- /(o). \n\nTherefore, fx = Ax +/(o). \n\n\n\nRATE OF CHANGE OF A FUNCTION. \n\n\n\n45 \n\n\n\nHence, all functions which change uniformly with respect \nto a variable are algebraic, and of the first degree with \nrespect to that variable, and all algebraic functions of the \nfirst degree with respect to a variable change uniformly \nwith that variable. \n\nThe graphs of such functions are right lines, and any \nfunction whose graph is a right line changes uniformly \nwith respect to the variable. \n\nTo illustrate, the right line AD is the graph of a function. \nConsider any state, as that represented by the ordinate PA, \n\n\n\n\n\n\n\nc \n\n\nc \n\n\n\n\nB \n\n\n/ \n\n\nQ" \ns" \n\nP" \n\n\nA \n\n\ny^ \n\n\nQ\' \n\n\n\n\n^ \n\n\nP \n\n\np\' \n\n\n?\'" X \n\n\n\nIncrease the corresponding value of the variable, repre- \nsented by OP, by any increments, as PP\' and PP" . Q\' B \nand S"C will represent the corresponding increments of \nthe function, and the similar triangles AQ\' B and AS"C \ngive \n\nAQ; : AS\'\' :: Q\'B : S"C. \n\n\n\nThat is, the corresponding increments of the variable and \nfunction are proportional. \n\nBy giving to jc = OP any equal increments, as PP\' , \nP\' P" ,P" P"\\ in succession, the corresponding increments \n\n\n\n4^ DIFFERENTIAL CALCULUS. \n\nof the function, Q\' B, Q" C, and Q\'\'D, are equal to each \nother. \n\nIt is also evident that the ratio of any increment of the \nfunction to the corresponding increment of the variable, as \nQ\'BjPP\', or S"C/FF\'\\ or \'q"\'DIP"F\'\'\\ is constant. \n\n58. Having the function 2.v, \xe2\x80\x94 \xe2\x96\xa0 \n\n\n\n"=\'<x \n\n\ngives \n\n\n2X =^ 2. , \n<2 \n\n\nX^ 2 \n\n\n" \n\n\n2X \xe2\x80\x94 4. \n\n\n<I \n\n\n\n\n2X = t.<^ \n\n\n^==3 \n\n\n\n\n\netc. etc. \n\nHence, the function 2x increases two units v/hile the \nvariable increases one ; in other words, twice as fast. \nHaving the function 5^, \xe2\x80\x94 \n\n\n\nx \n\n\n= \'<! \n\n\ngives \n\n\n5^: \n\n\n=-- 5- \n\n\n<5 \n\n\nX \n\n\n= 2 \n\n\n\n\n5^= \n\n\n= 10. \n\n\n\n\n\n\n<I \n\n\n<< \n\n\n\n\n\n\n<5 \n\n\nX \n\n\n= 3 \netc. \n\n\n\n\n5^= \n\n\n= 15- \netc. \n\n\n\n\n\nTherefore, the function ^x changes fives times as fast as \nthe variable. \n\nIn general, different functions change, with respect to \ntheir variables, with different degrees of rapidity. \n\nThe measure of the relative degree of rapidity of change of \na function and its variable at any state is called the rate of \nchange of the function^ with respect to the variable^ correspond- \ning to the state. \n\nA rate of change of a function with respect to a variable, \ncorresponding to a state, is an answer to the question: At \nthe state considered, how many times as fast as the variable \nis the function changing ? \n\n\n\nRATE OF CHANGE OF A FUNCTION. 47 \n\n59. Since, from all states, any uniformly varying function \nreceives equal increments for equal increments of the \nvariable, its rate, from state to state, is constant, and equal to \nthe ratio of any incre77ient of the function to the correspondiiig \nincrement of the variable. \n\n2(X -\\- H) \xe2\x80\x94 2X \n\nThus, rate of 2x \xe2\x80\x94 = 2. \n\nK,,e of 3. + 3 --= M^\xc2\xb1^^\xc2\xb1|l^Il^\xc2\xb1^ = 3. \nRate of 5^ - 3 = ^^^" + ^^ ~ ""} ~ ^^" " \'^ - 5- \n\n\n\nIt follows that the rate of any uniformly varying func- \ntion is equal to the tangent of the angle which its graph \nmakes with the axis of the variable, also that the product \nof the rate by any increment of the variable is equal to the \ncorresponding increment of the function. \n\n60. It follows from \xc2\xa7 57 that algebraic functions not of \nthe first degree with respect to the variable, and all \ntranscendental functions, are, with respect to the variable, \nnon-uniformly varying functions. Thus, having ay^ , in- \ncrease X by any amount as //. 2axh -j- ah^ , which varies \nwith X, is the corresponding increment of ax^. Also the \nratio {2axh -{- ah\'^)/h = 2ax -\\- ah =^ (p{x, h). Hence, ax^ \ndoes not change uniformly with respect to x. \n\nHaving /(jv"), in which n is not equal to i, then [f{x -f- /?)" \n\xe2\x80\x94 f{x^)yh = (p{x, h), and/(:v:") is therefore a non-uni- \nformly varying function with respect to x. \n\nThe graphs of such functions are curves, and any func- \ntion whose graph is a curve does not change uniformly \nwith respect to the variable. \n\n\n\n48 \n\n\n\nDIFFERENTIAL CALCULUS, \n\n\n\nTo illustrate, the curve MNT is the graph of a func- \ntion. Increase, in succession, \nany value of x, as 0P\\ by any \namounts, as F\' P" and F\' R. \nF\'F"IF\'R = QB/ST is the \nratio of the increments of the \nvariable, and it differs from \nQN/ST, which is the ratio of \nthe corresponding increments of \nthe function. Hence (\xc2\xa7 57), the ordinate of MNT, and \nthe function represented by it, do not change uniformly \n\n\n\n\n\n\n\n\n\n\n\n\\ \n\n\n/ \n\n\nY \n\n\n\n\n/^ \n\n\n\n\n\n\n/ \n\n\n/y \n\n\ni \n\n\n\n\n\n\n\nU \n\n\n^ \n\n\n^ \n\n\nQ \n\n\ns \n\n\n\'^ \n\n\n?\' \n\n\np" \n\n\nR X \n\n\n\nwith X. \n\n\n\n\n\n\n61. In the function \n\n\n:x\'- \n\n\n\n\nX = "i \n\xe2\x96\xa0y : \xe2\x80\x94 A \n\n\ngives \n\nit \n\na \nli \n\n\n2X\'\'= 2. \n2 <I4 \n\n\n\n2X =32. \n\n\n\nWhich shows that at different states the function 2^* has \ndifferent rates wHh respect to x. \n\nA non-uniformly varying function, in general, receives \nUnequal increments for equal increments of the variable. \nIt follows that its rate varies from state to state. \n\nAny particular rate is, therefore, designated as the rate \ncorresponding to a particular state. \n\nIf a function has two or more states corresponding to any \nvalue of the variable, each state will have a rate. \n\nIf a function has equal states for different values of the \nvariable, it may have a different rate at each ; in which \ncase it is necessary to indicate the value of the variable \ncorresponding to the state considered. \n\n\n\nRATE OF CHANGE OF A FUNCTION. \n\n\n\n49 \n\n\n\n62. Rate of Change of any Function. Let fx be any \nfunction. Denote by R its rate, \nwith respect to x, corresponding \nto any state, as PA. Increase \nX = OP \\yy h = PP\', and let \nP\' represent the rate of the func- \ntion at the new state f{x -f- ^) = \nP\'B. QB = f{x ^K)-fx will \nrepresent the corresponding increment of the function. \n\n\\i fx changes uniformly, \n\n\n\n\nAx^h)-fx \n\n\n\n^, = tan QAP \n\n\n\nwill (\xc2\xa7 59) be its constant rate, P = P\', and its graph will \nbe the right line AP (\xc2\xa7 57). \n\nIf fx is a non-uniformly varying function, its graph is a \ncurve, as AEB^ and in general P 4^ P\' . \n\nIn this case let PP\' = /^ be taken so small that while x \nvaries from OP to OP\' ^ the rate oi fx will either decrease \nor increase in order from P to P\\ \n\nIn either case, [/(-^ + h) \xe2\x80\x94 foc\\/h, which is the con- \nstant rate of the uniformly varying function whose graph \nis the right line AB^ will be between P and P\' \\ for other- \nwise the uniformly varying function would change between \nthe states considered by an amount greater or less than \nQB. \n\nIn other words, \\_f{x -^ K) \xe2\x80\x94 foc\\/h is the rate oi fx at \nsome state, as P" E, between PA and P\'B. Let PP"= Oh, \nin which B is the proper fraction PP" /PP\', \n\n\n\nThen \n\n\n\nP"E = fix + Oh), \n\n\n\n50 DIFFERENTIAL CALCULUS. \n\nand y(x + Ji) - fx\\/h = rate oi /{x + dA). . (i) \nThis relation will always exist as h\'m-^o. \n\n\n\nHence, "^"^\'^ \n\n\n\n- f{x + ^) - /r l \n7 = rate oi fx. (2) \n\n\n\nThat is, //^^ rate of change of any ftmctioji with respect to a \nvariable^ corresponding to any state ^ is equal to the limit of the \nratio of any increment of the function^ from the state consid- \nered^ to the- corresponding increment of the variable^ under the \nlaw that the increment of the variable approaches zero. \n\nThe above principle enables us to find the rate of any \nfunction with respect to a variable, corresponding to any \nstate, by the following general rule : \n\nGive to the variable any variable increment^ and from the \ncorresponding state of the functio7i subtract the primitive. \nDivide the retnainder by the increment of the variable, and \ndetermi?ie the limit of this ratio, under the law that the incre- \nment of the variable approaches zero. In the result substitute \nthe value of the variable corresponding to the state. \n\nIt should be observed that a rate, determined by the \nabove method, is equal to a limit of a ratio of two infinitesi- \nmals, which limit is determinate ; and that it is not equal to \nthe ratio of their limits. \xc2\xa7 39. \n\nTo illustrate, let the curve AB" B\' in the figure on page \n51 be the graph oi fx,d,Xi^ let PA be the state at which the \nrate is required. \n\nWhen h = FP\\ \n\nf(x-^rh)-fx _ Q\'B\' _ \n\n^ \xe2\x80\x94 -ppT \xe2\x80\x94 tan Q AB , \n\nwhich is the rate of the function whose graph is the secant \nAB\\ \n\n\n\nRATE OF CHANGE OF A FUNCTION, \nWhen h = FP", \n\n\n\n51 \n\n\n\npp, \n\n\n\ntan Q"AB", \n\n\n\nwhich is the rate of the function whose graph is the secant \nAB\'\\ \n\nAs /^\xc2\xab\xc2\xbb-\xc2\xbbo, the above ratio is always the rate of a function \nwhose graph is a secant approaching the tangent ^T as a \n\n\n\n\nlimit. Hence, the limit of the ratio is the rate of the func- \ntion whose graph is the tangent ^7", and is equal to the \ntangent of the angle Q\'AT. \n\nThat is, the limit of the ratio of any increment of any \nfunctio7i of a single variable to the corresponding increment \nof the variable^ tinder the law that the increment of the vari- \nable approaches zero, is equal to the tangent of the angle made \nwith the axis of abscissas by a tangeiit at the corresponding \npoint of the graph of the function. \n\nThe same result was obtained in \xc2\xa7 39, The angle which \nthe tangent AT makes with X is called the inclination, and \nthe numerical value of its tangent is called the slope of the \ngraph at M. \n\nThe rate of any function at any state is therefore the \nsame as that of a uniformly varying function whose graph \n\n\n\n52 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nis the tangent to tjie graph of the given function at the \npoint corresponding to the state considered. \n\nThis agrees with previous conceptions and definitions, \nfor the direction of the motion of the point generating the \ncurve at any position is along the tangent at the point, and \nthe ordinate of the curve representing the state considered \nis changing at the same rate as the ordinate of the corre- \nsponding tangent. \n\nIt follows that the product of the rate of a non-uniformly \nvarying function by an increment of the variable is not, in \ngeneral, equal to the corresponding increment of the func- \ntion. \n\n63. Let jj/ \xe2\x80\x94 PA represent any state of any increasing func- \ntion of x\\ andy the new state corresponding to an incre- \n\n\n\n\nment PP\' = h, of the variable, {y* \xe2\x80\x94 y)/h will be the ratio \nof the increment of the function to that of the variable. If \nh is assumed sufficiently small, this ratio will be positive and \nremain so as /^^-\xc2\xbbo. \xc2\xa7 15. \n\n\n\nHence (\xc2\xa733), \n\n\n\nlimit \n\n\n\n[(y ~ y)l^ \xe2\x80\x94 tan XEA is positive. \n\n\n\nLet y = Pi^i represent any state of a decreasing function ; \nand y\' its new state due to an increment of the variable \nequal to P,P/ = h. Then (/ \xe2\x80\x94 y)lh will be negative, if h \nis small enough, and will remain so as /^ ^^-\xc2\xbb o. \n\n\n\nHence, \n\n\n\nlimit \n\nhWHfO \n\n\n\n\\ky\' ~ y)/^^] \xe2\x80\x94 ta^ XE^A is negative. \n\n\n\nJ^A TE OF CHANGE OF A FUNCTION. \n\n\n\n53 \n\n\n\nTherefore, the rate corresponding to any state of aji mcr eas- \ning fu nctio7t is positive, and of a decreasing function is 7iega- \ntive. \n\nIt follows that a function is an increasing one when its \nrate is positive, a?id a decreasing one when its rate is negative. \n\nWhen, for any state of a function, as the one represented \nby PA or P\' A\\ the function is neither increasing nor de- \n\n\n\ncreasing, its rate is zero, and the tangent at the correspond- \ning point of its graph is parallel to the axis of x. \n\nIf for any state of a function, as the one represented by \nP"A", the rate is unlimited, the corresponding tangent to \nits graph is perpendicular to the axis of x. \n\nEXERCISES. \n\nFind the rate of change of each of the following functions: \n\n\n\nI. 2ax. \n\n\n\n2. X\\ \n\n\n\n3. ax^ -f- bx. \n\n\n\nlimit Y 2a{x - ^ h) \xe2\x80\x94 2ax \nh \n\n\n\nA\xc2\xb0^-i^;L H ""\' Y"\'- \n\n\n\nAns limit r(^^_)lz-_f.n=^, \nh-M-^o\\ h J \n\n(a a\\ \n\n\xe2\x80\x94 TT ~ \\ ^ \n\n\n\nf5. lax\\ \n\n\n\nAns \n\n\n\nAns i\\ax. \n\n\n\n6. x^. \n\n\n\nAns. "ix"^. \n\n\n\n54 DIFFERENTIAL CALCULUS. \n\n\n\n7. ^x^. Ans. \\t>o<;\'^. 8. \xe2\x80\x94 \xe2\x96\xa0 \xe2\x80\x94 Ans. \xe2\x80\x94 \n\n\n\ni-\\-x \xe2\x96\xa0 {i-\\-xf \n\n\n\nAns. \n\n\n\n3+^ (3 -f-^y \n\n\n\n10. How is the ordinate of a parabola, corresponding to \n\n^ = 3, changing with respect to the abscissa ? \n\n\n\nRate of \n\n\n\n_ . limit yV^Pi^-^ ^) - V^px~^ , ,\xe2\x80\x94 [~(x-^/i)^ - x^~] \n\ny-^ /,^o [_ \\ J -\xc2\xb1|/2/z[_ ^ J \n\n11. Same corresponding to focus? Ans. i. \n\n12. Find the abscissa of the point, of the parabola y = \n4x, where the ordinate is changing twice as fast as the \nabscissa. Ans. x = 1/4. \n\n13. At the vertex of a parabola, how is the ordinate \nchanging as compared with the abscissa ? \n\n14. Find the rate of change of the abscissa of a parabola \n\n\n\nwith respect to the ordinate ? j\\ns. y/p = \xc2\xb1 Vix/p. \n\n15. Find the coordinates of the point of the parabola \ny = 8x, where the abscissa is changing twice as fast as the \no\'-dinate. Ans. (8, 8). \n\n16. Find the rate of change of the ordinate of the right \n;ine 2y \xe2\x80\x94 -^x ^= 12, with respect to the abscissa. Ans\xc2\xbb 3/2 \n\n17. A point moves from the origin so that y always in \ncreases 5/4 times as fast as x\\ find the equation of the line \ngenerated. \n\n5/4 = tan of angle line makes with X. .\' . Ans. ^,y \xe2\x80\x94 5x. \n\n\n\nRATE OF CHANGE OF A FUNCTION. S5 \n\n\n\ni8. Find the slope of the graph of \xc2\xb1 r i2jc when x = \n1/2. Ans. \xc2\xb12.4495. \n\n19. Find the abscissa of the point of the graph of V 2px \nwhen the slope is i. Ans. x = p/2. \n\n20. Find the angles which the lines y^ =\xe2\x96\xa0 2>x, and 3jj^ \xe2\x80\x94 \n2X = 8, make with each other at their intersections. \n\nAns. 11\xc2\xb0 18\' 35", and 7\xc2\xb0 7\' 30". \n\n21. Find the angles which the lines y^ = /^x, and 2y = \nX -]- 2, make with each other at their intersections. \n\nAns. 10" 14\', and t,7,\xc2\xb0 4\'. \n\n64. A function of two or more variables is a uniformly \n.varying function with respect to all of its variables when \nit changes uniformly with respect to each. It follows \n(\xc2\xa7 57) t^"^^t ^11 uniformly varying functions are algebraic, \nand of the first degree with respect to each variable, and all \nalgebraic functions of the first degree with respect to each \nvariable are uniformly varying functions. \n\nLet u = Ax -\\- By -\\- Cz -{- etc., in which A, B, C, etc., \nare constants, be any uniformly varying function. Increase \nthe variables x, y, z, etc., respectively, by any increments, as \nAf k, /, etc., giving a new state, \n\nu\' = A{x + /^) + B{y-{-k) + C(z-i-/) + etc. \n\nu^ \xe2\x80\x94 u = Ah + Bk -\\- CI -\\r etc., is the corresponding in- \ncrement of the function. It is independent of the state of \nthe function, dependent upon the increments of the variables, \nand is equal to the sum of the increments due to the \nincrease of each variable separately. \n\n65. A uniformly varying function of two variables is \nsome particular case of the general expression Ax + By + \nC^ in which A^ B and C are constants. Its graphic surface \n\n\n\n56 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nis a plane (\xc2\xa7 27), and any function of two variables whose \ngraphic surface is a plane is a uniformly varying function. \nTo illustrate, take any ordinate, as NM^ of any plane, as \nMLH. Through NM pass the planes MNR and MNP \nparallel, respectively, to ZX and ZF, intersecting the given \nplane in the lines J/ZTand ML. Assume iVi? as the incre- \nment of Xy and NF as the increment of y. Complete the \n\n\n\n\nparallelogram NS, the parallelopipedon NQ, and in the \ngiven plane the parallelogram MLTH. Produce RK to ZT, \nSQ to T, and PA to L. KH is the increment of MN due \nto the increment, NR, of x alone, and AL is its increment \ndue solely to NP^ the increment of ji^. ^T^is the entire \nincrement of MN due to the increase of both variables \ntogether. \n\nDraw HD parallel to KQ, and draw AD\\ it will be par- \nallel to LT, because it is parallel to J/ iT, which is parallel \ntoZr. Hence, i?r= ^Z. \n\n\n\nFA TE OF CHANGE OF A FUNCTION. 57 \n\nTherefore, QT=QD-\\-nT= KH ^ AL. \n\nThat is, the total increment of any uniformly varying \nfunction of two variables from any state is equal to the sum \nof the increments from that state due to the increment of \neach variable separately. \n\nIt is important to notice that, while a uniformly varying \nfunction of two variables has a constant rate with respect \nto each variable alone, it has no fixed total rate with respect \nto both variables changing simultaneously. In the case \nillustrated, \n\nOT increment of J/iV^ ^,^_ \n\n^\xe2\x80\x94 = \xe2\x80\x94j=^=^ = tan QMT. \n\nis the corresponding total rate of MJV, but in general any \nchange in the relative value of JVJ? and J^S will cause a \nchange in the total rate. Thus as the ratio NR/RS \nchanges through all possible values, 0, the angle which the \nvertical plane MNS makes with the plane ZX changes, \nand the right line MT, cut from the given plane by the \nplane MNS revolving about MN as an axis, will in succes- \nsion coincide with all right lines in the given plane which \npass through M. Hence, depending upon the ratio of in- \ncrements of the variables, the total rate of any uniformly \nvarying function of two variables with respect to both \nvariables changing simultaneously, may have any value \nfrom zero to the tangent of the angle made by the graphic \nplane of the function with XK, the numerical value of \nwhich is called the slope of the plane. \n\nAll functions of two variables not of the first degree with \nrespect to each variable do not vary uniformly with respect \nto both variables, and their graphic surfaces are curved. \n\n\n\nSB DIFFERENTIAL CALCULUS, \n\n66. The Calculus is that branch of mathematics in which \nmeasurements, relations, and properties of functions and \ntheir states are determined from their rates of change. It \nis generally separated into two parts, called, respectively, \nDifferential and Integral Calculus. \n\nDifferential Calculus embraces the deductions and ap- \nplications of the rates of functions. \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nPART I. \n\nDIFFERENTIALS AND DIFFERENTIATION. \n\n\n\nCHAPTER IV. \nTHE DIFFERENTIAL AND DIFFERENTIAL COEFFICIENT. \n\nFUNCTIONS OF A SINGLE VARIABLE. \n\n67. An arbitrary amount of change assumed for the in- \ndependent variable is called the differential of the variable. \n\nIt is represented by writing the letter d before the symbol \nfor the variable ; thus dx^ read " differential of ^," denotes \nthe differential of x. \n\nIt is always assumed as positive, and remains constant \nthroughout the same discussion unless otherwise stated. \n\n68. The differential of a function of a single variable is \nthe change that the function would undergo from any state^ \nwere it to retain its rate at that state, while the variable \nchanged by its differential. \n\nThe differential of a function is denoted by writing the \nletter d before the function or its symbol. \n\n59 \n\n\n\n6o DIFFERENTIAL CALCULUS. \n\nThus, d2ax^^ read " differential of 2^;\\;V\' ii^dicates the \ndifferential of the function 2ax^. \n\nHaving J* = log ^ ax^^ we write dy =^ d log Vax^. \n\n~~dx denotes the differential of j; regarded as a function \ndx \n\nof X ; and \xe2\x80\x94dy is a symbol for the differential of the in- \n\ndy \n\nverse function ; that is, of x regarded as a function of y. \n\nThe differential of a function which varies uniformly \nwith its variable is equal to the change in the function cor- \nresponding to that assumed for the \nvariable, because its rate is con- \nstant. \n\nThus, let PA be any state of the \nuniformly varying function whose \ngraph is the right line AB. As- \nsume J^J^ =dx. Then QB = dy, the corresponding change \nin the function, is the differential of the function. \n\nThe differential of a function which does not vary uni- \nformly with its variable is not, in general, equal to the cor- \nresponding change in the function, because its rate varies ; \nbut it is equal to the corresponding change of a function \nhaving a constant rate equal to that of the given function \nat the state considered ; or, in other words, it is the change \nthat the function would undergo were it to continue to \nchange from any state, as it is changing at that state, uni- \nformly with a change in the variable equal to its differen- \ntial. \n\nThus, let PA be any state of a given function whose \ngraph is the curve AM. Assume PR = dx. \n\nQM is the corresponding change in the function ; but \nQB, the corresponding change in the function represented \n\n\n\n\nDIFFERENTIAL\xe2\x80\x94 Differential coefficient. 6i \n\n\n\n\nby the ordinate of the right line AB drawn tangent to \nAM at A, is the differential of \nthe given function correspond- \ning to the state PA. The func- \ntion whose graph is AB has a \nconstant rate equal to that of the \ngiven function at PA., and QB is \nthe change that the given func- \ntion would undergo were it to \ncontinue to change from the state PA., as it is changing at \nthat state., uniformly with a change in x equal to dx. \n\nThe differential of a function which does not vary uni- \nformly with its variable may \nbe less than the correspond- \ning change in the function. \nThus, QB, < QM, is the dif- \nferential of the function rep- \n\n^^ Q resented by the ordinate of \n\nthe curve AM, correspond- \n\xe2\x80\x94 ^ ing to PA. \nA train of cars in motion affords a familiar example of a differen- \ntial of a function. \n\nA a; B CD E \n\n> \n\nSuppose that a train of cars starts from the station A, and moves \nin the direction AE with a continuouslv increasing speed. Let x de- \nnote the variable distance of the train from A at any instant ; it will \nbe a function of the time, represented by t, during which the train has \nmoved, giving x \xe2\x80\x94 f{i). \n\nSuppose the train to have arrived at B, for which point x = AB. \nLet BD represent the distance that the train will actually run in the \nnext unit of time, say one second, with its rate constantly increasing. \n\nLet BC represent the distance that the train would run it it were \nto move from B with its rate at that point unchanged, in a second. \n\n\n\n\n62 DIFFERENTIAL CALCULUS. \n\nThen will the distance ^C represent the differential of x regarded as \na function of /, corresponding to the state x \xe2\x80\x94 AB ; and one second \nwill be the differential of the variable. \n\n69. From the definition of a differential of a function, \nand from \xc2\xa7 59, it follows that a differential of a function is \nthe product of two factors, one of which is the i-ate of \nchange of the function at the state considered, and the other \nis the assumed differential of the variable. Hence, the dif- \nferential of any given function may be determined by find- \ning its rate, by the general rule, \xc2\xa762, and multiplying it by \nthe differential of the variable. Thus, having the function \n2x\'^, we find \n\n\n\nlimit \n\n\n\nz(x^h) \n\n\n\nhy - 2x\'~\\ \n\n\\:=^x=^ rate corresp. to any state. \n\n^xdx is, therefore, a general expression for the differen- \ntial of 2jr^, and is written d 2x^ = ^xdx. \n\nIts value corresponding to any particular state is obtained \nby substituting the value of the variable corresponding to \nthe state ; thus, for :r = 2, we have {d 2J\\:^)^=2 \xe2\x80\x94 ^dx. \n\n70. Since, in the expression for the differential of a func- \ntion, the rate of change of the function is the coefficient of \nthe differential of the variable, it is, in general, called the \n" differential coefficient^\'\' and may be determined by the \ngeneral rule, \xc2\xa7 62. \n\nThe differential of a function is therefore equal to the \nproduct of the differential coefficient by the differential of \nthe variable. \n\nIt follows that the differential coefficient is the quotient \nof the differential of the function by the differential of the \nvariable. Thus, having d 2x^ = /^xdx., 4X is the diff;. rential \ncoefficient. In general, having y^=f{x), and representing \n\n\n\nDIFFERENTIAL\xe2\x80\x94 DIFFERENTIAL COEFFICIENT, (^l \n\nits differential by dy or \xe2\x80\x94dx^ its differential coefficient is \n\ndy/dx. \n\nThe expressions {dy/dx)^^\'^ and dy\' /dx\' are used to de- \nnote the particular value of dy/dx corresponding to x = x\' \nand 7 = j\'. \n\nThus, having jF = 2^^, then dy/dx = 4X, and (<^/^x)(2j = 8. \n\nHaving _y ==/(-^), in which jv is any function of any vari- \nable X., let y\' denote the new state of the function corre- \nsponding to any increment of the variable, as h or /\\x^ and \nlet A J = _y\' \xe2\x80\x94 jj^ represent the corresponding increment of \ny. Then (\xc2\xa7 62) \n\nlimit /(^ + ^) - /(\xe2\x80\xa2^ ) ^ j-^^/jZJ\' 3^ limit Aj ^ 4^ \n/^;^^o /; h Ax:^^o/xx dx\' \n\nSince the increment of the variable, represented by h or \nA^, varies, it may happen that h = A^ = dx. It is exceed- \ningly important to observe, however, that the correspond- \ning value oi y\' \xe2\x80\x94 y or l\\y is 7iot, in general, equal to dy ; \nfor that would give \n\n\\ h Ih^dx \\^xl ^:c=.dx dx\' \n\nwhich, in general, is impossible, since dy/dx is not a value \nof the ratio {y\' \xe2\x80\x94 y)/h, but is its limit under the law that \nh vanishes. \n\nIf, however, the function changes uniformly with respect \nto the variable, {y\' \xe2\x80\x94 y)/h will be constant for all values of \n^^ (\xc2\xa759)) and^\' \xe2\x80\x94y will be equal to dy when h is equal to \ndx. \n\n71* The following are important facts in regard to a dif- \nferential coefficient : \n\n\n\n64 \n\n\n\nD IFFEREN TIA L CAL CUL US. \n\n\n\nIt is zero for a constant quantity. In other words, a \nconstant has no differential coefficient. \n\nIt is constant for any function which varies uniformly. \n\nIt varies from state to state for any function which does \nnot vary uniformly. \n\nIn general, therefore, it is a function of the variable, and \nhas a differential. \n\nIt is positive for an increasing function, and negative for \na decreasing one. \xc2\xa7 d^t\' \n\nIt may have values from \xe2\x80\x94 oo to + <^\' \n\nHaving represented a function by the ordinate of a curve, \nthe differential coefficient is equal to the tangent of the \nangle made with the axis of abscissas, by a tangent to the \ncurve at the point corresponding to the state considered. \n\xc2\xa762. \n\nThus, assuming PR = dx, the differential coefficient of \nthe function whose graph is the curve AM, at the state \nPA, is \n\ndy/dx = tan X\xc2\xa3A \xe2\x80\x94 tan QAB. \n\nIt should be noticed that dy/dx is independent of the \n\n\n\n\n\n\n\n\n\n\n\nMi \n\n\n\n\n\n\n\n\niY \n\n\nA \n\n\n^ \n\n\n^ \n\n\nI \ndy \n\n\n\n\n\n\n\n\n\n\nQ \n\n\nE^ \n\n\n^^^ \n\n\n\n\n\nP \n\n\ndx \n\n\nR \n\n\nr\' X \n\n\n\nvalue assumed for the differential of the variable ; for if \nPP\' = dx, then Q\'D = dy, and we have, as before, dy/dx = \ntan XEA. \n\nIn this illustration the function is an increasing one, its \n\n\n\n\nDIFFERENTIAL\xe2\x80\x94 D IFFERENTIAL COEFFICIENT. 05 \n\ndifferential coefficient is positive, and the angle XEA is \nacute. \xc2\xa7 63. \n\nIn case the function represented by the ordinate of \nAM is a decreasing one, its \ndifferential coefficient corre- \nsponding to PA is negative, \nand the angle XEA is then \nobtuse. \xc2\xa7 d^\' \n\nIf for any value of the \nvariable the differential coefficient is zero, the function is \nneither increasing nor decreasing, and the tangents at the \n\ncorresponding points of the \ngraph of the function are \nparallel to the axis of x. \n\nIf the differential coeffi- \ncient is infinite, the rate of \nthe function is infinite ; and \nthe tangents at the corresponding points of the graph of \nthe function are perpendicular to the axis of x. \xc2\xa7 63. \n\nIf for a finite value of the variable the state of a function \nis unlimited, its corresponding differential coefficient will \nalso be unlimited. Thus, having ji; =/(^) and/(^) = co, \n\nlimit \n\n\n\n\' \n\n\nA \n\n\n\n\nA" \n\n\n\n\n^ y \n\n\n\n\n^^/ \n\n\ny \n\n\n\n\np y \n\n\nP\' \n\n\nP" X \n\n\n\n(^yUx). - \'^Zl [(/{a + h) -Aa))/h\\ \n\n\n\nHence, \n\n\n\n(dy/dx). + e = [/(a + h) -f{d)\\/h. \n\n\n\nin which e vanishes with h. f{a -\\-}i) is not, in general, un- \nlimited, therefore {dy/dx)a = 00 \xe2\x80\x94 ^ = 00 . \n\nThe principle is not necessarily true for an infinite state \ncorresponding to an infinite value of the variable, for in \nthat case /(a -|- A) will also be unlimited. \n\n\n\n66 DIFFERENTIAL CALCULUS. \n\n72. The following facts concerning a differential of a \nfunction should now be apparent : \n\nIt is zero for a constant. \n\nIt is constant for any function which varies uniformly. \n\nIt is a function of the variable for any function which \ndoes not vary uniformly ; and in such cases it has a differ- \nential. \n\nIts value depends upon that of the differential coefficient \nand that assumed for the differential of the variable. \n\nIt may have values from \xe2\x80\x94 00 to -I- 00. \n\nIt will be numerically greater or less than the differential \ncoefficient depending upon whether the differential of the \nvariable is assumed greater or less than unity. \n\nIt has the same sign as its differential coefficient, and \ntherefore is positive for an increasing function and negative \nfor a decreasing one. \n\nFunctions which are equal in all their successive states \nhave their corresponding differentials equal. \n\n73 \xe2\x80\xa2 The differential coefficient of aiiy function is equal to \nthe reciprocal of the corresponding differential coefficient of its \ninverse function. \n\nLet J =/(^) . . . (i) and x = F(y) ... (2) be any direct \nand inverse functions. Let L^x and l^y represent, respect- \nively, any set of corresponding increments of x andjv in (i). \nIt follows (\xc2\xa7 4) that they will represent a set of the same in \n(2), and we have \n\nAy/ l\\x = i/{/\\x/ /\\y). \nHence (\xc2\xa7 70) \n\n^y_ ^ limit A^ = lini \xe2\x80\x94- = \xe2\x80\x941\xe2\x80\x94. \n\ndx Ax-m^o /\\x l\\x/ l\\y dx/dy \n\nTo illustrate, let the function y be represented by the \n\n\n\nDIFFERENTIAL\xe2\x80\x94 DIFFERENTIAL COEFFICIENT, 6^ \n\n\n\nordinate of the curve AM. Assume dx=^FR^ and from \nthe figure we have, corre- \nsponding to the state PA^ \n\nQB dy ^ ,^ \n\nThe inverse function will \nbe represented by the ab- \nscissa of the curve AM re- \ngarded as a function of the \nordinate; and assuming dy \n= J^Z, we have for the state XA, corresponding to A, \n\nHE/ AH = dx/dy ^ tan EAH, EAH = 90\xc2\xb0 - QAB, \nHence, \n\n\n\nL H dx ^/ \n\n\n3 \n\n\ndy \nY \n\n\nA \n\n\n>/ M \n\n\ndy \n\n\n\nX \n\n\n\n\n/ \n\n\n\nP da: R \n\n\n\ntan QAB = cot EAH = \n\n\n\ntan EAH\' \n\n\n\nor \n\n\n\ndy \ndx \n\n\n\ndx/dy \n\n\n\nIt should be observed that, in general, dy in the first \nmember of the above equation is not the same as dy in the \nsecond; for the first is the differential of ^j\' as a function, \nand the second is a differential of y as the independent \nvariable. The same remarks apply to dx^ in the two menii- \nbers, taken in reverse order. \n\nThe figure illustrates the differences referred to. \n\n74* The differential of the su?n of any finite nur?iber of func- \ntions is equal to the sum of their differentials. \n\nLet y^^v \xe2\x80\x94 s-\\-w-\\- etc., in which v, s, w, etc., are func- \ntions of any variable, as x. Increasing x hy ^x, we have \n\ny + l\\y = 27+ Az;\xe2\x80\x94 (s -\\- /\\s) -\\- w -^ Aw -\\- etc. \nWhence \n\n\n\nAy \n\n\n\n= Av \xe2\x80\x94 As -{- Aw -{- etc., \n\n\n\n68 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nand (\xc2\xa7 70) \n\ndy \n\n\n\nlimit \\ ^_ \n\n\n\nIsw , ^ ~\\ dv ds , dw , \n\n\n\ndx \n\n\n\nAs \n\n_ Ax Ax \n\nTherefore \n\nd{v \xe2\x80\x94 s + w + etc.) = dv \xe2\x80\x94 ds 4- dw -f etc. \n\nIt follows that ^/le differential of the sum of any finite \nnumber of functions and constants is equal to the differential \nof the sutn of the functions. Thus, C being constant, \n\nd[f(x) + C]= df(x). \n\nIf corresponding differentials are equal it does not follow \nthat the functions from which they were derived are equal, \n\n75* T^^ differential of the product of any number of func- \ntions is equal to the sum of the products of the differential of \neach function by all of the other functions. \n\nLet V = yz be the product of any two functions of any- \nvariable, as X, then \n\nV -{- Av = {y -\\- Ay) {z-\\- A z) =yz-\\-z . Ay-\\-y . A z-\\- Ay, A 2, \n\nwhence Av ^ z . Ay -\\- y . Az -\\r Ay. Az \\ and (\xc2\xa7 70) \n\n\n\n\xe2\x80\x94 \xe2\x80\x94 ^^^^^^^ - \xe2\x80\x94 - \xe2\x80\x94 hm z -^^ + V h A V \n\ndx A^^->o ajc LAj*:"^AJt: "^ \n\n\n\n= zdy/dx -\\- ydz/dx. \n\nTherefore, dyz = zdy + ydz. \n\nTo illustrate, let ONPM be a state of a rectangle with a \nvariable diagonal represented by \nX. Two adjacent sides, denoted \nby z and y respectively, will be \nfunctions of x^ and yz will be the \nvariable area of the rectangle. \nAssume dx^^PR^dc^di complete the rectangles PT\'^jf^s \n\n\n\ndz \nV \n\n\n^dy \n\n\n\n\nz \n\nS \n\n\n\nDIFFERENTIAL\xe2\x80\x94 DIFFERENTIAL COEFFICIENT. 69 \n\nand FS=zdy. Then, since dyz = ydz -\\- zdy^ we have \n^ (rect. OF) = rect. FT-{- rect. FS, which is the amount \nof change required by the definition of a differential. \xc2\xa7 68. \nIt follows that t^e differential of the product of a function \nand a constant is equal to the product of the cotistant and the \ndifferential of the function. Thus, C being constant, \n\ndCf(x) = Cdf(x). \n\nLet vsu be the product of any three functions of the same \nvariable. Place vs =\xe2\x96\xa0 r, giving vsu = ru. \n\nDifferentiating, we have dvsu =^ dru = rdu + udr^ in which \n\ndr = vds + sdv. Hence, by substitution. \n\ndvsu = vsdu + vuds + sudv. . . . (i) \n\nIn a similar manner the principle may be established for \nany number of functions. \n\nDividing each member of (i) by vsu^ we have \n\ndvsu _ du ds dv \n\nvsu ~ u ~^ s \' V* \n\nSimilarly, it may be shown that the differential of the \nproduct of any nuinber of functions divided by their product \nis equal to the sum of the quotients of the differential of each \nfunction by the function itself. \n\nEXAMPLES. \n\n4(a -1- x){b + x)\\ = {b-^x) d{a^ X) -f {a-\\- x) d{b + x) = (a + b-^2x)dx. \n\nd[2,{c \xe2\x80\x94 ^)J = \xe2\x80\x94 2dx. d[(a -\\- x)x] = xd(a -\\-x)-\\-ia^ x)dx. \n\nd \\{a -f- x)x\'\\ _ d{a -\\- x^ dx _ dx _, ^ \n(a -\\- x)x a -\\- X X a -{- x x \' \n\n76. The differe7itial of a quotient of two functions is equal \nto the denominator into the differential of the manerator^ \n\n\n\n70 DIFFERENTIAL CALCULUS. \n\nminus the numerator into the differential of the denominator^ \ndivided by the square of the denominator. \n\nLet y = v/s^ in which v and s are functions of any varia- \nble. Then v \xe2\x80\x94 sy, and \n\ndv = sdy + yds = (s\'^dy -|- vds)/Sj \n\nwhence dy \xe2\x80\x94 (sdv \xe2\x80\x94 vds)/sl \nC being a constant, we have \n\nd{C/s) = - Cds/s^ and d(v/C) = dv/C. \n\nEXAMPLES. \nd [jc/ii + x)] = dx/{i + x)\\ dis/x) = - 3dx/x^. \nd\\x{x + i)/(x - I)] = {pc" -2.x- \\)dx/{x - if. \nd{x/3) = dx/3. d{2x/sa) =2dx/Sa. \n\n77* ^^^ differential coefficient of y regarded as a function \nof X is equal to the product of the differential coefficient of y \nregarded as a functioji of u, by the differential coefficient of u \nregarded as a function of x. \n\nHaving jv =/( 2^), and u = 0(.^), let A^, Hu and t^y \nbe corresponding increments of x., u and y, respectively. \nThen (\xc2\xa7 4) A u is the same in both cases, and \n\nAj/A^ = ( A;v/A?^) X ( A2//A:r). Hence, \naJ.\'^o V^yl ^^~\\ = lini [aVA2/] X lim [A^A^], \nand (\xc2\xa770) dy/dx = {dy/du) X {du/dx). \n\nSimilarly, having ji^ z= f(ji)^ u = (p{x), x = i^is), we find \ndy/ds = (dy/du) X (du/dx) X (dx/ds) ; \n\nand the same form holds true whatever be the number of \nthe intermediate functions. \n\n\n\nDIFFERENTIAL\xe2\x80\x94 DIFFERENTIAL COEFFICIENT. 7 1 \n\nHaving y =f{u), and x = ^\'{u), we may write u = <p{x), \nand dy/dx^^ {dy/dii) X {du/dx), but (\xc2\xa7 73) du/dx^= i/{dx/di/). \nHence, \n\ndy/dx = {dy/du)/{dx/du). \n\nThat is, ^/z^ differential coefficient of y regarded as a func- \ntion of X is equal to the quotient of the differential coefficient \nof y regarded as a function of u, by the differential coefficient \nof X regarded as a function of u. \n\n\n\nEXAMPLES, \n\n\n\nGiven \n\ny = au^, u = bx. . \n\n\n\ny = f{^u\\ X = (p{u), X = ^{s). \ny \xe2\x80\x94 u^, X = 3u, X = 2/. . \ny = f{ii), u \xe2\x80\x94 F{s), z = 1p{s). \n\n\n\ndy/dx = 2a 6^ X. \n\ndz/dx = 2ap. \n\ndy _ dy/du dx \nds dx /du ds \' \n\ndy/ds = i6s^/g. \n\ndy du/ds dy \ndz dz/ds du \' \n\n\n\ny \xe2\x96\xa0= f{u), V \xe2\x80\x94 0{u), V \xe2\x80\x94 ip{s), z = F{s), z = Fi{x) \n\n\n\n7. y=f{s)=f{x^h),%^ \ndy _ dy \n\n\n\ndy^ __ dy/du dv/ds dz \ndx ~ dv/du ^ dz/ds ^ d^\' \n\n.\'. s = x-}-A. \ndy dy \n\n\n\nThen i\'^- _ ^ X A and ^ - ^ ^ _\xc2\xb1 \n\ndx ds ^ dx\' dh ~ ds ^ dk \n\nPj ds d> dv dv dy \n\nBut \xe2\x80\x94 = \xe2\x80\x94 = I, hence -- =r - = \xe2\x80\x94 . \n\ndx d/i ax dh ds \n\n\n\n^2 DIFFERENTIAL CALCULUS. \n\n\n\nCHAPTER V. \nDIFFERENTIATION OF FUNCTIONS. \n\nFUNCTIONS OF A SINGLE VARIABLE. \n\n78. The differential of any function of a single variable \nmay be determined by applying the general rule, \xc2\xa7 70, \xc2\xa7 62, \nand multiplying the result by the differential of the variable; \nbut by applying the general rule, \xc2\xa7 70, \xc2\xa7 62, to a general \nrepresentative of any particular kind of function, there will \nresult a particular form, or rule, for differentiating such \nfunctions, which is generally used in practice. \n\n79. The differential of any power of any function with a \nconstant exponent is equal to the product of the exponent of the \npower, the function with its exponent diminished by unity, and \nthe differential of the function. \n\nLet J/ = x^, in which x is any variable and n is any \nconstant. Then, increasing x by h, we have j[\xc2\xa7 70) \n\n\n\ndy __ limit \n\n\n\nHx + hY - ^" "I \n\n\n\nPlacing X -{- h = s, whence h = s \xe2\x80\x94 x, and as ^ b-> o, \ns B-> X, we have (\xc2\xa7 40) \n\n^ ^ limit p"-^n = ^^n-1 ^^^ dx" = nx"-idx. \n\nHaving y\'^, in which y is any function of any variable, \nas xy, we have (\xc2\xa7 77) \n\ndy^dx = {dy^\'/dy) X (dy/dx). \n\n\n\nDIFFERENTIA HON OF FUNCTIONS. 73 \n\nHence, \n\ndy-^/dx = nf-^(dyldx), and tT ^ ny^-^dy. (i) \nSubstituting \\/n for \xc2\xab in (i), we have \n\ndyV\xc2\xab = -y/\'*-!^ = ^-y\'^dy -= dy/n |/y^^. \n\nHence, //^^ differential of the n^^ root of any function is \nequal to the differential of the function divided by n times the \nn^^ root of the n \xe2\x80\x94 1 power of the function. \n\nEXAMPLES. \n\n1. ^ V^ = dx/2. VJ. 12. a\' f ^ = dx/l ^7K \n\n2. dx^ = 2xdx. 13. dx^^ = \xe2\x80\x94nx-^-^dx. \n\n3. dx^ = 3x^dx. 14. </jf-* = \xe2\x80\x94 4x-^dx. \n\n4. ^/4x* = i6x^dx, 15. aT^r"^ = \xe2\x80\x94 \xe2\x80\x94x\'^dx. \n\n2 \n\n5. ^a^2 _ 2axdx. 16. </;\xc2\xab:" \xc2\xbb^= - -;p ^~ dx. \n\nn \n\n6. </(log xY = 2 log X d\\og X. 17. </(a\'\xc2\xbb)2 = 2a^da^. \n\n7. ^ sin^ ;t = 3 sin\'^ x d sin jf. 18. ^ tan\xc2\xab ;r = w tan\xc2\xbb-i ;t i/ tan x. \n\n8. 0^3;^^ _ g^VAT. 19. fl\'[3(fl + a:2)3] = i8(\xc2\xab + Ar2)2;r^;p. \n\n9. dl^^Tlx^/\'i) = 4;rxVjc. 20. d{27tax\'^ / z) = ^iiaxdxji. \n\n,^ ^ "/ \xe2\x80\x94 T";. / I \\~n~v / 21. di^xY^ = 20(2xydx. \n\nII. ^ V;(;3_^3 _ 2x^^x/2 Vx^\xe2\x80\x94a\\ 22. ^[2x \'^ /7] = 3^x/i4;t ^ . \n\n23. dia-^x^y = 3(a + x2)2 ^(a+;\xc2\xabr2) = sia+x^ 2xdx = 6x{a-\\-xydx. \n\n\n\n24. d^a-{-x^ = d{a + ;*:2)/3 f (a + x\'^f = 2xdx/3 ^{a -\\- x^f. \n\n25. d{2xy = 2{2x)d{2x) = 8xdx. \n\n\n\n74 DIFFERENTIAL CALCULUS. \n\n26. d^Zx\'f - 2{2x\')d{2x\') - Idx^dx. \n\n27. d{ax^f = 2(ax^fd{ax^) = 6a^x^dx. \n\n28. d{3x)~^ = - 2(3xy^d{3x) = - 6(3xf^dx. \n\n29. dia^- x-\'f = l{a^ - x^y^dia-" - x^) = - (a\'^ \xe2\x80\x94 x^)~^xdXo \n\n30. d[{x^ -\\- a)(3x^ + b)] = (15^\' + 3^^\' + 6ax)dx. \n\n31. d[xy V{i - xy] = 3x\'dx/{i - x^)"\\ \n\n32. d{a -\\- X \xe2\x80\x94 3x^ 4" 4^^] = (i \xe2\x80\x94 6x + I2x^)^x, \n\n33. ^[(i + x2)(i \xe2\x80\x94 x^)] = (2x \xe2\x80\x94 3X\'\' \xe2\x80\x94 t^x^)dx. \n\n34. d{a -|- bx^Y = bmn{a -f ^;c\'")\xc2\xab-ia:\xc2\xbb\xc2\xbb-i^^. \n\n\n\n35. d\\{2x\'\' - i)/x Vi -\\-x\'] = {4x\'^-\\-i)dx/x\'^ii-\\-x\'\'yi\\ \n\n36. d[x / Va\' - x\'] = a\'^dx/ia\'\' - x^", \n\n37. ^(l + ^H= - dx/2{l + X)\'. \n\n\n\n38. d\\/a\'-\\-x^ = xdx/ Va\'^-\\-x^. \n\n40, ^[;t:"/(i + ;\xc2\xbb:)"] = nx\'^-\'^dx/{\\ + ;t;)w+l. \n\n\n\n41. d\\i/ Vl - ;^2] = xdx/{i - ^2)3/2. \n\n\n\n42. ^[x/ Vi-x] = {2- x)dx/2(l - xf\'K \n\n\n\n43. d[x/ Vl - x^ = dx/(i - xy^. \n\n44. dlx^/{i - x^Y\'"\'] = 3x\'\'dx/{i - xyi\\ \n\n\n\n45. 4 Vl -^x/ Vi-x\'\\=^ dx/{i -x)Vi- xK \n\n46. d{x\'^^/{i +;c2)\xc2\xab] = 2nx\'i^--^dx/{i +;t:2)\xc2\xab+l. \n\n\n\n47. 4(1 - x)/ f I + x\'^j = - (I + xW(i + ^y""\' \n\n\n\n48. ^[x/(x - i^I - ^2)] := _ ^;j:/ |/i _ jc\\x- Vl - x\'% \n\n\n\n49. dVx\'^/{2a \xe2\x80\x94 x) = {3a \xe2\x80\x94 x) Vxdx/{2a \xe2\x80\x94 x)^ \n\n\n\n50. d{- Vd^ - x^/d^x) \xe2\x80\x94 dx/x Vd^ - x2 \n\n\n\n51. d[ax/{x -{-Va-\\- x\')] = a\'dx/{x -{- Va -{- x\'f Va + x\'^. \n\n\n\nDIFFERENTIA TION OF FUNCTIONS. 75 \n\n\n\n52. d{x/a\' \\\'x^ + a") = dx/ ^{x\' + d^f. \n\n53. ^[x/(i +x)]w \xe2\x80\x94 ^jr"-Vjr/(i + jc)"+i. \n\n54. ^[(\xc2\xab + bx^l\'\')/c- ^x\'] = il;x^l^-sa)dx/4c i/I^. \n\n55. d[x^/{a -\\-x^y] = 2x(\xc2\xab - 2x^)dx/{a + ^3)3. \n\n\n\n56. dxid" + x\xc2\xab) l^a-^ - x^ = (\xc2\xab4 + aV2 - 4x4)^x/{ /^ - x^)\'/\'. \n\n57. 4x^/=\'(i + xyi\\/{i - x)^/^j = (3+2X - 3x2) v^^;^/2(i _ xyi\\ \n\n\n\n58. \xc2\xab\'(i -|- \'^) \'^^i \xe2\x80\x94 -^ = (i \xe2\x80\x94 3x)dx/2 Vi \xe2\x80\x94 X. \n\n\n\n59. ^[ Vr - Vx/ Vi -^Vx]=- dx/i{\\ + Vx) 4/^r^^^. \n\n\n\n60. ^ i^x + vFT^ ^ dx s/x^ Vi -f ;rV2 vr+^. \n\n61. M = [a \xe2\x80\x94 V^^H- (^\' - ^\')\'"]\'/\'\'. \n\n1\xc2\xb0. d{a-d/ Vr+(^2_^2)2/3^ ^ |-^/2x V^-{-2{c\'\'-x\'\')-H-2x)/3]dx \n= \\bl2x Vx- 4x/3{c\' - xy]dx. \nHence, du = ^-[a - 4+(.^- x^\'^l -^[ -\xc2\xb1-_ ^-^^\xe2\x80\x94Idx \n\n^[_ Vx J \\_2xVx 3(^2-.v^)i/3j \n\n62. In the parabola y^ = gx, find the rate of y with respect to x \nwhen X = 4. What value will x have when rate of y equals that of jf ? \nWhen rate of y is the greatest ? When the least? \n\ndy/dx = g/2y = \xc2\xb13/2 V x. .-. (dy/dx)x=4 = \xc2\xb1 3/4. \n\ndy/dx = \xc2\xb1 3/2 \\/x = I gives x = 9/4. \n\ndy/dx is the greatest when x = o, and the least when x = cc, \n\n63. Find the slope of the curve y = \xc2\xb1 \\/g \xe2\x80\x94 x^ when x = 2. Find \nvalues of x and ^ when the slope is i. \n\nAns. \xc2\xb1 0.894. x\' = \xc2\xb1 1/4.5, y = :F 4/4.5. \n\n64. Find the angle which the curve y = x/(i -\\- x\'^) makes with X at \ntheir point of intersection. Ans. 7t/4. \n\n65 Find the angle at which the curves^ = lox and x^ -{-y^ = 144 \nintersect. Ans. 71\xc2\xb0 o\' 58". \n\n\n\n^6 DIFFERENTIAL CALCULUS. \n\n66. Find the value of x at the point where the slope oi y^ \xe2\x80\x94 /^x^ is \nunity. Ans. i/g. \n\n67. At what rate does the volume of a cube change with respect to \nthe length of an edge ? Ans. 3 (edge)"^ \n\n68. Find the angle that a tangent to the curve x^ = 6y^ -\\-3y-\\- i, \nat the point (8, 3), makes with the axis X. Ans. tan-\'(i6/39). \n\n69. Find the rate of change of ( \\^x -j- 3/ax\'^) when x = 3. \n\nAns. 1/2I/3 \xe2\x80\x94 2/ga. \n\n70. Find the rate of change of the ordinate of a circle with respect \nto the abscissa. Ans. "^x/ ^ R\'\' \xe2\x80\x94 x\'^. \n\n71. Find the rate of change of the ordinate of an ellipse with re- \n\n\n\nspect to the abscissa. Ans. T bx/a \\/a^ \xe2\x80\x94 x^. \n\n80. Tke differential of the logarithm of any function is \nequal to the differential of the function divided by the function. \nLet jj\' = log X, then (\xc2\xa7 70, \xc2\xa7 42) \n\nx-\\-h \ndy_ ^ limit log(^ ^-h)- log x _ ^^ x \n\ndx h-m^o h h \n\n= lim , = Iim ; \n\n\n\n= - lim log ( I + -)^ = \xe2\x80\x94 log <? = \xe2\x80\x94 . \nX ^ \\ \' xJ X ^ X \n\nHence, d log x = dx/x. \n\nSince loga e = Ma , it follows that d loga x = Ma dx/x. \n\nHaving >\' =f(x), we write (\xc2\xa777) \n\n\n\nDIFFERENTIA TION OF FUNCTIONS. 77 \n\nd\\ogy __ dXogy^ ^ \xe2\x80\x94 L x -^ \n\ndx . dy dx y dx \n\nHence, d log y = dy/y. \n\n\n\nEXAMPLES. \n\nI. flTlog ^ = {de^)/e\'\'. 2. ^ log sin jf = (aT sin x)/sm x, \n\n3. d log x\'^ = dx\'^/x^ = 2xdx/x^ = 2dx/x. \n\n4. </log |/^= d^x l^~x = (ar;\xc2\xabr/2|/7)/|/J=^V2-^\xc2\xab \n\n5. d log jf"\' = dx\'^/x^ = nx^-\'^dx/x^ = ndx/x. \n\n6. d log i/x2 - 1 = d |/x2 -iZ-l/jt^-i = ;>:^;<:/(;c2 \xe2\x80\x94 I). \n\n7. ariog[(i +^)/(i - x)] =4(1 + xj/{i - x)]/[(l + ^)/(l - AT)] \n\n=z2dx/{i \xe2\x80\x94 x"^). \n\n8. </[log log x"] =d log Ar/log jf = (^j;/;c)/log x = dx/x log x. \n\n9. fi?[Ar log jr] = jroT log ^c \xe2\x96\xa0\\- log JCdfx = (i -|- log x)dx. \n\n10. flT log (x2 + X^) = d(x^ + X^ix\' -t- X3) = (2X + 3X^)dx/(x^ -f ^3). \n\n11. ^(log x)^ = m(\\og x)^-\'^ dlog X = m{\\og x)^-^dx/x \n\n12. d(i/log x\'*) = \xe2\x80\x94 d log ;t\xc2\xab/(log x\'^f = \xe2\x80\x94 ndx/x(\\og x^)\\ \n\n13. d log |/(i + x)/(i - X) = dx/{i - ^2). \n\n14. ^log [(I +;t;V(i - ^^)] = 2dx/3x^i-xh \n\n15. /^"\xc2\xbb (log xY = [w(log x)\xc2\xab + \xc2\xab(log xY-^x^-^dx. \n\n16. a\' log [y^3qr-i/ ^^z _ I] = _ 2,x\'\'dx/{x^ - I). \n\n17. dT log [(I + |/i^^2)/^-j ^ _ ^^/^ 4/1 _ ;p\xc2\xab. \n\n18. ^log [(I + |/^/(i - f\'^)] = c/V(i - ^) 4/^. \n\n19. a\' log [x 4/^ + Vr^^"\'] = - dx/X^x\'^ ^ I. \n\n- , i/;*;\' -I- I \xe2\x80\x94 I 2^;k: \n\n20. ^log -^-^==3ir = . \n\nV;c\xc2\xab + I -f- I ^ 1/^2 _|_ J \n\n\n\nyS DIFFERENTIAL CALCULUS^ \n\n\n\ni/x"^ -\\- a- \xe2\x80\x94 X \xe2\x80\x94 2dx \n21. d log \' \xe2\x80\x94 \n\n\n\n22. ^ log \n\n\n\n23. a\' log \n\n\n\n^ \\ -\\- x -\\- \\/\\ \xe2\x80\x94 X _ \xe2\x80\x94 dx \n\\/i-^x \xe2\x80\x94 \\/i - X ^\\/\'i~ \n\n\n\na/Vi + x^\xc2\xb1x_ ^ d^ \n\n\n\nVi + x^ \xe2\x80\x94X Vi + ^ \n\n24. d log ^\'^-\'^i = _ xdxjia} \xe2\x80\x94 x\\ \n\n\n\n25, d log |/\xc2\xab2 _|_ ^-2 :^ x^x/0?-^ -f- X""). \n\n26. ^ (log [(a - ;tr)/(a + ^)]/2\xc2\xab) = dx/{x\'^ - flS). \n\n\n\n27. ^nog (x/|/i +x\'^) = ^x/x(i +x^). \n\n\n\n28. a\' log (x\xc2\xb1 |/x2 \xc2\xb1ii^) \xe2\x80\x94 \xc2\xb1dxl \\/x\'^ \xc2\xb1 a^. \n\n29. \xc2\xab\'[log log... (repeated n times) ofjr] = i/r/xlogx(log)\'x...(log)\'^-\'x. \n\n30. d log [log {a + bx\'^)\'\\ ~ nbx\'\'^-^dx/{a -\\- dx^) log {a -\\- dx^). \n\n31. zi = (log ;*:\xc2\xab)"*. Put log x^ = jj/, then (\xc2\xa7 77) \n\n^z< = d{log x\'^)\'^ = 7ny\'"^-\\n/x)dx = f/in(\\og x\'\')^-\\dx/x. \n\n32. Find the rate of change of a logarithm in the common system \nwith respect to the number. Ans. J/10/number. \n\n33. Find the slope of the curve y = \\oga x at the point (i, o). \n\nAns. Afa. \n\n81. T/ie diffei\'etitial of any exponential function with a \nconsta7it base is equal to the product of the function^ the loga- \nrithm of the base, a?id the differential of the exponent. \n\nLet y ^ < then (\xc2\xa7 70, \xc2\xa7 43) \n\ndy limit ^*\'^\'\' \xe2\x80\x94 ^"^ r 1- a^ \xe2\x80\x94 \\ \n\ndx h-mH>o h h ^ \n\nHence, da^ \xe2\x80\x94 a^ log adx. \n\n\n\nDIFFERENTIATION OF FUNCTIONS. Jg \n\nHaving a^, in which j- = /(x), we have (\xc2\xa7 77) \n\nda^ _ day ^V _ y ^ ^y \n\ndx dy dx dx \' \n\nTherefore da^ = a^ log ady. \n\nIt follows that d^y = eydy. \n\nEXAMPLES. \n\n1. da^"^ = a^^ logadx^ = 2a*\' Jt: logadx, \n\n2. c/J""^ ^ = a^\xc2\xb0^ ^ log a ^ log X = a^\'^S ^ log a dx/x, \n\n3. da^^ = a"^ \\oga d ^\'x \xe2\x80\x94 a ^^ log a dx/2 Vx, \n\n4. da ^ \xe2\x80\x94 a ^ \\ogad{l/x) = \xe2\x80\x94 a ^ [ogadx/x"^. \n\n5. da^\'x\'^ = a^dx-^ + x\'^da^ \xe2\x80\x94 a^x\'^-\\x log a + \xc2\xab)^. \n\n1 _ 1 \n\n6. de~ "" = e ^dx/x"". \n\n7.^ a\'[(^^^ \xe2\x80\x94 e-x)/2\\ = (^* + e-^)dx/2. \n\nX _X X X \n\n8. ^[a(^" + ^ \xc2\xab)/2] = (.\'\xc2\xab - e\'"^)dx/l, \n\n9. ^i?* log X = (i/x -f- log x)e^dx. \n\n10. \xc2\xab\'[(^* - \\)/{e^- + I)] = le^\'dxl^e^ + i)^ - \n\n11. a\' log \\[e^ - i)/{e^ 4- i)] = 2e\'\'dx/(e^^ \xe2\x80\x94 l), \n\n12. ^" log (^* 4- ^-a-) = (^a^ _ e-\'\')dx/(e\'\'-]-e-^). \n\n13. a\' log {e^ \xe2\x80\x94 e-^) = {e^ + e-^)dx/{e^ - e-% \n\n14. 4(a* \xe2\x80\x94 i)/(a* 4- i)] = 2a* log a a\'x/(o* -1- l)\xc2\xab. \n\n15. a\'(a* + xf = 2(a* + x)(aa; log a -\\- \\)dx. \n\n16. a^^*(i \xe2\x80\x94 A^) = ^*(i \xe2\x80\x94 3^2 \xe2\x80\x94 x3y^_ \n\n17. flt[(^ - e-^)/{e^ + ^-^0] = 4^V(^* + ^"^)*- \n\n18. a[x/{e=\' - i)] = [^^(i ~ x) - \\\\dx/{e^ - 1)2. \n\n19. dx\'^{\\ + xY \xe2\x80\x94 nx\'^-\\\\ + x)"-i(i + \xe2\x96\xa0zx)dx. \n\n\n\n8o DIFFERENTIAL CALCULUS. \n\n20. When jr \xe2\x80\x94 o, find the inclination of the curve jv = lo\'" to X. \n\nAns. 66\xc2\xb0 31\' 30". \n\n82. Logarithmic Differentiation. \xe2\x80\x94 The differentiation \nof an exponential function, or one involving a product or \nquotient, is frequently simplified by first taking the Na- \npierian logarithm of the function. \n\nThus, let u = f^xvi which y and z are functions of the \nsame variable. \n\n.\xe2\x80\xa2. log u=^ zXogy \n\nand (\xc2\xa7 80, \xc2\xa7 75) \n\ndu/u = zdy/y -f- log ydz. \nHence, du = dy\'\' = zy^-^dy -{- y"" log ydz^ \n\nwhich is the sum of the differentials obtained by applvinq \nfirst the rule in \xc2\xa7 79, then that in \xc2\xa7 81. \n\nEXAMPLES. \n\n\n\ni/{x 1)\' log \xc2\xab = I log (x - I) \n\n\n\nV{x - 2f V{x - 3)\' \n\n- I log (^ - 2) - I log {x - 3), \ndu ^ dx 3 </x 7 dx \'jx^ -\\- 30X \xe2\x80\x94 97 \n\n\n\nu 2x\xe2\x80\x94 I /^x\xe2\x80\x942 3x\xe2\x80\x942> i2{x-i){x\xe2\x80\x942){x\xe2\x80\x942) \n\ndu=- (^-1)^7^; +30^ -97)^^ \n\n2. dx" = jr*(i + log x)dx. \n\nI I-2JC \n\n3. dx\'^ \xe2\x80\x94 X -^ (i \xe2\x80\x94 log x)dx. \n\n4. dx^ = x^ X (log\' X -f- log X -f- l/x)dx. \n\n\n\ndx, \n\n\n\n5. dx\\/i \xe2\x80\x94 x{i 4- x) \xe2\x80\x94 (2 -j-x \xe2\x80\x94 <sx\'^)dx/2\\/i \xe2\x80\x94 X. \n\n{x-if^^ ^ _ (x-i)^/\'(7x\'-j-3ox-g7)dx \n\n(;, _ 2)3/4(^ - 3)\'/\' i2{x-2y^\'\\x-3y\xc2\xb0^^ \' \n\n\n\nDIFFERENTIA TION OF FUNCTIONS. 8 1 \n\n7. d{x/ny^ = \xc2\xab(V\xc2\xab)"^Li + log {x/n)]c/x. \n\n8. ^x^/^ = x^^"" \\og{e/x)dx/x\\ \n\n9. de^"^ = e\'^^e^dx. - 10. dx^ = x\xc2\xabV^;tr(i + x log x)/x. \n\n11. d\'-f\'"^ = ^\xc2\xbb^ji;*(l -\\- log jr)i/^. \n\n1-2X \n\n12. dix\'^ + a;^/*) = [;c* log ex \xe2\x80\x94 x \'^ \\og{x/e)]dx. \n\n13. M = :\xc2\xab\xe2\x96\xa0\'<\xe2\x80\xa2\xc2\xbb \'\xc2\xbb. Put log X = z, then \n\n</\xc2\xab = ^;t;2 \xe2\x80\x94 (logx ^loga;-! _|_ ^^logj; [og jc/xy^tr = 2;ci\xc2\xb0g^-i logxdfj;, \na\'(Iog ;t:)^ \xe2\x80\x94 [(log ;c)*-i + (log x)"^ log log x]^;c. \n\n\n\n14. d \n\n\n\ni/2.a:(r - x\')3/-^ _ (- 8 ;c\' + 24;^:^ - Jtr - 6K;t: \n(j, _ 2)^/3 ~ 3(^)\'/\'(l - x^W\\x - 2)5/3* \n\n\n\nTrigonometric Functions, \n83. d sin X =: COS X dx. \nLet ^ be the increment of x^ then (\xc2\xa7 70) \n\n\xe2\x96\xa0^sin^ _ limit \ndx hm^ \n\n\n\nit Fsin (:<[: + i^) \xe2\x80\x94 sin or\\ \n^o L li J \n\nI 2 sin - cos [x A I \n\n\n\ncos^. \n\n\n\nHaving sinj, in which j =f(x)^ we have (\xc2\xa7 77) \n\n^sinj\' _ ^sin_y ^y _ dy \n\ndx dy dx dx\' \n\nHence, ^sin^ = cosjf^. \n\n\n\n82 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nSimilarly, by applying the general rule, \xc2\xa7 70, and the \nprinciple in \xc2\xa7 77, the differential of any trigonometric func- \ntion may be determined ; but it is perhaps simpler to make \nuse of the relations existing between the functions. \n\n\n\nd COS X = ^sin {jt/2 \n\n\n\nx) \xe2\x80\x94 cos {7t/2 \xe2\x80\x94 x)d{7r/2 \xe2\x80\x94 x) \n= \xe2\x80\x94 sin X dx. \n\n\n\nd tan \'x.\xe2\x80\x94 d \n\n\n\nsm^ \n\n\n\ndx \n\n\n\ncos X \n\n\n\nCOS\' X \n\n\n\nsec\' xdx =r (i -)- tan\' x)dx. \n\n\n\nd cot X = ^ tan ( ;r/2 \n\n\n\nx)= \xe2\x80\x94 dx/sin\' X = \xe2\x80\x94 cosec\' x dx \n= - (i 4- cot\' x)dx. \n\n\n\nd sec X = ^(i/cos x) \xe2\x80\x94 sin x dx/cos" x = tan x sec x dx. \n\nd cosec X 1= ^sec {n/z \xe2\x80\x94 x) = \xe2\x80\x94 cotiL cosec xdx. \n\nd vers x = ^(i \xe2\x80\x94 cos x) = sin x dx. \n\nd covers x= ^vers (rr/i \xe2\x80\x94 x) \xe2\x80\x94 \xe2\x80\x94 cos xdx. \n\nIn order to illustrate the formulas for the differentials of the \n\nsine and cosine of any angle, \nlet ACB = be any given \nangle. Assume BCN = d(p, \nand with any radius, as CO \n= R, describe an arc, as \nOMN. Then \n\nFM/R = sin <p, \n\nA_ CF/R = cos 0, arc MJV = Rd(p. \n\nThe definition of a differential (^6;^), in this case, requires \nthat sin and cos 0, retaining their rates at the states \ncorresponding to = y4CB, shall continue to change from \nthose states while increases by the angle BCJV = d(p. \n\n\n\n\nDIFFERENTIA TION OF FUNCTIONS. \n\n\n\n83 \n\n\n\nDraw the tangent line to the arc at M^ and lay off MT \nequal to the arc MN = Rdcp. Through T draw TQ paral- \nlel to MP^ and through M draw MQ parallel to OC, \n\nThen QT a.nd \xe2\x80\x94 MQ are, respectively, the changes that \nthe lines PM and CP would undergo were they to continue \nto change with the rates they have when 0= ACB, while \n<p increases by d<p. \n\nHence, QT/R, and \xe2\x80\x94 MQ/R are, respectively, the \nchanges that sin and cos <p would undergo under the \nsame requirements. \n\nThe angle M TQ \xe2\x80\x94 0. Hence, \n\nQT=MT cos = ^cos (pd0, and QT/R=ds\\n 0=cos (pd(p. \n\xe2\x80\x94MQ=MT s\'m 0=/? sin 0d(p, and MQ/R=d cos 0= \xe2\x80\x94 sin (pd(p. \n\nSimilarly the formulas for the differentials of the othei \ntrigonometric functions may be illustrated. \n\n\n\nRegarding the right lines \nPM, CP, OE, O\'B, etc., as Q \nfunctions of the variable angle \n0, we have \n\n\n\n\ndPM=dR sin (})=R cos (pd0. \ndOE=dR tan (p---Rdcp /cos"^ (p. \ndCE \xe2\x80\x94 dR sec \n\n=i? tan sec (pd(p. \n\n\n\ndCF=dR cos (p= \xe2\x80\x94 R sin (pd(p. \ndO\'B=dR cot (p^-Rd(p/s\\n^ (p. \ndCB = dR cosec <p \n\n=\xe2\x80\x94R cot cosec (pd<p. \n\n\n\ndOP = dR vers = /? sin 0^0. dO\'Q=dR covers (p= \xe2\x80\x94 R cos 0^<?>. \n\n\n\nIt is important to notice the difference between the dif- \nferentials of the above lines, which depend upon the radius \n\n\n\n84 DIFFERENTIAL CALCULUS. \n\nof the circle used, and the differentials of the trigonometric \nfunctions which do not depe-^d upon any radius or circle. \n\n\n\nEXAMPLES. \n\n1. d sin x^ = ix ros x^ dx, 12. d cos x^ = \xe2\x80\x94 2x sin x^ dx. \n\n2. d sin^ ;c = 2 sin X cos x dx. 13. d sin^ ^ = 3 sin* jt cos xdx. \n\n3. d zo^ X =-^izo^ xsinx dx. 14. ^cos^ jc = \xe2\x80\x94 3 cos" j;sin ;f ^jf. \n\n4. ^ tan\' j; = 2 tan jf dx/cos^ x. 1 5. </ tan^ ^ = 3 tan^ x dx/cos^ x. \n\n5. </cot\' ;f = \xe2\x80\x94 2 cot j;^;r/sin^ A-. 16. ^cot^^f =\xe2\x80\x94 3 cot\';r^jr/sin* at. \n\n6. d sec^ jr = 2 sec^ ;p tan x dx. ly. d sec^ ;r = 3 sec"^ x tan jra^x. \n\n7. ^cosec\'^=\xe2\x80\x94 2 cosec^ j;cot;r </x.i8. ^cosec^;c= \xe2\x80\x94 3cosec^CDt;f ^;i;. \n\n8. d vers\'-^ ^ =2 vers x sin ;if dfjc. 19. d vers^ jc =3 vers\' sin x dx. \n\n9. d CO vers2x= \xe2\x80\x94 2 covers x cos ^ ^^. 20. d cos ;r^ = \xe2\x80\x94 3;^\'\' sin x^ dx. \n\n10. </ tan :r\' = 2;i(r^;ir/cos\'\' Jf2. 21. </covers^;ir= \xe2\x80\x94 3 covers\'^ ^ cos ;\xc2\xab:</;\xc2\xbb:. \n\n11. de^^^ ^ = r^^\xc2\xb0^ ^ cos X dx. 22. d log cos x = \xe2\x80\x94 tan x dx, \n23. </ sin\'* x^ = 2 sin ^V sin ^\' = ^x sin a-" cos x^ dx\xc2\xbb \n\n\n\n24. ^|/tan 2jr = sec\' 2x dx/ |/tan 2x, \n\n25. d cos{i/x) = (i/^") sin (i/x)dx, \n\n26. d? tan" ;r\' = 4^ tan jc\' dx/cos^ x\'*. \n\n27. </ cos mx = \xe2\x80\x94 sin wjf d(mx) = \xe2\x80\x94 m sin mxdx\xc2\xbb \n\n28. ^ sin 2>x = cos 3;); ^ 3;^ = 3 cos "^x dx. \n\n29. </ sin\' 2;i: = 4 sin 2x cos 2x dx. \n\n30. i/ sin^ajf = na sin^-^ajc cos ax dx, \n\n31. d tan"^ ^ = \xc2\xbb tan"^~^ x dx/cos^ x, \n\n32. ^(tan X \xe2\x80\x94 j:) = tan\' x dx. \n\n33. ^[(;r \xe2\x80\x94 sin X cos ^)/2] = sin\' x dx. \n\n34. </ tan X sec ;f = (sec\' x \xe2\x96\xa0\\- tan\'* x) sec x dx, \n\n35. t/ tan a\'"" = - a-"" sec\' a"^ log a flJr/-^\'^\'. \n\n\n\nDIFFERENTIATION OF FUNCTIONS. 85 \n\n36. ^[(i \xe2\x80\x94 tan j:)/sec x] = \xe2\x80\x94 (cos x -j- sin x)dx. \n\n37. ^[sin nx/cos\'^ xl = \xc2\xab cos (\xc2\xab \xe2\x80\x94 i):r arx/cos"+^ ^. \n\n\n\n38. d tan Vi \xe2\x80\x94 X = \xe2\x80\x94 (sec r i \xe2\x80\x94 xydx/2 Vi \xe2\x80\x94 x. \n\n\n\n39. df T sin \\/ X = cos Vxdx/4 Vx sin V;*:. \n\n40. d^(cos jf)^\'\xc2\xb0 ^ = (cos jr)^\'" ^ (cos X log COS X \xe2\x80\x94 sin\' ;r/cos x)dx. \n\n41. d sin (sin ^) = cos x cos (sin jr)^jr. \n\n42. d COS (sin ;r) = \xe2\x80\x94 cos x sin (sin j:)d!vXr. \n\n43. 0^ sin (log nx) = cos (log nx)dx/x. \n\n44. ^;c^\'" ^ = ^sin a; ^^^^ ^ log x + sin x/x)dx. \n\n(sin ajr)^_ a3 (sin ax)^~\'^ cos (^;r \xe2\x80\x94 ax)dx \n^^\' (cosJxY\' ~ (cosl^f+1 \xe2\x80\xa2 \n\n46. i/ sin^ :v COS X = sin\'* x(3 \xe2\x80\x94 4 sin" x)fl&P. \n\n47. d cos^ X sin\' x = (2 cos\'* x \xe2\x80\x94 s sin\'* x) cos* x sin ;ir di\'x. \n\n48. </[(sin X -|- cos x)/(sin x \xe2\x80\x94 cos j^)] = \xe2\x80\x94 2dx/{s\'m x \xe2\x80\x94 cos xy. \n\n49. flr(sin at)^^" ^ = (sin x)\'^" ^ (i + sec\'* x log sin ;\xc2\xabry;c. \n\n50. Determine the manner in which the sine of an angle varies with \nthe angle. \n\nThe rate of change of sin x is (\xc2\xa7 70) d sin x/dx = cos x, from which \nwe see that as (p increases from o to 7t/2, the rate is +\xc2\xbb but diminish- \ning; hence, the sine increases, but its increments decrease. \n\nFrom 7r/2 to tt the rate is \xe2\x80\x94 , and diminishing; hence, the sine di- \nminishes and its decrements increase numerically. \n\nFrom 7t to 37r/2 the rate is \xe2\x80\x94 , and increasing. That is, the sine \ndecreases, but its decrements diminish numerically. \n\nFrom 37r/2 to 27r the rate is -{-, and increasing. That is, the sine \nincreases, and its increments increase. \n\nIn a similar manner the circumstances of change of each trigone^ \nmetric function with respect to the angle may be determined, \n\n51. Assuming dx = 7t/4, we have, corresponding to :r = 7t/6, \n\n^ . V3 , ^ y ^ \n\nasiax^=\xe2\x80\x94- \xe2\x80\x94 7t. dcosx = \xe2\x80\x94\xe2\x80\x94. dtanx=\xe2\x80\x94. \n\n8 83 \n\n\n\n86 DIFFERENTIAL CALCULUS. \n\nd cot X =^ \xe2\x80\x94 7t. d sec x = \xe2\x80\x94 . d cosec jc = \xe2\x80\x94it. \n\n6 2 \n\n52. Corresponding to x = 7r/4, we have \n\na\'sinx I ^cosx i ^ tan ;\xc2\xbb: \n\n\n\ndx j^2 ^^ 4/2 ^^ \n\nd cot X _ d sec -^ _ /\xe2\x80\x94 d cosec Jic /\xe2\x80\x94 \n\ndx \' dx \' dx \n\n53. What is the value of x when tan x is increasing twice as fast \nas X? Ans. n/^. \n\n54. Find the rate of change of the tangent, regarded as a function \n\nof the sine of an angle. \n\nSince tan x =/r, and sin x = Fx, we have (^5 77) \n\nd tan x/dsin x\xe2\x80\x94{d tan x/dx)/{d sin x/dx) \n\n\xe2\x96\xa0= (i/cos\'^ ;r)/cosx = i/cos^^. \n\nIn a similar manner the rate of change of any trigonometric func- \ntion regarded as a function of any other may be found. \n\n55. Find the slope of the curve y = sin x when x= o, x := 7r/4, \nX = Ti/i. Ans. I, |/i/2, o. \n\n56. When X = 7r/3, find the inclination of the curve y = tan x to X. \n\nAns. 75\xc2\xb0 57\' 50". \n\n57. Find the angle which the curves^ = sin x and j = cos x make \nwith each other at their point of intersection. Ans. tan"^ 2^2^ \n\nInverse Trigonometric Functions, \n84. d sin-i X = dx/|/i - x^ \n\nLet = sin~^ x ; then Jt: = sin and dx/d(p = cos 0. \nHence (\xc2\xa7 73), \n\n^sin~^ x/dx = i/cos = i/\xc2\xb1 r i \xe2\x80\x94 x\'\\ \n\n* The sign depends upon that of cos (p. Formulas involving the \ndouble sign in this article are generally written with the plus sign \nonly, which corresponds to angles ending in the first quadrant. \n\n\n\nDIFFERENTIA TION OF FUNCTIONS. \'^\'J \n\nBy applying the principle in \xc2\xa7 77, it may be shown that \n\n\n\nd^vcT^ y = dy/Vi \xe2\x80\x94 y, in which j^ =/x. \n\n\n\nd cos-i X = d{n/2 \xe2\x80\x94 sin-^ x) = \xe2\x80\x94 dx/ 4/1 \xe2\x80\x94 x\'. \ndtan-ix = dx/(i + x\'). \n\nLet = tan"-^ x ; then \n\nX = tan 0, and dx/d(p = i + tan" <f>. \n\nHence (\xc2\xa7 73), ^tan~^ x/dx = 1/(1 -|- jc*). \n\nd cot-i X \xe2\x80\x94 d(n/2 \xe2\x80\x94 tan-* x) = \xe2\x80\x94 dx/(l + x"), \n\n\n\nd sec-i X = dx/x Vx\' \xe2\x80\x94 i. \n\nLet = sec~* x, \nthen jc = sec 0, and dx/d(^ = sec tan 0. Hence (\xc2\xa7 73), \n\n^sec"* XI I I \n\n\n\ndx sec tan sec (pVsec\'\' \xe2\x80\x94 i xVx\' \xe2\x80\x94 i \n\n\n\nd cosec-i X = d{7r/2 \xe2\x80\x94 sec~^ x) = \xe2\x80\x94 dx/xVx" \xe2\x80\x94 i. \n\n\n\nd vers"\'^ x = dx/y 2X \xe2\x80\x94 x^ \n\n.et = v\xe2\x82\xac \nHence (\xc2\xa7 73), \n\n\n\nLet = vers"* x ; then ;^ = vers 0, and -^ = sin \n\n\n\nI \n\n\n\n^^ sin |/i _ cos\' 1^1 - (i - vers 0)\' \n\nI I \n\n\n\nr 2 vers \xe2\x80\x94 vers\'^ t 2x \xe2\x80\x94 j^^ \n\n\n\n88 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nd covers-^ x = d{n/2 \xe2\x80\x94 vers"^ x)= \xe2\x80\x94 dx/l/2x - x". \n\nRegarding as a function of \nthe line PM^ denoted by% we \n\n\n\n\n. y \nE have = sin"^-^. Hence, \n\n\n\nd(t) \n\n\n\n4 \n\n\n\ndy \n\n\n\n\n\n\nSimilarly, having \n\n\n\nCP =y, .-. = cos-i ^ \n\n\n\nOE =y, .\xe2\x99\xa6. = \n\n\n\nR* \n\n\n\nR\' \n\n\n\n0\'B=y, .-. = cot-i^, \n\n\n\nCE =y, .\'.0= sec \n\n\n\n-il \n\n\n\nCB =y, o \'o = cosec \n\n\n\n.i2 \n\n\n\nwe have </0 = \nwe have dip = \nwe have d<p = \nwe have ^0 = \nwe have dif) = \n\n\n\n-4/ \n\n\n\nRdy \n-Rdy_ \n\n^4/^ \xe2\x80\x94 i?a\' \n\n\xe2\x80\x94 Rdy \ny^fZTR-^ \n\ndy \n\n\n\nPO =1\', .*. = versin-i^, we have ai^ = - ; \n\nR ^2Ry\xe2\x80\x94y^ \n\ny \xe2\x80\x94 dy \n0\'0= V, .*. = coversin-i ^^ yyg have 0^0 = - , . \n\n\n\nEXAMPLES. \n\n\n\n1. dsin-^ x^ = 2xdx/ Vi \xe2\x80\x94 .r*. \n\n2. ^ sin-i 2;\xc2\xab: Vi \xe2\x80\x94 x^ = 2dx/ Vi \xe2\x80\x94 X*. \n\n3. d tan-i \\ia + a:)/(i - a^)] = dx/{i + x^). \n\n\n\nDIFFERENTIA TION OF FUNCTIONS, 89 \n\n\n\n4. d sin-i \\sx + I)/ V2] = ^V ^i - 2;c - ^\xc2\xbb. \n\n\n\n5. a^ vers-i jt\'^ = 2dxJ V2 \xe2\x80\x94 x\'^ \n\n\n\n6. d Vsin~^ ;f = dx/2 Vshi~^ x{i \xe2\x80\x94 x^). \n\n7. d tan-i (\xe2\x80\x94 \xc2\xab/x) = adx/{a^ + x\'*). \n\n8. dcos--^ [(i - ^2)/(i H--^\')J = 2dx/(i-\\-x% ^ \n\n9. ^ tan-i ( Vi - x/ Vi -\\-x) = - dx/2 Vi \xe2\x80\x94 x\\ \n10. ^sec-i [i/(2j:2 _!)] \xe2\x80\x94 _ ^.dxj Vi \xe2\x80\x94 x^. \n\n\n\nII. ^tan-i [x/ Vi - x\'\'] = dx/ V \n\n\n\n:/ y 1 \xe2\x80\x94 X-, / \n\nI - x\\ \\/^ \n\n\n\n12. d cos~^ [x/{a \xe2\x80\x94 x)] = \xe2\x80\x94 adx/{a \xe2\x80\x94 x) Va^ \xe2\x80\x94 2ax, \n\n13. d tan-i [x Vi7(2 + x)] = Vjdx/2{x\' + -^ + l). \n\n14. i/cos-i Vi \xe2\x80\x94 jc-\'^ = dx/ Vi \xe2\x80\x94 x^. ^ \n\n\n\n15. ^sec-l [n/ Va"" - x\'^ = dx/ iV - x \n\n\n\n/^2 _ v-2 \n\n\n\n[6. d cos-i [(4 \xe2\x80\x94 3x\'*)/x^] = \xe2\x80\x94 })dx/x Vx^ \xe2\x80\x94 i. \n\nI-], d \xe2\x80\x94 cot-i {x/a)/a = d tan-^ {x/a)/a = a\';t:/(a\'+ a:\'). \n\n[8. a\' tan-i [2^(1 + ^^)] = 2(1 - x\'\')dx/{i + 6;c\' + x*). \n\n\n\n19. d sec~^ (x/a)/a = ^\xe2\x80\x94 cosec"! {x/a)/a = dx/x Vx^ \xe2\x80\x94 a\'. \n\n20. ^ sin-i [(I - ;\xc2\xbb:2)/(i + x^)] = - 2dx/{i + ;c\'). \n\n21. ^tan-i [(i - x)/(i + x)] = dx/ii -f- jc"). \n\n\n\n22. ^sin-l [x/ Vi + ^i;\'\'] = dx/{l 4- ;\xc2\xab:\xc2\xab). \n\n\n\n23. ^cos-i {2x \xe2\x80\x94 i) = \xe2\x80\x94 dx/ Vx{i \xe2\x80\x94 ;r). \n\n24. ^tan-i [2x/(i - ^*)] = 2dx/{i -{-x\'\'). *4 \n\n25. oT sin-i (1/ Vr^T?) = - dx/{i + ^\xc2\xbb). \n\n\n\n26. fl^sin-i ^{i -x)/2 = \xe2\x80\x94 dx/2 |/i \xe2\x80\x94x^. \n\n27. 0^130-1(1/ 4/x\'^ \xe2\x80\x94 i) = \xe2\x80\x94dx/x \\^x^ \xe2\x80\x94 I. \n\n28. fl\'cos-\'(4^3 \xe2\x80\x94 2)X) \xe2\x96\xa0= \xe2\x80\x94 idx/ 4/1 \xe2\x80\x94 x2. \n\n29 ^ tan - 1 a^\'^\' = - \xc2\xabi\'^ log a^. /x2 ( , + a^\'^\'). \n\n\n\n90 DIFFERENTIAL CALCULUS, \n\n\n\n30 \n\n\n\n. d tan-i (|/i J^x\'-x)=- dx/2{i -\\-x\'\'). \n\n\n\n31. d cos-\'(l \xe2\x80\x94 2x\'^) \xe2\x80\x94 2dx/\\f\\ \xe2\x80\x94 x"^ \n\n\n\n32. d /^" ^ = /\'" -^ ./x/f/i - Jt^ \n\n33. d vers-^ (lA) = \xe2\x80\x94 dx/x \\^2x \xe2\x80\x94 i. \n\n34. d\\r vers-^0//r) \xe2\x80\x94 |/2ry \xe2\x80\x94 ^J \xe2\x80\x94 y dyj ^2ry \xe2\x80\x94y^. \n\n35. t/;^^\'^"\'-^ = x^\'""\'-^(sin-\' V-^-f log x/|/r=-"^Vx \n\n\n\n36. jj/ = 2 tan-\' \'j/(i - ;t:)/i -\\-x. (i \xe2\x80\x94 ;r)/(i + Jr) = tan\'(>\'/2). \n\n\n\nHence, x^cosy, and ^ = \xe2\x80\x94 dx/\\/i \xe2\x80\x94 x^ \n\n37. ;j/ = cos-i[(4 - 3x\'^)/x\'^]. Put (4 - 3x\'\')/x^ = z. \n\nThen dy/dz = (- i/|/i \xe2\x80\x94 z~)(dz/dx) = - 3/;>:V\'x=\' \xe2\x80\x94 I. \n\n38. Wheny = o, andjj/ = 2r, find the slope of the curve \n\n\n\n1. d log sin X = \n\n2. d log tan X = \n\n\n\n;t: = r vers\xe2\x80\x94 ^jv/\'\') \xe2\x80\x94 y2ry \xe2\x80\x94 y^. \n\nAns. 00, and o. \n\nMISCELLANEOUS EXAMPLES. \nd sin X cosxdx dx \n\n\n\nsin X sin x tan ji: \n\n^ tan X dx 2dx \n\n\n\ntan X cos jr sin ;r sin 2x \n\n\n\nX dx \n\n3. d log tan \xe2\x80\x94 = \n\n\n\nsin X \n\n\n\nX dx = d\\ \xe2\x80\x94 log . \n\n[_2 I + cos Xj \n\n\n\n(7t , X \\ dx dx \n\n4" ~\' ~ \n\n\n\n2/ sin (;r/2 -|-x) cos jc \n= sec X dx =^ d \n\n\n\nI , I + sin X \nlog \n\n\n\n2 I \xe2\x80\x94 sin ;p \n\n\n\n]\xe2\x96\xa0 \n\n\n\n5. d \xe2\x80\x94 log cos X = d log sec x = tan ;r dx. \n\n6. rt\'f?^ cos j; = ^^(cos X \xe2\x80\x94 sin x)dx. \n\n7. dxc^\'"" -^ = /\'^ ^ (I + -^ cos x)dx. \n\n\n\nDIFFERENTIA TION OF FUNCTIONS. QI \n\n8. dxe\'\'\'\'^ ^ = ^^\xc2\xb0^ -^ (I - X sin x)dx. \n\n9. (at\' sin (log x) = cos (log x)dx/x. \n\nID. u = sin"* x/cos** X. .\'. log u = m log sin ;i; \xe2\x80\x94 \xc2\xab log cos ;t:, \n\n, du ( cos jf , sin;c\\ \n\nand \xe2\x80\x94 = \\?7i \\- n \\dx. \n\nu \\ s\\n X cos xj \n\n[ms\\n^-^x ns\\n^ + ^x\\ \n\xe2\x80\xa2*\xe2\x80\xa2 ^^ = \\ cosn-ix + cos^ + i;. j^\' \n\n11. \xc2\xab = jf/^" \xe2\x80\xa2^. .-. log \xc2\xab = log ;i; ~f tan-i x. \n\ndu = u{i/x + 1/(1 + x\'\'))dx = /^""\'^ (I + ^ + x\')/{i -f- x^). \n\n12. tf\'^^^" ^\'"-^ = \xc2\xabx\xc2\xab-V\xc2\xab ^^\xc2\xb0 ^(i +;c cos x)dx. \n\n13. ^^^\xc2\xb0^ -^ sin X = /\xc2\xb0^ -^ (cos ;r \xe2\x80\x94 sin*^ x)dx. \n\n14. ^[sin nx/s\\n^x\'\\ = \xe2\x80\x94 n sin(;? \xe2\x80\x94 i)x dx/sin^\'^^x \n\n15. ^cos log (i/x) = sin log {i/x)dx/x. \n\n16. i/cos sin X = \xe2\x80\x94 cos ;t sin sin x dx. \n\n17. fl?\'^^ log ;r = (?* log (ex^)dx/x. \n\n18. ^log (;\xc2\xbb:/\xc2\xb0^-^) = (i \xe2\x80\x94 X sin x)dx/x. \n\n19. fl\'^\'^ (log x)\'^ = jc\'^-\' (log x)""-^ (m log jc + n)dx, \n\n20. ^x\xc2\xab^l\xc2\xb08^^ = (\xc2\xab + i);c\xc2\xab-^^^\xc2\xb0^^^x. \n\n\n\n21. ^log \\/(i -\\- sin ^)/(i \xe2\x80\x94 sin x) = dx/cos x. \n\n\n\n22. ^log y (i \xe2\x80\x94 cos x)/{i -\\- cos jr) = ^;i:/sin x. \n\n23. <^ sin tan x \xe2\x80\x94 cos tan x dx/cos^ x. \n\n24. ^ \xe2\x80\x94 log ^-^, \xe2\x80\x94 - -\\ \xe2\x80\x94 tan-i ^- = \xe2\x80\x94 \xe2\x80\x94 \xe2\x80\x94 , \n\n25. d cos log sin ;r = \xe2\x80\x94 cot x sin log sin x dx. \n\n26. d sin~ f^sin x = t/x/2 4^1 -f- cosec x. \n\n27. a\' log (x/a\'\') = log {e\'^-^/a)dx. \n\n\n\n28. ^ log sin ;if = (a\'jc\'/sin ^ ^i \xe2\x80\x94 jc* \n\n\n\n92 DIFFERENTIAL CALCULUS, \n\n29. d sin~ (tan x) \xe2\x80\x94 sec* x dx/ \\/i \xe2\x80\x94 tan\'\' x. \n\n30. d log cos" ^ = \xe2\x80\x94 dx/cos" X )/i\xe2\x80\x94x^. \n\n31. ^ log tan~^;i: = dx/{i -^ x"^) tan""^;i:. \n\n32. d tan ~ log X = dx/x[i -\\- (log xY]. \n\n33. 0^ cos [a sin~ (i/-\'^)] = \xc2\xab sin (\xc2\xab cosec~ x)dx/x \\/x^ \xe2\x80\x94 i. \n\n34. </ cot \'^ (cosec x) = cos x dx/(\\ -\\- sin\'* jf). \n\n35. d sin-\'O^^"\'\'-^) = /^"\'\'\xe2\x80\xa2^^V(i+-=^\')\'^i-*\'\'^\'\'~\'-*\' \n\nHyperbolic Functions, \n85. d sinh X = ^i(^^ - ^-^) \n\n= W"" + ^-\'=)^^ = cosh X dx. \nd cosh X = d\\(f + ^-*) \n\n= i(^^ \xe2\x80\x94 ^"^)^^ = sinh X dx. \nd tanh x = ^sinh j^/cosh x) \xe2\x80\x94 sech^ x dx. \nd coth X = ^(cosh jc/sinh x) = \xe2\x80\x94 cosech\' x dx. \nd sech X = 4 1 /cosh x) = \xe2\x80\x94 sech x tanh x dx. \nd cosech x = ^(i/sinh x) = \xe2\x80\x94 cosech x coth x dx. \n\nEXAMPLES. \n\n\n\n1. ^4/cosh X = sinh X ^x/2y cosh x. 3. d log sinh x = coth x dx. \n\n2. d log cosh X = tanh x dx. 4. <^(;c \xe2\x80\x94 tanh x) = tanh" xdx. \n\n5. fl^[(sinh 2x)/4 + \xe2\x80\xa2^/2] = cosh\'* j:^;f. \n\n6. a^[(sinh 2x)/4 \xe2\x80\x94 x/2] = sinh\'* xdx. \n\n\n\nDIFFERENTIA TION OF FUNCTIONS, 93 \n\n(x^ x^ \\ \n\nX -\\- . h 7\xe2\x80\x94 + ... I = cosh X dx. \n|3- li / \n\nI + I h -; h \xe2\x80\xa2 \xe2\x80\xa2 \xe2\x80\xa2 ) = sinh xdx, \n\nll ll / \n\n\n\nInverse Hyperbolic Functions, \n\n\n\n86. d sinh-i x = d log (x + Vi + x\') = dx/Vi + x\' \nLet J\' = sinh"^ x ; then \n\nX = sinh J, and (\xc2\xa7 85) dx/dy = cosh^. \nHence (\xc2\xa7 73), \n\n^/sinh~^^ _ ^ _ ^ _ I \n\n^^ ~ cosh;; ~ Vi + sinh\'^ ~ vT+^* \n\n\n\nd cosh-* x= d log (x + i/x^ - i) = dx/Vx\' - i. \n\nLet^ = cosh"*^ ; then \n\nX = coshji^, and (\xc2\xa7 85) dx/dy = sinhj;. \n\nHence (\xc2\xa7 73), \n\n^cosh-*jc_ I _ I _ I \n\ndx "sinh;;" Vcosh\'y- i ~ Vx\' - i* \n\nd tanh-i x = d - log i\xc2\xb1^ = \xe2\x80\x94 ^(^ < i). \n2 *I\xe2\x80\x94 X I \xe2\x80\x94 x\'^ ^ \n\nLet j; = tanh--\':^;; then \n\nX = ta.nhy, and (\xc2\xa7 85) dx/dy = sech\'^. \n\n\n\n94 DIFFERENTIAL CALCULUS. \n\nHence (\xc2\xa7 73), \n\ndx.2,x\\\\C^ X III \n\n\n\ndx sech y i \xe2\x80\x94 tanh y i \xe2\x80\x94 ^ \n\nd coth-i X = d - log ^-i-^ = \xc2\xa3^:r^(^ > ^)\xc2\xab \n\n\n\nLet J = coth~^ x\\ then \n\n:v = coth J, and (\xc2\xa7 85) dx/dy = \xe2\x80\x94 cosech^j/. \nHence (\xc2\xa7 73), \n\nd coth-^ X \xe2\x80\x94 I \xe2\x80\x94 I \xe2\x80\x94 I \n\n\n\ndx cosech^ y coth^ y \xe2\x80\x94 \'^ x^ \xe2\x80\x94 \\ \n\n\n\n. ,\', ,, ii\xc2\xb1i/i-x^ -dx \nd sech-i X == d log \n\n\n\nLet J = sech"^ x; then \n\njc = sech jF, and (\xc2\xa7 85) dx/dy = -- sech j tanhj^. \nHence (\xc2\xa7 73), \n\nd sech~^ X \xe2\x80\x94 I _ \xe2\x80\x94 I \n\ndx sech jv tanh J j^\\/j _ ^2* \n\n\n\nI \xc2\xb1 y I + x\' \xe2\x80\x94 dx \nd cosech-i x = d log \xe2\x80\x94 \n\n\n\nX xVx^ 4- I \n\nLetjj^ = cosech"^ x; then \n\nX = cosech_y, and (\xc2\xa7 85) dx/dy = \xe2\x80\x94 cosechjv cothji^. \nHence (\xc2\xa7 73), \n\n\n\nDIFFERENTIATION OF FUNCTIONS. 95 \n\nd cosech"^ x \xe2\x80\x94 i \xe2\x80\x94 i \n\n\n\ndx \n\n\n\ncosechj/coth^ ^|/^^ -j- i\' \n\n\n\nEXAMPLES. \n\n\n\nr. d cosh-1 {x/d) \xe2\x80\x94 dx/ Vx\'i \xe2\x80\x94 a^. \n\n\n\n2. d sinh-i (x/a) =,dx/ Va" + x\\ \n\n3. ^tanh-i (x/d) = a dx/ia"^ \xe2\x80\x94 x"^), (x < a.) \n\n4. ^coth-i (x/a) = a dx/{d \xe2\x80\x94 x-^), {x > a.) \n\n5. ^ tan-i (tanh x) = sech 2x dx. \n\n\n\n6. ^[^tanh-i^ 4-|tan-ix] = ^ logy _ \'^ -f i tan-^ .r \n\n7. ^ [^ cot-i jf \xe2\x80\x94 ^ coth-i x] = ^ I cot-i^ \xe2\x80\x94 logy ^ J^ ^ \n\n\n\n8. ^[^x |/x^ - d" - \\a^ cosh-i(x/a)] = i/x^ - d" dx. \n\n\n\n9. ^ [|x \\/d + x^ + ia" sinh-i (x/a)] = ^d\' + x* ^x. \n\nGeometric Functions. \n\n87. Differential of an Arc of a Plane Curve.\xe2\x80\x94 Let s \n\nrepresent the length of a varying portion of any plane curve \nin the plane XY. It will be a func- \ntion of one independent variable \nonly (\xc2\xa7 18), which we may take to \nbe X. \n\nAssume any point of the curve, \nas M, and increase the correspond- \ning value oi X -^^ OF, by FF\' = \nAx. /\\s = MM\' vfiW be the cor- \nresponding increment of s^ and A/ = QM\\ \n\n\n\n\n9^ DIFFERENTIAL CALCULUS, \n\nrrr, /e e \\ ^-f limit ^^ i- ch J/ilf\' \nThen (\xc2\xa7 70, \xc2\xa7 44) \xe2\x80\x94 = ^^"^\'- \xe2\x80\x94 = hm \n\n\n\n\n\n\nds \n\n\n\n= Vdx\' + ^V^x. Similarly \xe2\x80\x94 = Vdx\' + df/dy. \n\n\n\nHence, ds = l^dx\' + dy\'.* \n\nThe double sign is omitted because s may always be considered as \nan increasing function of x. \n\nThat is, f/ie differential of an arc of a plane curve is equal \nto the square root of the sum of the squares of the differentials \nof the coordinates of its extreme point. \n\nIf s were to change from its state corresponding to any \npoint, as M^ with its rate at that state unchanged, the \ngeneratrix would move upon the tangent line at M\\ hence, \nMT = \\/dx^ -\\- dy^ represents ds in direction and measure. \n\nIn order to express ds in terms of a single variable and \nits differential, find expressions for dy in terms of x and dx^ \nor of dx in terms of y and dy^ from the equation of the \ncurve, and substitute them in the formula. \n\nThus, let s be an arc of the circle whose equation is \nx^ +y \xe2\x80\x94 4. Solving with respect to j, and differentiating, \nwe have \n\n\n\n^\' = qi xdx/ r 4 \xe2\x80\x94 x^ . \n\n\n\n* The square of the differential of a variable represented by a single \nletter is generally written as indicated in the above formula, and is \nsimilar in form to the symbol for the differential of the square of the \nvariable. Similarly, the \xc2\xabth power oidx is generally written dx^. \n\n\n\nHence, \n\n\n\nDIFFERENTIATION OF FUNCTIONS, 97 \n\n2 dx \n\n\n\nds \n\n\n\n/ \n\n\n\ndx" ^ \n\n\n\nX \n\n\n\n\'dx\' \n\n\n\nX \n\n\n\nv.- \n\n\n\nX \n\n\n\n88. Differential of any Arc. Let s represent the length \nof a varying portion of any curve in space. It will be a \nfunction of one independent variable only (\xc2\xa7 i8), which we \nmay assume to be x. \n\n\n\n\nThrough any assumed point of the curve, as M, draw \nthe ordinate MN \\ and through N^ the point where it \npierces XF, draw NP parallel to Y. OP will be the value \nof X corresponding to M. Increase x = OP by PP\' =^ l\\x, \nand through P\' pass a plane parallel to YZ, intersecting the \ngiven curve at M\'. As = arc MM\' will be the increment \nof s corresponding to the assumed increment of x. \n\nDraw the chord MM\' and the ordinate M\' K, Through \nM draw MQ\' parallel to a right line drawn through N and \nK\\ and through N draw NN\' parallel to X. Then \nN\' K \xe2\x80\x94 Ay and Q\'M\' = As will be the increments of \ny and z corresponding to A^ ; and we have \n\n\n\nchord MM\'= V^Axf + {Ayf + (As)^ \n\n\n\n98 \n\n\n\nDIFFERENTIA L CAL CUL US. \n\n\n\nHence (\xc2\xa7 70, g 44), \nds_^ limit 2.YCMM\' ^ lii^it ch. MM\' \n\n\n\nlimit \n\n\nV{ \n\n\nAA;)\' + (Aj.r + (A2r \n\n\nAJfS^^c \n\n\n) \n\n\nAa: \n\n\nlimit \n\n\n\n\n\n5 may be a curve of single or of double curvature. The \nincrement A^ may or may not lie in the projecting plane \nof the chord MM\'. If not, the projection of the chord \nMM\' on the plane XY will change direction as Ajj: ap- \nproaches zero, but the above relations will not be affected \nthereby. \n\nLet \xc2\xab\'\', P\' and y\' represent the angles made by the \nchord MM\' with X, Y and Z, respectively, then chord \nMM\' / l\\x ^= i/cos a\'. Let a^ (3 and y, respectively, \nrepresent the corresponding angles made by the tangent at \nM. Then \n\nds/dx = ^^J^o[^/^^^ ^\'] ~ i/cos a, and dx = ds cos a. \n\nSimilarly dy = ds cos /3, and dz = ds cos y, in which x, y, z \nor s may be considered as the independent variable. \n\n89. Differential of a Plane Area.~Let u represent the \nY area of the plane surface in- \n\nM____j\\/\' eluded between any varying por- \n\ntion of any plane curve, as AM^ \nthe ordinates of its extremities, \nand the axis X. \n\n\n\n\nDIFFERENTIATION OF FUNCTIONS. \n\n\n\n99 \n\n\n\nRegarding u as a function of x (\xc2\xa7 21), let x \xe2\x80\x94 OP\' be in- \ncreased by P\' P" \xe2\x80\x94 Ax. P\' MM\'\' P" will he the correspond- \ning increment of zi. Hence (\xc2\xa7 70, \xc2\xa7 46), \n\nlimit \n\n\n\ndu/dx \n\n\n\nAx-m^ o \n\n\n\nIP\'MM"P"I Ax\'\\ =y, \n\n\n\nwhich gives du=: ydx. \n\nThat is, the differential of a plane area is equal to the ordi- \nnate of the extreme point of the boufiding curve into the differ- \nential of the abscissa. \n\nTo illustrate, let u represent the \narea BAMP, and PR \xe2\x80\x94 dx; then \ndu =ydx = rect. PQ, which ful- \nfils the requirements of the defi- \nnition of a differential (\xc2\xa7 68). \n\nSimilarly, it may be shown that \n^^ is the differential of the plane area included between \nany arc, the abscissas of its extremities, and the axis of F. \n\nIn case the coordinate axes are inclined to each other \nby an angle B, we have du = y sin 6dx, or du = x sin f^dy. \n\nIn order to express du in terms of x and dx, "substitute \nforji^, or dy, its expression determined from the equation of \nthe bounding curve. \n\nThus, if ^y + b\'^x^ = a^b\'^ is the equation of the bounding \n\n\n\n\ncurve, we have jF = \xe2\x80\x94 ^ c^ \xe2\x80\x94 x^ \na \n\n\n\nand du = - Va^\xe2\x80\x94x\'dx. \na \n\n\n\n90. Differential of a Surface of Revolution. \xe2\x80\x94 Let the \n\naxis of X coincide with the axis of revolution; and let \nBM = i- be any varying portion of the meridian curve \n\n* It is important to notice and remember that ydx is the differen- \ntial of a plane area bounded as described ; and that it is not, in \ngeneral, the differential of a plane area otherwise bounded. \n\n\n\nLofC. \n\n\n\n100 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nin the plane XY. Through M draw the tangent MT, \n\nthe ordinate MF^ and the right \nline MR\' parallel to X. Let u \nrepresent the surface generated by \ns\\ and regarding it as a function \nof X (\xc2\xa7 24), let X = OF be in- \ncreased by FF\' = Ax. MM\' \n= As will be the corresponding \nincrement of s ; and the surface \ngenerated by it will be the increment of the function u cor- \nresponding to A^. Hence (\xc2\xa7 70, \xc2\xa7 56), \n\n\n\n\n^^ _ limit ^^^\' S^"- ^y ^\'"^ -M^M\' \n\n\n\n27Cy \n\n\n\ndx \n\n\n\ncos R\'MT\' \n\n\n\nAssume FR = dx\\ then R\' T \xe2\x80\x94 dy^ MT = dsy and cos \nR\'MT=dx/ds. Substituting this expression for cos R\'MT \nin above, we have \n\nu _ _Z_i. and du = 27ryds = aTTyf^dx" +dy\xc2\xab. \n\nHence, the differential of a surface of revolution is equal \nto the product of the circum. of a circle perpendicular to the \naxis and the differential of the arc of the generating curve. \n\nSimilarly, it maybe shown that 27tx^dx\'^^dy\'^ is the \ndifferential of a surface of revolution generated by revolv- \ning a plane curve about the axis of Y. \n\nIn order to express du in terms of a single variable and \nits differential, find expressions iox y and dy in terms of x \nand dXy or of dx in terms of jv and dy^ from the equation of \nthe generating curve ; and substitute them in the formula. \n\nThus, ify = 2px is the equation of the generating curvr, \n\n\\^2px and dy = ^-=^. \nV2px \n\n\n\nwe have y \n\n\n\nHence, \n\n\n\nDIFFERENTIA TION OF FUNCTIONS. \n\n\n\nlOI \n\n\n\ndu \n\n\n\n= 271 V2j>x\\/dx\' -^t^ = 2n{2px -{-p\')^dx. \n\n\n\n2pX \n\n\n\n91. Differential of a Volume of Revolution Let the \n\naxis of X coincide with the axis of revolution ; and let BM \n\nbe any varying portion of the \n\nmeridian curve in the plane \n\nXY. Through M draw the \n\nordinate MP^ and the right \n\nline MQ\' parallel to X. Let v \n\nrepresent the volume generated \n\nby the plane surface included between the arc BAf, the \n\nordinates of its extremities, and the axis of X. Regarding \n\nz/ as a function of x (\xc2\xa7 29), let x be increased by BB\'= Ax. \n\nThe volume generated by the plane surface PMM\' P\' will \n\nbe the corresponding increment of the function v. Then \n\n(\xc2\xa770, \xc2\xa748) \n\n\n\nY \n\n\nm\' \n\n\nM \n\n\n^^^\'^\' \n\n\nP\' X \n\n\nb/ \n\n\n\n\nP t^x \n\n\n\ndx \n\n\n\nlimit vol. gen, by PMM\'P\' __ \n\nAx ^\' \n\n\n\nAJf\xc2\xab\xc2\xbb-\xc2\xbb0 \n\n\n\nand \n\n\n\ndv = Try\'di:, \n\n\n\nHence, the differential of a volume of revolution is equal to \nthe area of a circle perpendicular to the axis into the differen- \ntial of the abscissa of the 7?ieridian curve. \n\nSimilarly, it may be shown that nx^dy is the differential \nof a volume of revolution generated by revolving a plane \nsurface about the axis of Y. \n\nIn order to express dv in terms of a single variable and \nits differential, determine an expression for jj^ in terms of x^ \nor of dx in terms of _y and dy^ from the equation of the me- \nridian curve, and substitute them in the formula. \n\nThus, if jc\' + / \xe2\x80\x94 2Rx = o is the equation of the me- \n\n\n\n102 DIFFERENTIAL CALCULUS, \n\nridian curve, we have dv = 7t{2Rx \xe2\x80\x94 x^^dx ; or since dx \n= q= ydy/ ^R\' -y\\ dv = ^ nfdy/ VR\' -/. \n\n92. Differential of an Arc of a Plane Curve in Terms \nof Polar Coordinates \xe2\x80\x94 Let r = /{?;) be the polar equation \n\n\n\n\nof any plane curve, as BMM\\ referred to the fixed right \nline PD, and the pole P. Let BM = i- be any varying \nportion of the curve, and PM = r the radius vector corre- \nsponding to M. Regarding i" as a function of v (\xc2\xa7 19), let \nV be increased by MPM\' \xe2\x80\x94 /\\v. The arc MM\' = As will \nbe the corresponding increment of s. With Z\' as a centre \nand PM as a radius, describe the arc MQ\', Denote PM\' \nby r\'; then Q\'M\' = r\' \xe2\x80\x94 r will be the increment of r corre- \nsponding to Az\'. Through i^ draw the tangent^/\', and \nthe chords MM\' and MQ\' . Then (\xc2\xa7 70, \xc2\xa7 55) we have \n\n\n\ndj_ \ndv \n\n\n\nlimit a rc MM \' _ Xxxv^xl .//r^-^y .yA/dr" ^, \n\n\n\nHence. ds = Vdr^+rMv^ \n\n\n\nDIFFERENTIATION OF FUNCTIONS. \n\n\n\n103 \n\n\n\nAlso, \n\n^\'_ limit e^^\'^iim-^\'-^- \n\ndv Av^^o ^i, arc Q\'M \n\n\n\n= r lim \n\n\n\nQ\'M\' \n\n\n\n\nch. Q\'M \n\nIf the radius vector J^M coincides with the normal to \nthe curve at M, the corresponding tangent to the arc \nMQ\' will coincide with M T; \nand (\xc2\xa7 55) \n\n^ _ limit arc MM^ _ \n\ngiving ds = r^^^. \n\nIn this case dr = o, because \nthe motion of the generatrix at P B\\\\ \n\nthe point considered is perpendicular to the radius vector. \nAn important example of this case is a circle with the \nT pole at its centre. \n\nv^ Let BM = ^ be any arc \n\nof a circle, and BCM = v \nthe subtended angle. Then, \nsince the radius is always \nnormal to the arc, we have \nds ~ rdv. \n\nThat is, the differential of \nan arc of a circle regarded as a function of the corresponding \nangle at the centre y is equal to its radius into the differential \nof the angle. \n\nTo illustrate, assume MCQ = dv ; then will the arc MQ \n\xe2\x80\x94 rdv. The direction of the motion of the generatrix at \nany point is along the corresponding tangent to s ; hence, \nby laying off from M upon the tangent at that point a dis- \ntance M T =^ ds =z rdVj we have ds represented in measure \nand direction. \n\n\n\n\n\'104 DIFFERENTIAL CALCULUS. \n\nIn order to represent graphically the general case when \nds = Vdr\'^ + r^dv^, let BM be the given curve, I* the pole, \nM the assumed point, and MPM\' = dv. If r were con- \nstant, as we have seen in the case of a circle, MT\' = rdv \n\n\n\n\nwould be ds ; but, in general, ds is affected by a uniform \nchange in r, in the direction FM^ equal to dr. To deter- \nmine it we have \n\ndv Az^^^Och. (2 J/ \n\nAt T\' draw T\' T parallel to PM\\ then T\'T/T\'M = \ntan T\'MT = T\'T/rdzK Hence, dr/dv \xe2\x80\x94 rT\'T/rdv = \nT\'T/dv, and dr = T\' T. MT = ds=Vdr\' -f- rW\\there- \nfore, represents ds in measure and direction. \n\nIn order to express ds in terms of a single variable and \nits differential, find expressions for r and dr in terms of i> and \ndv, or an expression for dv in terms of r and dr, from the \npolar equation of the curve ; and substitute in the formula. \n\n93. Differential of a Plane Area in Terms of Polar \nCoordinates. \xe2\x80\x94 Let u represent the area of a varying portion \nof the surface generated by the radius vector PM revolving \nabout the pole P. Regarding zi as a function of v (\xc2\xa7 2;^), \n\n\n\nDIFFERENTIA TION OF FUNCTIONS. IO5 \n\nlet MFM\' = Av. The area MPM\\ represented by Hu, \nwill be the corresponding increment of u. Hence (\xc2\xa7 47), \n\n\n\ndu _ \n\n\nlimit ^u_ \n\n\nr\' \n\n\nand \n\n\nrMv \ndu = , \n\n\ndv \n\n\nAz^m->o /\\i) \n\n\n2\' \n\n\n\n\n2 \n\n\n\nTo illustrate, with FM = r, describe the arc of a circle \nMQ = rdv corresponding to MPQ = dv ; then du = r\'\'dv/2 \n= area of the circular sector MPQ. \n\n\n\n\ndu may be expressed in terms of v and d7\\ by substituting \nfor r its value in terms of v^ determined from the polar \nequation of the bounding curve. \n\n94. Motion. \xe2\x80\x94 When a point changes its position with re- \nspect to any origin it is said to be in motion with respect to \nthat origin. \n\nIn general, the distance from any origin to a point in \nmotion continually changes, and is a continuous function \nof the time during which the point moves. \n\nWhen the distance changes so that any two increments \n01 it whatever are proportional to the corresponding inter- \nvals of time, the distance changes uniformly with the time. \n\n* Motion, without regard to cause, is generally discussed under the \nhead of Kinematics, but many important applications of the Calculus \ninvolve motion, therefore some of the definitions and principles of \nKinematics are here and elsewhere introduced. \n\n\n\nI06 DIFFERENTIAL CALCULUS. \n\nThe point is then said to be moving uniformly^ or with uni- \nform motion with respect to the origin. \n\nIf the distance does not change uniformly with the time \nthe point is said to be moving with varied motion with \nrespect to the origin. \n\nA train of cars moves from a station with varied motion \nuntil it attains its greatest speed, after which its motion \nalong the track is uniform while it maintains that speed. \n\nWith uniform motion equal distances are passed over in \nany equal portions of time, and with varied motion unequal \ndistances are passed over in equal portions of time. \n\nLet s in both figures represent the variable distance from \nany origin, as A^ to a point moving on any line, as MNO \\ \n\n\n\n\nand let t denote the number of units of time during which \nthe point moves ; then s =f{t). \n\nli f(t) is of the first degree with respect to /, the distance \ns will change uniformly ; otherwise the point approaches \nor recedes from the origin with varied motion. \xc2\xa757. \n\nThe rate of change of s^ regarded as a function of /, cor- \nresponding to any position of the moving point, is called \nthe rate of motion of the moving point with respect to the \norigin ; and since uniform motion causes i- to change uni- \nformly with /, the rate of motion, in such cases, is constant. \n\xc2\xa7 59- \n\n\n\nDIFFERENTIA TION OF FUNCTIONS. \n\n\n\n107 \n\n\n\nIn varied motion the rate varies .with /, and is therefo/e \na function of /. \n\n95. If the differential of the variable is assumed equal to \nthe unit of the variable, the differential of a function and \nthe corresponding differential coefficient will have the same \nnumerical value. \n\n\n\nThus, if %- = 2, and dx \n\n\' dx \n\n\n\nI inch, we \n\n\n\nhave , dx^^ 2 \ndx \n\n\n\nB \n\n\n\ninches. In such cases the differential of the function ex- \npresses the rate in terms of the unit of the variable ; and \nsince it is more definite, it is frequently used instead of the \ndifferential coefficient. \n\nTo illustrate, let s denote any variable distance regarded \nas a function of time, giving \ns\xe2\x80\x94f{t). Assuming any con- \nvenient length to represent \nthe unit of t, we may, by sub- \nstituting s for y and t for x \n(\xc2\xa7 20), determine a line, as \nAM^ whose ordinate repre- \nsents the given function. \n\nds \nIf PR = dt represents one hour, \xe2\x80\x94dt = QB represents \n\nthe change that s would undergo in one hour, from the state \nrepresented by PA, were it to retain its rate at that state ; \nand is more definite than the corresponding abstract value \nof ds/dt. \n\n96. Velocity. \xe2\x80\x94 The differential coefficient of the variable \ndistance from any origin to a point in motion, regarded as \na function of the time of the motion, is called the velocity \nof the moving point with respect to that origin. \n\n\n\ny^ \n\n\n\n\nd8 \n\nQ \n\nR T\' \n\n\n\n\n\nP dt \n\n\n\nI08 DIFFERENTIAL CALCULUS. \n\nRepresenting the variable distance by s, and the velocity \nby v^ we have v = ds/dt. \n\nFor the reasons given above, velocity is measured by the \nproduct of ds/di and the distance assumed to represent the \nunit of time. \n\nThat is, the measure of the velocity of a point in motion \nat any instant, in any required direction, is the distance in \nthat direction that the point would go in the next unit of time^ \nwere it to retain its rate at thai instant. \n\nIt should be noticed that the distance referred to above, \n\nand represented by s, may or \nmay not be estimated along \nthe line or path upon which the \nbody moves. Thus, if a point \nmoves from A towards B^ and \nthe velocity at any point, as C, \nin the direction AB is required, \nthe distance s is estimated along the path described ; but \nif the rate or velocity with which a point, moving from \nA to B, is approaching D is required, s must represent the \nvariable distance from the moving point to D^ in order \nthat ds/dt shall be the required velocity. \n\nSince velocity is a rate of motion, it is constant in uni- \nform motion, and a variable function of time in varied \nmotion. \n\n97. Acceleration. \xe2\x80\x94 The differential coefficient of velocity \nregarded as a function of time is called acceleration. It is \ndenoted by dv/dt., in which v represents velocity. Since \nacceleration is the velocity of a velocity, it is generally \nexpressed in terms of the distance which represents the \nunit of time. \n\n\n\n\ni^^ \n\n\n\nDIFFERENTIA TION OF FUNCTIONS. IO9 \n\n98. Angular Motion. \xe2\x80\x94 Let C be a fixed point, CA a fixed \nright line, and B a point in motion so that the angle ACB, \n\n\n\n\ndenoted by 0, is changing. Then the line CB is said to \nhave an angular motion with respect to, or about, C. \n\nLet s represent the length of the varying arc, of any con- \nvenient circle, subtending 6^, giving B = s/r. \n\nBoth B and s are functions of the time during which CB \nmoves. \n\nAngular motion is uniform when any two increments of \nthe angle, or arc subtending the angle, are proportional to \nthe corresponding intervals of time ; otherwise it is varied. \n\n99. Angular Velocity.\xe2\x80\x94 The differential coefficient of \nany varying angle regarded as a function of the time is \ncalled angular velocity. \n\nRepresenting any varying angle by 6^, and its angular \nvelocity by 00^ we have 00 \xe2\x80\x94 d6/dt. \n\nIf s denotes the varying arc of a circle whose radius is r, \nwhich subtends ^, we have \n\n6 = s/r; hence, go = dB/dt = ds/rdf. \n\nThat is, angular velocity is equal to the actual velocity \nof a point describing any convenient circle about the \nvertex of the angle as a centre, divided by its radius. \n\nIt is customary in applied mathematics to consider the \nradius equal to the unit of distance used in any particular \ncase. Angular velocity will then be measured by the \n\n\n\nno DIFFERENTIAL CALCULUS. \n\nactual velocity of a point at the unit\'s distance from the \nvertex. \n\n100. Angular Acceleration.\xe2\x80\x94 The differential coefficient \nof angular velocity regarded as a function of time is called \nangular acceleration. It is denoted by dod/dt^ when gd \nrepresents angular velocity. \n\nPROBLEMS. \n\n1. The side of a square increases uniformly 3 in. a \nminute ; find the rate per minute of its area when its side \nis 6 in. \n\nLet X = side of square in inches, and u = area = x^; then dx/dt = \n3 in. min. and du/dx = ix. Hence (\xc2\xa7 77), du/dt = (du/dx)(dx/dt) = \n3 X 2x = 6x sq. in. min. and {du/di)x = 6 = 36 sq. in. min. \n\n2. The radius of a circle increases uniformly .01 in. per \nsecond ; find the rate of its area when the radius is i in. \n\nLet r = radius, and u \xe2\x80\x94 area = Ttr"^; then dr/d:( = .01 in. sec. \nand du/dr = 27m Hence (\xc2\xa7 77), du/di = .01 X 27tr = .027tr sq. in. \nsec; and {du/dt)r = i = .027t sq. in. sec. \n\n3. Find the rate of the radius when the area of a circle \nincreases uniformly at the rate of 2 7rr sq. in. sec. \n\nAns. I in. sec. \n\n4. The radius of a sphere increases uniformly .0491 in. \nsec; find the rate of its volume when the radius is 1.5 ft. \n\nAns. 200 cu. in. sec. \n\n5. The volume of a sphere increases uniformly 500 cu. \nin. sec; when its radius increases at the rate of 2 in. sec, \nfind the radius. Ans. 4.45 in. \n\n6. The area of a rectangle increases uniformly 100 sq. in \nmin. Its base and altitude are increasing at the rates of \n3 and 7 in. per min. respectively ; find the area when the \naltitude is double the base. Ans. 118.34 sq. in. \n\n\n\nDIFFERENTIATION OF FUNCTIONS. \n\n\n\nIll \n\n\n\n7. The diameter of a circle increases uniformly 3 in. \nsec; find the difference between the rates of the areas of \nthe circle and its circumscribed square when the square \nis I sq. ft. Ans. 15.45 sq. in. sec. \n\n8.* A man 6 feet in height walks away from a light 10 feet \nabove the ground at the rate of 3 mi. per hour. At what \nrate is the end of his shadow moving, and at what rate does \nhis shadow increase in length ? \n\nLet X = AM \xe2\x80\x94 distance from foot of light to man, y \xe2\x80\x94 AB = dis- \ntance from foot of light to end of shadow, and s = MB = length of \nshadow. Let ( = number of hours. \n\n\n\n\nThen we have c/x/dl\xe2\x80\x943 mi. hr. ; ^Z = io ft.; MC= 6 ft. \nand it is required to find dy/di and ds/dt. \n\nThe similar triangles ABI^nA DCL give x ; y \\: ^ \n\ny = 5x/2, and dy / dx == 5/2. Therefore (\xc2\xa777) \n\n\n\nDL=4 ft.; \n10; hence, \n\n\n\ndy/dt = (dy/dx){dx/dt) = (5/2)(3) = 7.5 mi. hr. \nAlso, X : s :: 4 : 6 ; hence, j- = 3^/2, and ds/dx = 2/^> \nand ds/dl ~ {ds/dx){dx/dl) = 1.5 X 3 == 4-5 mi. hr. \n\n9.* A vessel sailing south at the rate of 8 mi. per hour is \n20 mi. north of a vessel sailing east at the rate of 10 mi. an \nhour. At what rate are they separating at the time ? At \nthe end of i^ hrs. ? At the end of 2^ hrs. ? When are they \nneither separating from nor approaching each other ? \n\n\n\n* Rice and Johnson\'s Calculus. \n\n\n\n112 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\n\nLet t = time in hours from the given epoch. \n\nLet AB = y = 20 \xe2\x80\x94 8^ = distance of first ship \nfrom EC t hours after the given epoch. \n\nLet^C\xe2\x80\x94 x=io/=clistance of second ship from \nBA at the same time. \n\n\n\nLet M =AC \nGiven, \n\n\n\nVx^+/ = V400- 32CV+l64/a. \n\n\n\ndt \n\n\n\nhr. \n\n\n\ndx \ndt \n\n\n\nmi. \n\n10\xe2\x80\x94-. \n\nhr. \n\n\n\n^ . , du \nRequired, \xe2\x80\x94 \n\ndt \n\n\n\n\xe2\x80\x94 160 -|- 164/ \n\n\n\nldu\\ mi. ldu\\ _ 1 mi. ldu\\ __ \n\n\\^// = o^~ hr7\' \\di)t=^i\'^K-]hr~.\' \\dtJt = ^o-^\' \n\n\n\n1/400 \xe2\x80\x94 320/ -f- 164/\'* \n\nmi. ldu\\ \n\n7 hr. \\aiit = ^o \n\nThe follovi^ing general outline of steps may assist the stu- \ndent in solving similar problems : \n\n1\xc2\xb0. Draw a figure representing the magnitudes and direc- \ntions under consideration ; and denote the variable parts \nby the final letters of the alphabet. \n\n2\xc2\xb0. Write, with tjie proper symbols, all known data ; and \nindicate the symbols for the required rates. \n\n3". From the relations between the magnitudes find an \nexpression for the function whose rate is required, in terms \nof the variable. \n\n4\xc2\xb0. Differentiate and determine values or expressions for \nthe required rates. \n\nIn case an explicit function of a variable cannot be \nfound, make use of the principles in \xc2\xa7 77. \n\n10. x^ \xe2\x80\x94 2pz is the equation of q Xl \n\na parabola OM. A point starting \nfrom O moves along the curve in \nsuch a manner that z = 16.1/^; in \nwhich z is expressed in feet, and / in \nseconds. Find the rate of x with \nrespect to /. \n\n\n\n\nDIFFERENTIATION OF FUNCTIONS. II3 \n\n\xe2\x96\xa0b dz \n\n\n\n\n\ndx \ndz " \n\n\np \n\n^/opz \n\n\ndx \n\ndt \n\n\ndx \n~ dz \n\n\n^ df \n\n\n\n1/32. 2/ /2 di \n\n\n\n32.2t. \n\n\n\nHence, ^ = \xe2\x80\x94 X -j,= --=== X 32. 2^ \xe2\x80\x94 4/32.2/. \n\nyi2.2pf\'- \n\n11. One ship was sailing south 6 mi. per hour, another \neast 8 mi. per hour. At 4 p.m. the second crossed the \ntrack of the first at a point where the first was 2 hrs. be- \nfore. How was the distance between the ships changing \nat 3 P.M. ? When was the distance between them not \nchanging? Ans. 2.8 mi. hr. ; 3 hr. 16 min. 48 sec. \n\n12. A ship is sailing south 60\xc2\xb0 east, 8 mi. per hour ; find \nthe rate of her latitude and longitude. \n\nAns. 4 mi. hr.; 4I/3 mi. hr. \n\n13. A point P moves in a straight line away from a point \nB dX the rate of 8 mi. hr.; find its velocity with respect to \na point C situated upon the perpendicular to the line BP \nthrough B and at 100 ft. from B when BP \xe2\x80\x94 50 ft.; when \nBP \xe2\x80\x94 150 ft. Ans. 8/V\'5 mi. hr.; 24/V13 mi. hr. \n\n14. If the diameter of a sphere increases uniformly at the \nrate of i/io inches per second, what is its diameter when \nthe volume is increasing at the rate of 5 cubic inches per \nsecond ? Ans. \\o/y n in. \n\n15. If the diameter D of the base of a cone increases \nuniformly at the rate of i/io inch per second, at what rate \nis its volume increasing when D \nbecomes 10 inches, the height being \nconstantly one foot ? \n\nAns. 271 cu, in. sec, \n\n16. The base of a right triangle \nis 4 mi. ; its altitude is variable and ^/_ \ndenoted by j\', and is the variable \nangle opposite to 7. Corresponding to _;^ = 2 mi. find the \n\n\n\n\n114 DIFFERENTIAL CALCULUS, \n\nrate of 0, first as a function of j, then as a function of tan \n0. Explain the difference between the two results. \n\nAns. 1/5; 4/5. \n17. A train is running from A to B at the rate of 20 mi. \nan hour. The distance from ^4 to C on a perpendicular to \nAB is 2 mi. Find the rate of the angle at C included \nbetween CA and a right line from C to the train. \n\nLet (p = variable angle at C, and j = mi. from A to train. \nThen CA \xe2\x80\x94 1 mi., and dyjdt = 20 mi. hr. \n\n\n\ny = CAta.n(p. .\', (p= ta.n-\'^ -p\xe2\x80\x94 and \n\n\n\n^/ yiC\'+y 4+r* \n\n\n\n18. Find the rate of the surface and volume of a sphere \nwhen its radius decreases at the rate of 2 ft. per minute. \n\nAns. \xe2\x80\x94 i67rr sq. ft. min.; \xe2\x80\x94 Zttt^ cu. ft. min. \n\n19. A ball of twine rolls along a floor in a right line at \nthe rate of 4 mi. per hour. One end of it is 30 feet above \nthe floor and is attached to the top of a pole. At what rate \nis the ball unwinding when it is 40 feet from the bottom of \nthe pole on the floor ? Ans. 3.2 mi. hour. \n\n20. A ladder 20 ft. in length leans against a wall ; if the \nbottom is drawn out at the rate of 2 ft. per second, at what \nrate will the top descend when the bottom is 8 ft. from the \nwall? Ans. 10.5 in. sec. \n\n21. The side of an equilateral triangle increases at the \nrate of 2 in. per minute. Find the rate of its altitude, and \nthe rate of its area when the side is 10 in. \n\nAns. 1^3 in. mi. ; 10 1/3 sq. in. min. \n\n\n\nDIFFERENTIAl\'lON OP PUNC7F0NS, I15 \n\n22. Two straight\' railways intersect at an angle of 60\xc2\xb0. \nAn engine approaches the intersection on one of the tracks \nat the rate of 25 mi. per hour, and on the other track an \nengine is leaving it at the rate of 30 mi. per hour. At what \nrate are the engines separating when each is 10 mi, from \nthe intersection ? Ans. 2.5 mi. hr, \n\n23. A man walking on a horizontal plane approaches the \nfoot of a pole 60 ft. in height, with a constant rate. When \nhe is 40 feet from the foot how will the rate with which he \napproaches the top compare with that with which he ap- \nproaches the bottom ? How far will he be from the foot \nwhen he is approaching it twice as fast as he is the top ? \n\n^ Ans. 2 I/1/13, V1200 ft. \n\nFUNCTIONS OF TWO OR MORE VARIABLES. \n\nloi. The Partial Differential of a Function of Two or \nmore Variables, with respect to one of the variables, is the \nchange that the function would undergo from any state, \nwere it to retain its rate at that state, with respect to that \nvariable, while that variable changed by its differential. \n\nThe Total Differential of a Function of Two Variables \nis the change that the function would undergo from any \nstate, were it to retain its rate at that state, with respect to \neach variable, while both variables changed by their differ- \nentials. \n\nAny function of two variables which changes uniformly \nwith each variable has a constant rate with respect to each, \nand its form must be some particular case of the general \nexpression Ax -\\- By -f- C (\xc2\xa7 64). \n\nRepresenting such a function by z^ we have \n\nz^Ax^By^C, (i) \n\n\n\nIl6 DIFFERENTIAL CALCULUS. \n\nIncreasing x and y by their differentials, and denoting \nthe corresponding new state of the function by z\\ we have \n\nz\' = A{x-\\-dx)^B[y^dy)-^C. ... (2) \n\nSubtracting (i) from (2), member from member, we have \n\nz\' \xe2\x80\x94 z= Adx + Bdy. \n\nSince the function z changes uniformly with respect to \neach variable, the total differential of it, denoted by dz, is \nequal to the corresponding change in the function. \n\nTherefore, dz = Adx -\\~ Bdy. \n\nAdx is the corresponding partial differential of the func- \ntion z with respect to x ; and Bdy is the same with respect \nto y. \n\nHence, the total differential of any function of two vari- \nables^ which changes uniformly ivith respect to each^ is equal to \nthe sum of the corresponding partial differentials. \n\nThe total differential of any function of two variables \nwhich does not vary uniformly with each variable is not, in \nin general, equal to the corresponding change in the func- \ntion, but it is equal to the corresponding change of a func- \ntion having a constant rate with respect to each variable, \nequal to that of the given function at the state considered. \nIn other words, the total differential is equal to that of a \nfunction which changes uniformly with each variable, and \nwhich has at the state considered its partial differentials \nequal to the corresponding partial differentials of the given \nfunction. \n\nHence, the total differential of any function of two vari- \nables is equal to the sum of the corresponding partial differen- \ntials. \n\n\n\nDIFFERENTIATIOM OF FUNCTIONS. HJ \n\nIn a similar manner it may be shown that the total differ- \nential of any function of any number of variables is equal to \nthe sum of the corresponding partial differentials. \n\nIn order to distinguish between a total and a partial dif- \nferential the symbol 9 is used to indicate a partial differen- \ntial. Thus having z = f{x, _y), then \n\ndz = {\'dz/dx)dx -\\- {dz/dy)dy, \n\nin which dz represents the total, \'dz a partial, differential. \n\nEXAMPLES. \n\n1. d{xy) = xdy-\\-ydx. \n\n2. d{2ax\'^y \xe2\x80\x94 2y^ + S\'^-f^ \xe2\x80\x94 5) = taxydx -\\- c^bx\'^dx -\\- 2,ax^dy \xe2\x80\x94 ^ydy. \n\n3. d\\{x ^y)l{x -y)-\\ = [2{xdy - ydx)]/{x -y)\\ \n\n4. dixYz") = 2f2^xdx + 2x\'^zydy + HxYzdz. \n\n5. dt2in-\\y/x) = (xdy--ydx)/{x\'Jrf)\' \n\n6. ^sin (xy)] = cos {xy){ydx-{- xdy). \n\n7. d log {xy) \xe2\x80\x94 ydx/x -\\- log xdy. \n\n8 dy^^^^ =y^^^ [\\og y co^ xdx-\\- {sm xdy/ y)]. \n\n\n\n9, d vtrs\\n~\\x/y) = {ydx \xe2\x80\x94 xdy)/{y^2xy \xe2\x80\x94 x^), \n10. d sin {x -\\-y) = cos (x -\\-y){dx -\\- dy). \n\n\n\nII. Deduce the formula ds =ydr^ + rVz/\' (\xc2\xa7 92) from the formulas \n\n\n\n, [Anal. Geo.,] and ^^ = ^dx^ -\\- dyU%Sy). \n\n12. One side of a rectangle increases at the rate of 3 in. per second \nand the other decreases at the rate of 2 in, per second. Find the rate \nof the area when the first side is 10 in. and the second 8 in. in length. \n\nAns. 4 sq. in. sec. \n\n102. In order to represent ^^=(aV^jt:)^ji:H-(aV^)^-..(i) \ngraphically, let any state of the function z be represented \nby the ordinate JVM of a surface. \xc2\xa7 27. \n\n\n\nIi8 \n\n\n\nD IFFERENTIA L CA L Ct/L t/S. \n\n\n\nThrough JVM pass the planes J/7V7? and MJVP parallel, \nrespectively, to ZX and ZV. Let MC he the intersec- \n\nW \n\n\n\n\ntion of the surface by the plane MNR, and let MB"be the \ntangent to it at M. Let M\xc2\xa3 be the intersection of the sur- \nface by the plane MNP, and let ML be its tangent at M. \nMH and ML determine the tangent plane to the surface at \nM. \n\nAssume dx = NR, and dy = JVF. Complete the par- \nallelogram NS and the parallelopipedon NQ. Produce \nRK, PA and SQ. \n\nThen, \\.2.n KMH =dz/dx, and KH = {dz/dx)dx. \n\ntan AML = dz/dy, and AL \xe2\x80\x94 {dz/dy)dy. \n\nDraw HB parallel to KQ. Connect A and D by AD, \nand draw LT parallel to AD. \n\nThen, QD = KH, and DT=AL. \n\nTherefore, QT =^ {dz/dx)dx + idz/dy)dy = dz. \n\n\n\nDIFFERENTIATION OF FUNCTIONS. II9 \n\nZr is parallel to AD, which is parallel to MH. Z" there- \nfore lies in the tangent plane, and MT is the tangent to MB^ \nthe line of intersection of the surface by the plane MNS. \n\nWhile X and y are changing as assumed, the foot of the \ncorresponding ordinate passes with uniform motion from A \nto S, and QT vi the total differential required by definition. \n\nAs in the case of a function of a single variable, the ex- \npressions \'dz/dx and dz/dy are, respectively, independent of \ndx and dy, but for any state of a function of two variables \nthere is no fixed total differential coefficieitt. In the figure. \n\n\n\nQT/QM= tan QMT = dz/ Vdx\' + d/ \n\nis the total rate of change of z, corresponding to the values \nassumed for dx and dy, but any change in the relative value \nof dx and dy will change this total rate. For any values \nassumed for dx and dy, the total rate of change of the func- \ntion, and therefore its total differential, will be the same as \nthat of t/ie ordinate of the line cut out of the tangent plane \nby the plane through NM, whose trace on XY makes with \nX an angle = tan"\' dy/dx. \n\nIt is now apparent that the function represented by the \nordinate of the tajigent plane at M is the one with a constant \nrate with respect to each variable, whose total differential for \nthe same values of dx and dy is the same as that of the \ngiven function at the state corresponding to M. \n\ndz/ Vdx^ -{- dy\'\' = tan QMT is the tangent of the angle \nwhich a tangent to MB makes with XV, and (\xc2\xa765) its \nnumerical value is the slope of the surface at M along MB. \n\nFrom (i), we write \n\ndz/dx = dz/dx + (dz/dy) (dy/dx), \n\n(2) \n\ndz/dy = dz/dy + (dz/dx){dx/d_y), \n\n\n\nI20 DIFFERENTIAL CALCULUS. \n\nthe first members of which depend upon the relative value \nof dx and dy. \n\nDraw 7"/^ parallel to SR. Then MW is the projection \nof MT upon the plane MNR, dz/dx = tan KMW, and \ndz/dy is the tangent of the angle which the projection of \nMT upon the plane MNP makes with Y. \n\n\'dz/dx \xe2\x80\x94 tan KMH is the partial differential coefficient \nof the function with respect to x only, and is independent \nof dx and dy. It is equal to the tangent of the angle which \nthe intersection of the tangent plane, and any plane parallel \nto ZX, makes with X or the plane XY. Whereas dz/dx = \ntan KMW is dependent upon the quotient dx/dy (Eq. 2), \nwhich has no definite value because x and y are independ- \nent, and their differentials arbitrary. \n\nIf, however, by means of a second equation (p{x,y) = o, \nx and y are related, thus making one of them dependent \nupon the other and determining dy/dx, then z becomes a \nfunction of a single independent variable ; its graph becomes \nthe line of intersection of the surface z=^/(x,y) by the \ncylinder (p{Xfy) = o, dz/dx becomes the differential coeffi- \ncient of z regarded as a function of x and is equal to tan ^, \nd being the angle between the axis of X and the projection \nof the tangent to the graph on XZ. \n\nSimilarly both members of Eq. i may be divided by dt^ \nt being any independent variable, giving \n\ndz/dt= {\'dz/dx){dx/dt) + {dz/dy){dy/dt). \n\nThe values of both members will now depend upon the \nvalues assumed (\xc2\xa7 67) for dx, dy, and dt. \n\nBut if we also write (p{x, t) = o and ^{y, t) == o, thus \nrelating x and y to /, and determining dx/di and dy/di, \nthen z becomes a function of one independent variable, \nand the first member becomes the differential coefficient of \nz regarded as a function of /. \n\n\n\nSUCCESSIVE DIFFERENTIA riON, 121 \n\n\n\nCHAPTER VI. \nSUCCESSIVE DIFFERENTIATIONo \n\nFUNCTIONS OF A SINGLE VARIABLE. \n\n103. In general the differential coefficient, and therefore \nthe differential of any function of a variable, are functions \nof the variable and may be differentiated. \n\nThus, having ax^ ^ \n\ndax^ = ^^ax^dx^ and dax^/dx = T^ax^. \n\nDifferentiating again, denoting d{dax^) by d^ax^, read \n^^ seco?td differential of ax^," and representing {dxY by dx"^, \nwe have \n\nd{dax^) = d^ax^ = Saxdx"^, and d\'^ax^ /dx^ = dax. \n\n(iaxdx^ is the differential of -^ax^dx, which is the differ- \nential of ax^ . daxdx^ is therefore called the yfri-/ differential \nof T^ax^ix and the second differential of ax^. \n\nSimilarly, dax is the fii\'st differential coefficient of Ty\'^x\'^ \nand the second differential coefficient of ax:\\ \n\nDifferentiating again and extending the notation, we \nhave \n\nd(d\'\'ax^) = d^ax^ \xe2\x80\x94- Sadx"", and d^ax^/dx^ = 6a. \n\n6adx^ is theyfri/ differential of daxdx"^, the i"<fr^;z^ differ- \nential of T,ax\'^dx, and the //^/r^y differential o{ ax^. 6a is the \njfirsf differential coefficient of 6ax, the second of T^ax^^ and \n\nthe tJiird of ax\\ \n\n\n\n122 DIFFERENTIAL CALCULUS. \n\nRepresenting the function x^ by j\', we have j^ = x^. \nDifferentiating, we obtain \n\ndy = nx^\'\'^dx, and dy/dx = nx\'^~^, \n\nfor the ^r^/ differential and differential coefficient, respec- \ntively, of x^. \n\nDifferentiating again, we have \n\nd\'^y = n{n \xe2\x80\x94 i)x\'^~^dx^^ and d\'^y/dx^ = n(n \xe2\x80\x94 i)jc""^, \n\nfor the second differential and differential coefficient, \nrespectively, of x"^. \n\nAgain, d^y = n(n \xe2\x80\x94 i){n \xe2\x80\x94 2)x\'^~^dx^y \n\nand d^y/dx^ = n(n \xe2\x80\x94 i)(n \xe2\x80\x94 2)x\'^~^, \n\nfor the ^Ai\'rd differential and differential coefficient \nrespectively, of x\'^. \n\nSimilarly, the fourth, fifth, etc., differentials may be \nderived in succession. \n\nIt should be observed that the symbol d^y/dx^, which \nrepresents the second differential coefficient, is the differ- \nential coefficient of the symbol dy/dx, which represents the \nfirst differential coefficient. \n\n\n\nThus, -^^ = dV-^\\ I dx. \n\n\n\nSi.,i,\xe2\x80\x9e,\xe2\x80\x9e g = .(g)//.. \n\nSuccessive differential coefficients are also called derived \nfunctions or derivatives, and are frequently represented in \norder by accents on the functional letter. \n\n\n\nSUCCESSIVE DIFFERENTIATION. 1 23 \n\nThus, if/(^) represents the primitive function, then \n\nf\\x\\ f\\x\\ r\\x\\ etc, \n\ndenote respectively the first, second, third, etc., deriva- \ntives oi f(x). \n\nOther forms are also used. Thus, having jj^ =1 f(oc)y \n\n\n\nD^y, DxJ, Dxy, etc., \n\nrepresent the successive derived functions in order. \n\nEach successive differential coefficient or derivative in \norder is the rate of change of the immediately preceding \none, and d\'^~\'^y / dx^~^ is an increasing or decreasing function \nof Xj according as d^y/dx^ is positive or negative. \xc2\xa7 71. \n\nLet y =/(x) = x\'^f n being entire and positive. \n\nThen dy/dx = f{x) = D^y = nx^\'K \n\nd^/dx\' =f\\x) = Z>^> = n{n - i)x\'\'-\\ \netc. etc. etc. \n\ndy/dx"" = /"(jc) = D^\'y = n{n - i){n - 2) . . . 2.1. \n\nd\'\'+\'^y/dx\'\'+\' = r-^\\x) = JD/+\'y = o. \n\nIt should be observed that the symbols dy/dx^, /"(\xe2\x80\xa2^)> \nDx^, etc., serve only to represent expressions, and to \nindicate their relations to the primitive function. In the \nabove d\'^y/dx^ denotes that n{n \xe2\x80\x94 i)^""^ is the second differ- \nential coefficient of x^. \n\nThe differential of any order is the product of the cor- \nresponding derivative and power of the differential of the \nvariable. \n\nThus, dy = {d\'\'y/dx\'^)dx^=f\'\'{x)dx\'\' ... (a) \n\nrepresents a differential of the n^\'^ order, and d\'^y/dx\'^=^f\'\'{x) \n\n\n\n124 DIFFEKENTIAL CALCULUS. \n\nrepresents a differential coefficient or derivative of the \nn^^ order. \n\nDividing both members of (a) by ^x"~\\ dx^~^y etc., in \nsuccession, we have, in order, \n\n(ty/dx\'\'-\'^ = d^d^\'-^y/dx\'\'-^) = f\'\'{x)dx. \n\nd^y/dx""-^ = d\\d\'-\'\'y/dx\'\'-^) = /\'\'{x)dx\\ \netc. etc. etc. \n\nd^\'y/dx\'\'-\'\' = d\\d\'\'-\'\'y/dx^-\'\') = f\'\'{x)dx\\ \netc. etc. etc. \n\nd^y/dx = d^\'-^idy/dx) = f\'(x)dx\'\'-\\ \n\nFrom which we see that the product of a derivative of \nany order by the ^rsf power of the differential of the \nvariable is the Jirsf differential of the immediately preced- \ning derivative, and its product by the second power of the \ndifferential of the variable is the second differential of the \nsecond preceding derivative, etc. The product of a deriv- \native of the n^"^ order by the {n \xe2\x80\x94 i)"^^ power of the differ- \nential of the variable is the {ii \xe2\x80\x94 \\Y^ differential of the first \nderivative. \n\nEXAMPLES. \n\n\\, y \xe2\x96\xa0=. ax*, dy = ^ax^dx, d^y = \\2ax\'^dx\'^y \n\nd^y =z 2i\\ax dx^, d^y = 24a dx*. \n\n2. /(x) = (a- x)-K fix) = {a- x)-\\ f\'{x) r-. 2{a - x)-\\ \n\n/^(x) = 2.3 . . . n(a \xe2\x80\x94 jr)-"-i, etc. \n\n3. J = sin X. Dxy = cos x, Dxv = \xe2\x80\x94 sin x, \n\nDx^y = \xe2\x80\x94 cos X, L)x*y = sin x, \n\netc. etc. \n\nThe exponent of the power of a function is diminished by \nunity at each differentiation (\xc2\xa7 79), and when entire and \n\n\n\nSUCCESSIVE DIFFERENTIATION. 1 25 \n\npositive it will finally be reduced to zero. Hence, algebraic \nfunctions which do not contain fractional or negative ex- \nponents affecting the variable have a limited number of \nderivatives. All others, including transcendental functions, \nhave an unlimited number. \n\n4. f{x) = ax^ + bx"^. \n\nf\'{x) = 3^x2 -j- 2dx, f"{x) = tax -\\- 2^, /\'" {x) = 6a. \n\n5. y = ax^. \n\ndy ^ -h d-\'y ^ -i d^y ^ -f ^ \n\n~^^\\ax , _ = -iax , ^3 = 1-^ . etc. \n\nt. y \xe2\x96\xa0= a^. \nDxy = a^ log a, n/y = a^ (log a)^ etc., Dx\'y = a^ (log a)\xc2\xbb. \n\n\n\n7. \n\n\ny- \n\n\n= cos X. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\ndy \ndx \n\n\n= - \n\n\n- sin X \n\n\n\xe2\x96\xa0 = cos(x + \n\n\nly \n\n\nd^y \ndx^ ~ \n\n\n\xe2\x80\x94 cos Jf= \n\n\ncos( X \n\n\n+ \n\n\n^)- \n\n\nd^y \ndx\' \n\n\n= 5 \n\n\nsin X - \n\n\n= cos(. + ^), \n\n\netc., \n\n\ndy _ \n\ndx\'^ \n\n\n: COSj X \n\n\n^ + \n\n\nv)- \n\n\n8. \n\n\ny-- \n\n\n\xe2\x80\x94 log \xe2\x80\xa2 \n\n\nX. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\ndy \ndx \n\n\nI \nx\' \n\n\nd\'^y I \ndx\'\' ~ x^\' \n\n\nd^y \ndx^ \n\n\n2 \n\n\nd^y _ \ndx" ~ \n\n\n2.3 \n\nx"^ \' \n\n\netc. \n\n\n\n\n\n\n\n\ndy/dx^ \n\n\n= {- \n\n\ni)n-i\\n \n\n\n- l/xn. \n\n\n\n\n\n\n\n\n9- \n\n\n<P \n\n\n\xe2\x80\x94 cos- \n\n\n-^u. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nd(p/du = - 1/ \\/i-u\\ d\'\'<p/du^ = - u/{i - \xc2\xab2)3/2^ etc \n\n10. /{x) = sin mx. \n\nf\\x) = m cos ;\xc2\xabjr, f"{^) = \xe2\x80\x94 ^"^ sin /\xc2\xab;r, etc. \n\np\\x) = (- i)\'^w2\xc2\xab sin mx, /^^+\\x) = (\xe2\x80\x94 i)\'^w2\xc2\xbb^+i cos mx. \n\n11. y = e^^. \n\ndv d^v d^v \n\nJ_ _- ae<^x^ _JL \xe2\x80\x94 a\'igax^ etc. . . . \xe2\x80\x94 ^ = a\'^e\'^^. \n\ndx dx\'\' dx\'^ \n\n12. / = log {e^ + e-^). d^y/dx^ \xe2\x80\x94 - %{e^ - e-^)/{e^ \\- e-^f. \n\n13. jF = cos mx. dy/dx\'\'^ = m\'"\' cos {7?tx -\\- nil J 2). \n\n14. y \xe2\x80\x94 cos\'^ ;<:. dy/dx^ = 2^\'\'^ cos (2;c -|- nit 1 2), \n\n\n\n126 DIFFERENTIAL CALCULUS. \n\n15. y = sin X. dy/c/x\'^ = sin {x -\\- mt/2). \n\n16. JJ/ = (l + x)/{\\ \xe2\x80\x94X). d^/dx^ =r 2| V(l - xY+^. \n\n17. jj/ = tan X. d^yfdx^ = 6 sec* x \xe2\x80\x94 4 sec^ x, \n\n18. J = \\/2px. d\'^y/dx"^ = \xe2\x80\x94 //(2/>x)=*\'2 \xe2\x80\x94 _^2/y. \n\n19. /(^) = x\'/{i - x). /iv (^) = 24/(1 - xf. \n\n20. y = e^ cos ^. d^y/dx\'^ = 2^1\'^e^ cos (jc -|- nit/^. \n\n21. J = \xc2\xb1 j/i?^ \xe2\x80\x94 ^2. ^ \xe2\x80\x94 \xe2\x80\x94 \n\n\n\ndx\'\' \xc2\xb1 {R-\' - x-\'f^\'\' \xc2\xb1 y \n\n22. f{x) = tan X -f- sec :r. /"{x) = cos x(i \xe2\x80\x94 sin x)-^. \n\n23. jf/ =;>r^. dy/dx^ = ^^(i + log xf + X* -1. \n\n24. /(x) = x^ log X. /i^ (x) =r 6x-i. \n\n25. ^ = sin-y d^x/dy* = {qy + 6/)/(i -y)7/2 . \n\n26. jK = log sin X. Dxy = 2 cos ar/sin^ ;\xc2\xbb;. \n\n\n\n27. JJ/ = |/sec 2;f, d^/dx^ = 3(sec 2x)^\'^ \xe2\x80\x94 (sec 2xyi^. \n\n28. j)/ = {x^-\\-a\'\')ian-\\x/a). d^y/dx^ \xe2\x80\x94 ^a^/{a? + x^- \n\n29. _j/ = sec X. d\'^y/dx\'^ = 2 sec^ x \xe2\x80\x94 sec x, \n\nd^y/dx^ = sec Jf tan jr(6 sees x \xe2\x80\x94 l). \n\n\n\n30. \n\n\n;^ = \n\n\n^Jl-lJog X. \n\n\ndy/dx\xc2\xbb \n\n\n\' = \\n - \\/x \n\n\n\xe2\x80\xa2\xe2\x80\xa2 \n\n\n31. \n\n\n/(^) \n\n\n= (2ax)^\'b. \n\n\n/"(2) = \n\n\n4(a-&)/^(^ _ \n\n\n^)a(a+&)/&/^2. \n\n\n32. \n\n\n>\' = \n\n\ntan-i (i/x). \n\n\n.*. X = \n\n\ncoiy. \n\n\n\n\n\n\ndy _ \ndx \' \n\n\n,. I \n\n\n- sin^y, \n\n\nI \n\n\n\n\n\n\n-i + x\'^ - \n\n\n\xe2\x80\xa2\xe2\x96\xa0 (1+^ \n\n\n2)n/8 - Sin"j/. \n\n\n\n\ndiy \ndx-" \n\n\nd sin^ y \n\n~ dx \n\n\n2 siny cosy dy \n\n\n= sin 2y sin^jv. \n\n\n\n\n\n\ndx \n\n\n\nd^y ^(sin 2j)/ sin^ j) sin 2jj/ 2 sin y cos ;j/ <^^ 4" ^i"^ y cos 2;j/ 2c/i/ \n^jf*^ dx dx \n\n= 2(sin jF)(sin 2y cos j -|- cos 2y sin }^{dy/dx) \n=r \xe2\x80\x94 2 sin^_y sin 3^, \n\n\n\n12/ \n\n\n\nS UCCESSI VE D IFFEREN TIA TION. \n\nSimilarly, d^yldx^ = |3 sin* j sin 4^, \n\nand d^yldx^ = (\xe2\x80\x941)\'\' \\n \xe2\x80\x94 i sin\'^/ sin ny. \n\nSince tan-^ x = 7t/2 - tan"\' (i/x), \n\nwe have ^"(tan-^ x)/dx\'^ = (- i)\xc2\xab-\'|W\xe2\x80\x94 \xc2\xa3 sin\xc2\xbb^ y sin ^zy, \n\n1 n/2 \n\nor ^\xc2\xbb^(tan-J ;*:)/^x\xc2\xbb^ = (- jY-i \\n - i sin (w tan-ii)/(i +;f2) . \n\n\n\n34. J = aV(\xc2\xab\' + \xe2\x80\xa2^\')\' \n\n35. _y= aV2 + C"/+C". \n\n36. ;/=^ + ^(x-af/^ \n^j//^;p = 3^/5(x - \xc2\xab)2/5. \n\n\n\n^7/^;c = a + C", of V^;\xc2\xab^"^ = C- \nd\'^y/dx\'^ = - t>c/2SKx - dfl^. \n\n\n\n37. fx=x\'^\xc2\xb1 X^/\\ f\'x = 2X\xc2\xb1^X^\'\'\'l2, f\'x = 2\xc2\xb1 15x1/74. \n\n38. fx = e\'/\\ f\'x = - //7x2, f\'x = (2x + i)e\'^yx\'\' \n\n\n\n39. fx =g \n\n\n\n\xe2\x96\xa0l/x \n\n\n\n/\'x=e-\'/yx\'. f\'x = e-\'\'\\Y - 2x)/x\\ \n\n\n\n40. The relation between the time, denoted by /, and the distance, \nrepresented by s, through which a body, starting from rest, falls in \na vacuum near the earth\'s surface, is expressed very nearly by the \nequation s =16.1^\'^; s being in feet and i in seconds. Construct a table \ngiving the entire distance fallen through in i second; in 2 seconds; \nin 3 seconds; and in 4 seconds; the distance passed over during each \nof the above seconds; the velocity and acceleration at the end of each. \n\n\n\nTime in \nSeconds. \n\n\nEntire Distance \nin Feet. \n\n\nDistance each \nSecond. \n\n\nVelocity. \n\n\nAcceleration. \n\n\nI \n2 \n\n3 \n4 \n\n\n16.I \n\n64.4 \n\n144.9 \n\n257-6 \n\n\n16.I \n\n48.3 \n\n80.5 \n\n112. 7 \n\n\n32.2 \n\n64.4 \n\n96.6 \n\n128.8 \n\n\n32.2 \n32.2 \n32 2 \n32.2 \n\n\n\n128 DIFFERENTIAL CALCULUS, \n\n41. Having j- = 5/^, find the velocity and acceleration when / = \nseconds; if = 3 seconds. Ans, F\'i; = 2 = 3\'/io/2. Vt=2>= sV^S/\'^- \n\nAt = 2 = 31/572/4. ^^=3 = 31/573/4. \n\n\n\n42. _)/=sin (w sin-i jr). dy/dx\xe2\x80\x94m cos (w sin-^ \'^)/Vi \xe2\x80\x94 ^\'^\' Hence, \n\n(i - x\'\'){dyjdxf = m^ cos2 (m sin-i x) = m\\i -/). \nDifferentiating again and dividing by 2dy, we have \n(l - x^){d-^y/dx^) - x{dy/dx) + my = O. \n\n43. fx = sinh X. f"x = sinh ;f. \n\n44. fx = cosh X. /"x = cosh ;r. \n\n104. Leibnitz\'s Theorem \xe2\x80\x94 Let ji^ = uv, in which ?^ and v \n\nare any functions of x ; then (\xc2\xa7 75) \n\ndy/dx \xe2\x80\x94 2^ dv/dx + Z\' du/dx^ \nd\'^y/dx\'\' = u d\\^/dx\' + 2{du/dx)(dv/dx) + z\' d\'\'u/dx\\ \n\nd\'\'y _ d^ du d\'^v d\'^u dv d^u \ndx\' ~ ^dx\' ^^Jxd?~^^~d?dx~^ ^dx\'\' \n\nin which the numerical coefficients follow the law of those \nof the binomial formula. By a method similar to that used \nin deducing that formula for positive entire exponents it \nmay be shown that \n\ndx"" ~^ dx""^ ^dxdx\'\'-^\'^ 1.2 dx^ dx\'\' "\xc2\xbb + \' \' * \n\n,n{n\xe2\x80\x94\\)...{n \xe2\x80\x94 r^\\)d\'\'ud\'\'~\'\'v d\'^~\'^u dv d^u \n\n^ |7 di" dx^-^"- "^ \xe2\x80\xa2 * \xe2\x80\xa2 ^^.r"-!\' ^ "^ \'\'\'^\' \n\nEXAMPLES. \n\nI. f ^ /^"^X \n\nu \xe2\x80\x94 <f\xc2\xab^, du/dx = ae^^, . . . d\'*^u/dx^ \xe2\x80\x94 a^e^^, \nV = X, dv/dx = 1, ... d\'^v/dx\'*^ = o. \n\nd^y/dx"^ = ;m"-i^"^ + \xc2\xab"^\xc2\xab^;r. \n\n\n\nSUCCESSIVE DIFFERENT I ATION. 1 29 \n\n1. y =L e^^x"^. d\'^y/dx^ \xe2\x80\x94 a\'^-\'^ e"^\\(r x\'^ 4" "^nax -f- n{n \xe2\x80\x94 i)], \n\n3. jj/ = j;^ tan X. \n\nd^y/dx^ = 2x^ sec\'^jc(3 tan^ x -\\- 1) -\\- i^x\'^ sec^jc tan x \n-{- i%x sec^ X -\\- b tan x. \n\n4. jj/ = ^\xc2\xab^z/. \n\n^^\'^ \\ \' ^x^ I . 2 dx^^ ^ dxnj \n\n\n\nFUNCTIONS OF TWO OR MORE VARIABLES. \n\n105. Successive Partial Differentiation. \xe2\x80\x94 A partial dif- \nferential of a function of two or more variables is, in gen- \neral, a function of each variable and may, therefore, be dif- \nferentiated again with respect to each. \n\nThus, if z =/{x,y) we write (\xc2\xa7 loi) dz = \'T^^ ~l~ v~^\' \n\nin which dz/dx and dz/dy are symbols for t/ie partial \nderivatives of the first order \\\\\\\\\\\\ reference to x and _;)^ respec- \ntively. \n\nDifferentiating dz/dx with respect to x^ we write \n\n\n\n9(S/"\'\'^ = <I\xc2\xae) = (1?) \n\n\n\nfor the partial derivative of the second order taken twice \nwith respect to x. \n\nDifferentiating dz/dx with respect to j^, we have \n\ndrdz\\ ^ d\'z \n\ndy\\dxi dx dy \n\nfor the partial derivative of the second order taken once \nwith respect to x^ and then with respect to/. \n\n\n\n130 DIFFERENTIAL CALCULUS. \n\nSimilarly, we write \n\ndx \\dy I ~ dy dx* dy \\dy ) ~ dy\'^ \' \n\nfor the other partial derivatives of the second order. \n\nEach partial derivative of the second order will, in gen- \neral, admit of differentiation with respect to each variable^ \nand the differentiation may be continued, in general, to any \nrequired order. \n\nThe notation adopted is as follows: \n\n\n\ndx\\dxV~ dx\'\' \n\n\n\ndy\\dx\') ~ dx\' dy\' \n\n\n\ndx \\dx dy) \n\nd fd\'z \\ \n\ndy\\dx dy 1 \n\n\n\ndy) dx dy dx \n\n\n\ndy\\dx dy i dx dy^ \netc. etc. \n\n8^ + \' \n\n\n\ndyXdx\'\'-\'\' dy"-) ~ \n\n\n\ndy\\dx\'\'-\'\'dy-\'l ^jc^-^^^ + i* \n\nwhich represents the partial derivative of the {n -\\- if^ order, \ntaken {ii \xe2\x80\x94 r) times with respect to x and (r -\\- i) with re- \nspect \\o y. \n\nThe numerator indicates the number of differentiations, \nand the denominator the order of the successive operations \nwith respect to the variables. \n\nBy multiplying the symbol for any partial derivative by \nits denominator we obtain the symbol for the corresponding \npartial differential. Thus, d\'\'\'^\'\'z dx\'\'\'dy\'/dx\'"dy\'\' represents \n\n\n\nSUCCESSIVE DIFFERENTIATION, 13 1 \n\na partial differential of the {jn -\\- lif^ order, taken m times \nwith respect to x^ and n times with respect to y. \nHaving 2^ ^ f{x,y, z), \n\n\'du/dxy \'du/dy^ \'du/dz, \n\nrepresent, respectively, the partial derivatives of the first \norder, each being, in general, a function of x, y, and z. The \ndifferentiation may, therefore, be continued, and the suc- \ncessive operations indicated by extending the notation used \nabove. Thus, \'d\'^^\'^^^\'u/dx^^dy^dz^ represents the partial de- \nrivative of the {;m -\\- n -\\- rf^ order, taken ni times with \nrespect to x^ n times with respect to j, and r times with \nrespect to z. \n\nIn a similar manner, a partial derivative of any order of a \nfunction of any number of variables may be obtained and \nrepresented. \n\nlo6. Partial differentials and their correspofiding deriva- \ntives are independent of the order of differentiatio7i. \n\nAssume z = f{x,y), and increase x by h, giving (\xc2\xa7 70) \n\n\n\ndz _ limit \ndx ~ ti-B-^o \n\n\n\nfix -i-h,y) -f{x,y) \nh \n\n\n\n] \n\n\n\nIn this expression increase y by k, and we have (\xc2\xa7 70) \n\ndy \\dx I dx dy \n,.^.^ r limit r /(x+ ^, y^k)-f{x. yJ^U)-\\f{xJrh, y)-f{x, ;/) ]"jl \n\n\n\n_ J5"^\'\'t Y f{x^h, y\\-k)-f{x, yJ^k)-[f{x+h, y)-f{x, yyn \n\n\n\n(I) \n\n\n\n132 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nSimilarly, increasing J^^ in the primitive function by k^ we \nwrite \n\n?\xc2\xa3 ^ limit \\ A^^y + k) - Ax,y) ^^ \ndy k^^o\\_ k S \n\nfrom which, increasing x by h, we obtain \n\n\xc2\xa7\xe2\x96\xa0)- \'\xe2\x96\xa0\xe2\x96\xa0 - \n\n\n\n8_ \n\ndx \n\n\n\nlim \n\n\n\nIt r limit r \n\n\n\ndy dx \nAxJrh, y-^k)-f{x-^h, y)-\\_Ax, y\\k)-f{x. \n\n\n\nf]] \n\n\n\nlimit \n\n\n\noL- \n\n\n\n/(^ J^h,yJ^ k)-f{x ^k,y)- [f[^, y^k) -f{x , y)\\ \nhk \n\n\n\n]\xe2\x80\xa2 \n\n\n\nwhich compared with (i) gives \n\n\n\n^Jt: ^ dy\\dx \nFrom which we have \n\n\n\n/ dx\\dyl \n\n\n\n_a^ \n\ndy dx\' \n\n\n\n-d\'z \n\n\nd \n\n\ndx^dy \n\n\ndy \n\n\n\n\nd r \n\n\n\n\n^;^:L \n\n\n\n_dx \\dx )j dx\\_dy\\dx /J \n\n8 / a^ \\"i ^ a\'-g \n\n\n\nSimilarly, we derive \n\n8^^ \n\n\n\ndx\\ dy /J dydx^\' \nd\'z \n\n\n\nd\'z \n\n\n\ndx^ dy dx^ dydx dy dx^* \nand, in general, \n\nd\'^+\'^z/dx\'^dy\'\' = d\'^+^\'z/drdx\'^. \nSimilarly, having u = f{^x,y, z), it can be proved that \n\n\'d\'^u/dz dx dy = \'d^ti/dy dx dz, etc. \n\n\n\nSUCCESSIVE differentiation: 133 \n\nHence we infer that the order of diff.erentiation in all \ncases does not affect the result. \n\nEXAMPLES. \nI z \xe2\x80\x94 X sxn y-\\-y s\\x\\ X . = (zos y -^ zos x =. \n\n\n\ndxdy dydx \n\n2. 2 = 2xiy^ -f x^y. = _ \xe2\x80\x94 = 1 2(^2 +;/2). \n\ndxdydx dydx\'\' \n\n\n\n3. z = x logy. \n\n\n\nd\'^ I d\'^ \n\n\n\n4. z = x^y -\\- 4y^. = ^x \n\n\n\ndxdy y dydx \n\n\n\ndxdy dydx \n\n\n\n5.. = .an-MiV 3\' \n\n\n\nQ\\2 \xe2\x80\xa2 \n\n\n\ny I dxdy dydx (^^ + y\'^) \n\nt. z = sin {ax\'^ + by"^). \xe2\x96\xa0 ^ ^ = \xe2\x80\x94 abn^ixyY-\'^ sin (\xc2\xabx^ + ^v^). \n\n7. ^ =;\'^. -"T-f = j/^-Hi + -^ log;/) = --^ \n\n8. 2/ = ^^2/^. \xe2\x80\x94 ^:^ = (i + 2,^yz + x-y^\'\')^^^\'^ = \n\n9- ^ - ^2^37- -^^ - - 8-^^ (^2 _y^3- \n\nx^y d\'^u 2x\'^z \'d\'^u 2x \n\nII. U = o -^ , \xe2\x80\x94 \xe2\x96\xa0 ^ \n\nz \n\n\n\n). 2 = sin-1 \xe2\x80\x94 ). \n\n\n\n^ - ^^\' dydz {a" - z^f dxdy a^ - z"\' \n\n\'d^u /\\xyz \'^^u 4XZ \n\n\n\ndxdz \n\n\n\n12. z = sin \xe2\x80\x94 . \n\n\nd\'z \n\n\n2 , X , X X \n\n= -V Sin [- - cos \xe2\x80\x94 . \n\ny^ y y* y \n\n\ndydx\' \n\n\n13. z = sin (x +_)/). \n\n\nd/ \n\n\n- sin (X -\\-y). ^= -COS (x+y). \n\n\n\n134 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\n14. z = cos (x \xe2\x80\x94 y). \n\n\n\n:5. z = \\og{x -^y). \n\n\n\n\n\n\nsin {x A^ y). \n\n\n\ndx\'> \n\n\n\n= \xe2\x80\x94 cos [x \xe2\x80\x94 y) \ndx^ dx\'^ \n\n\n\ndy^ \n\n\n\n= \xe2\x80\x94 cos {x \xe2\x80\x94 y) \n\n\n\ndy^ \n\n\n\n= cos {x -\\-y). \n\n= sin {x \xe2\x80\x94 y). \n- = \xe2\x80\x94 sin(x\xe2\x80\x94 _y). \n\n\n\ndx-" {x+yY^\' dy^ {x-\\-yf\' \n\n107. Partial Differentials of a Surface. \xe2\x80\x94 Let A TL be \n\nany surface, and ABCD \xe2\x80\x94 it 2. portion of it included be- \n\n\n\n\ntween the coordinate planes XZ, FZ, and the planes DQR^ \nB f\'S, parallel to them respectively. From \xc2\xa7 26 we have \n\nIncrease OP = ^ by FP\' = h, giving (\xc2\xa7 70) \n\n\'du ^ limit r /jx^h^y) -f{x,y) ~\\ \ndx /^;^^o |_ /i J \n\n\n\nSUCCESSIVE DIFFERENriATION. 135 \n\nNow increase OQ \xe2\x80\x94 y\\>y QQ\' \xe2\x80\x94 k, giving (\xc2\xa7 70) \n\n\n\nRJL^ limit ^^^oL J, J \n\ndx dy k-m-^Q I \xe2\x96\xa0 \n\nlimit \n\n\n\n] \n\n^ J\'^\'q V A^^h, y+k)- fix, y-\\.k) - [/(^ + h, y) - f{x, j)] "| ^ \n\nIn which \n\nf{x J^ h,y^k)= AEGI, f{x, y^k)= ABHI, \nf{x + /i,y) = AEFD, f{x, y) = ABCD. \n\nHence, \nf{x ^h,y-{- k) -f{x, y-{-k):= A EG I - ABHI = EEGH. \nf{x + h,y) -Ax,y) = AEFD - ABCD = BEFC. \nf{x + h,y-\\-k) -f{x,y-\\- k) - [/{x + /i,y) -/{x^y)] \n= BEGH - BEFC = CFGH, \n\n\n\nTherefore (\xc2\xa7 50] \n\n\n\n?)\\i _ limit CFGH I \n\n\n\ndx dy k-^-^Q ^^^ cos /? \xe2\x80\xa2\' \n\nand a^udx dy/dx dy = wsec ft dx dy. \n\nio8\xc2\xbb Partial Differentials of a Volume.\xe2\x80\x94 Let ATL \n\n(figure \xc2\xa7 107) be any surface, and ABCD-ON = z^ a volume \nlimited by it, the three coordinate planes, and the planes \nB>QJ? and BBS parallel, respectively, to XZ and VZ. From \n\xc2\xa7 30 we have v = /{x, y). \n\nBy the method used in the last Article, considering the \ncorresponding volumes instead of the surfaces, we obtain \n\n\n\n3\\/ limit \n\n- \xe2\x80\x94 -- = /i-^^-^o \ndx dy j^^^^ \n\n\n\nfix Jrh, y+k) - fix, y-^k) - If^x-^h, y)~fix, y)^ \n\n\n\n]\xe2\x96\xa0 \n\n\n\n136 DIFFERENTIAL CALCULUS. \n\nIn which \n\nf{x^h, y-\\-k)-f{x,y-itk)-[f{x-{- h, y) -f{x, ;/)]=vol. CFGH-NM. \nHence (\xc2\xa7 49), \n\na^z; _ limit CFGH-NM ^^^ \ndxdy ^^o hk \n\nand 8\'vdx dy/dx dy = zdx dy. \n\n109. Successive Total Differentiation. \xe2\x80\x94 Having z = \n/fej), \xc2\xa7101 gives \n\n^^==1^^+!^^\' (^> \n\nin which the total differential of the first order dz, and the \n\ncorresponding partial differentials -^dx and -r dy, are, in \n\ngeneral, functions of x andjj^. \n\nHence, the total differential of the second order, denoted \nby d\'z, is obtained by differentiating each term in the sec- \nond member of (i) with respect to each variable, and \ntaking the sum of the partial differentials of the second \norder. \n\n. . dz . \n\nDifferentiating j-dx with respect to each variable, we \n\nhave \n\ndy \n\n\n\n^&^)==ll^"^ + l-i^"^^- \n\n\n\nSimilarly, d (|^^) = ^-dy dx + p^/. \n\n\n\ndy dx"\'^ "\'\'^ ^ dy\' \nTherefore \n\n\n\nd^z = |!? dx^ + 2 v^ dx dy + l^^dy^ . . (2) \ndx^ ^ dx dy -^ ^ dy^ -^ ^ ^ \n\n\n\nSUCCESSIVE D IFFERENTIATION. 1 37 \n\nDifferentiating again, since \n\n\n\n\n\n\ndx\' \' ^dx\'dy ^ \' dy\' dx ^ \' dy\' -^ \n\nSimilarly, formulas for the total differentials of the higher \norders may be obtained, the numerical coefficients of which \nwill be found to follow the same law as those of the bi- \nnomial formula. Thus the formula for the n^^ differen- \ntial is \n\n"^ -dx-"" ^""dx--^ dy^"" "^^^ 1.2 dx^-^df\'^\'\' ^-^ \notherwise written for abbreviation \n\n\'"\'=^\'\'\'+14\'\' \xe2\x80\xa2 \xe2\x80\xa2 \xe2\x80\xa2 \xe2\x80\xa2 ^"^ \n\nwhich form is not to be interpreted as usual, but as fol- \nlows : Expand as indicated, regarding each term as a single \n\n/ 9 \\" \nquantity, and in the result replace each term, as ( ;i\xe2\x80\x94 ^-^ J \xc2\xbb \n\nby {d"z/dx")dx\'^^ and each combination, as \n\n\n\n138 DIFFERENTIAL CALCULUS, \n\nIt is important to notice that in deriving eq. (2) from eq. \n(i) we write, in accordance with \xc2\xa7 106, \n\nd\'^zdx dy/dx dy = Z\'^zdy dx/dy dx ; \n\nand it follows that having any expression in the form Pdx \n+ Qdyy in order that it may be a total differential, it is \nnecessary and sufficient that \n\ndy dxdy dydx ~ dx \' \n\nThis is known as Euler\'s Test. It determines whether or \nnot the given expression is the result of the complete differ- \nentiation with respect to both variables of some other ex- \npression. Thus, having \n\n2xdx -\\-ydx -\\- xdy -f- 2ydy, \nwe obtain \n\nwhich shows that the given expression is a total differential \nof some function of x and_)^. \n\nHaving 2x dx -\\-ydx -\\- 27 dy, the test fails and the expres- \nsion is not a total differential of any function. By differ- \nentiating x\'^ -{- xy -\\-y^, the student may confirm both \nresults. \n\nThe successive total differentials of any function of any \nnumber of variables may be determined in a similar man- \nner. \n\n\n\nSUCCESSIVE DIFFERENTIATION, 1 39 \n\nThus, having u = /{x^y, z), then \n\n+ 2 1 \xe2\x80\x94 r^-^ 6^+2 - \xe2\x80\x94 ^dydz + 2-7 \xe2\x80\x94 r dx dz, \ndxdy -^ dydz-" dx dz \n\netc. etc etc. \n\n9 , . 9 . , 9 .N\'^ \n\n\n\n^"/^ = \\\xe2\x80\x94-dx 4- v-^v + -7-^^) 2^.\' \nV^jc \' dy dz I \n\n\n\nRule. Differentiate the function and obtain expressions for \nthe several partial coefficie fits to the desired order. Substitute \nthese in the proper formula for their respective symbols, \n\nEXAMPLES. \nI. 2 = x^y^. dH = txy\'^dx\'\' + I2x^ydx dy + 2x\'^dy\'\', \n\n3. 2 = ;tr\'\xc2\xbb_j/2. ^=2 = 2;\'V;tr2 -j- %xy dx dy + 2;t:\'^2^ \n\nd\'^z =i I2ydx\'^ dy -f- I2x^x^\'. \n* Extension of the symbolic form {a), \xc2\xa7 109. \n\n\n\nI40 DIFFERENTIAL CALCULUS, \n\n4. z = (x\'-^+j/^l\'S. \n\n\n\n^\'^ = (;,^+y)3/^ (/^^\' - ^^y^^^y + ^\'^/). \n\n\n\ne^, z = <?(a\xc2\xab+&2/). d^z \xe2\x80\x94 ,f(\xc2\xaba;+6j/)[^V;r2 + 2ab dx dy -\\- d^dy^]. \n\n6, 2 = x^y"^ -\\- y^x^. \ndh = (6-r/ + 2y\')dx\'\' -\\- I2{x^y + xy\'\')dx dy + {bx^^y + 2x^)dv\'\'. \n\n\n\nIMPLICIT FUNCTIONS, I4I \n\n\n\nCHAPTER VII. \n\nIMPLICIT FUNCTIONS AND DIFFERENTIAL EQUATIONS. \n\nIMPLICIT FUNCTIONS. \n\n1 10. With the exceptions considered in \xc2\xa7 73 and \xc2\xa7 77, \ndifferentiation hitherto has been limited to ^jc//zWV functions. \n\nLety = ax, and assume x to be independent, then (\xc2\xa7 14) \n^ is an implicit function ofx. \n\nSolving with respect to _y, we have _>\' = \xc2\xb1 Vax^ which \nexpresses y as an explicit function of x. \n\nDifferentiating, we have \n\ndy^ -^ adx/iVax (l) \n\nOtherwise, we may write \n\n^(/) = ^^-^j \xe2\x80\xa2*\xe2\x80\xa2 d{y^)/dx = a, . . , (2) \n\nin which (/) =/{y), and y = (p(x). Therefore (\xc2\xa777) \n\ndjf) ^4/),\xc2\xb1^ \xc2\xb1^ \ndx dy dx dx \n\nSubstituting in first member of (2), we obtain \n2y dy = adx, . . (3) /. ^ = \xc2\xb1 adx/2^/ax^ . (4) \n\nwhich result corresponds with (i). \n\nExamining (3) we see that it, and therefore (4), may be \nderived directly from y = ax, by differentiating (y) as an \n\n\n\n142 DIFFERENTIAL CALCULUS. \n\nexplicit function of y, y, and in general dy^ being functions \nof X. \n\nThat is, the equation may be solved with respect to y^ \nandy differentiated as an explicit function, or we may differ- \nentiate, regarding y as an implicit function, and then solve \nthe resulting differential equation with respect to dy. \n\nThis principle is general; for, having any equation con- \ntaining two variables, x and jf, it may be written \n\nf[x,y) = (p{x,y); \n\nand regarding ji^ as an implicit function of x, each member \nmay be regarded as an explicit function of x, equal to the \nother for all values of x; therefore (\xc2\xa7 72) their differentials \nare equal. It is not essential that the value of the depend- \nent variable shall be expressed in terms of the other, but \nit is necessary to remember which is assumed as the de- \npendent and which the independent variable. \n\nThe advantage of differentiating without solving with \nrespect to the implicit function beforehand increases, in \ngeneral, with the degree of that function. \n\nEXAMPLES. \n\n\n\n1. ay^ \xe2\x80\x94 x^ -{- dx = o. dy/dx = \xc2\xb1 (3^2 \xe2\x80\x94 3)/2 ^a{x^ \xe2\x80\x94 bx^ \n\n2. \xc2\xaby + b\'^x\'\' = a^^ dy/dx = - b^x/ay. \n\n3. (^ _|_ ay = 4bx. dy/dx = \xc2\xb1 {b/x)^^\\ \n\n4. cos y = a cos x. dy/dx = tan jr/tan y, \n\n5. cos (x -\\-y) = o. dy/dx = \xe2\x80\x94 i. \n\n6. xy \xe2\x80\x94 jj/* = o. dy/dx = {y^ \xe2\x80\x94 xy log y)/{x^ \xe2\x80\x94 xy log x). \n\n7. (y- xy = xK dy/dx =2x \xc2\xb1 c^x^\'\'^/2. \n\n8. ^y \xe2\x80\x94 x\'^-\\- bx\'^ = o, dy/dx = \xc2\xb1 (3;^ \xe2\x80\x94 2b)/ 2 V~a{x \xe2\x80\x94 b). \n\n\n\nIMPLICIT FUNCTIONS. 143 \n\n\xe2\x80\x9e , dv 3^^ \xe2\x80\x94 IX {b \xe2\x80\x94 c) \xe2\x80\x94 he \n\nax 2\\ax{x \xe2\x80\x94 b) {x -\\- c) \n\ncfy x{x^ -\\- 12X \xe2\x80\x94lb) \n\n\n\nlo. x\'^ \xe2\x80\x94 2x^y \xe2\x80\x94 2Jf\' \xe2\x80\x94 8/ = O. \xe2\x80\x94 = \n\n\n\ndx 2{x^ -\\- 4) \n\n\n\nIT./- 2yVa\' -{-x\'\'^x^=0. dy/dx = x/ ^ a" + x"". \n\n12. y s\\x\\ X - X sin_j/ 4"! = o\xc2\xab \n\n14. ;\xc2\xab;/-_>/ + I = o, dy/dx = e^/(2 - y). \n\n\n\ndy sin y \xe2\x80\x94 y cos x \ndx sin X \xe2\x80\x94 X cos y \n\n\n\na-[.^a\'-y\' x ^ ^ a\' - y\'\' dy V . \n\n15. log J =0. - = ~ ^^-^,\' \n\nBy continuing the differentiation in a similar manner, expressions \nfor d\'^y/dx\'^, and derivatives of the higher orders, may be obtained. \n\n16. x2/3+//3 = a2/3. dy/dx = -y/V^i/^ \n\nd^y/dx^ = (i/3x^/^)(i/y^/^+y/V^^/^). \n\n\n\n17. x =:r vers~i (y/r) - ^/2ry \xe2\x80\x94y^. \n\ndy/dx = {2r/y - i)^/^ d-^y/dx\' = - r/y\\ \n\n18. ay-^V= -aH\'\'. \n\ndy _ b\'^x _ bx dy \n\n\n\nab \n\n\n\ndx d\'y a i/x"" - \xc2\xab^ ^^\' \xc2\xab>\' (x^ - a^f\'^ \n\nIII. Having ti = /{^^ y) = o, and regarding y as an im- \nplicit function of x, we write (\xc2\xa7 102)* \n\n* Also u may be differentiated w^ith respect to x by differentiating \nit as a function of x and _y (\xc2\xa7 loi), and since y is a function of x \ndiu/dy must be multiplied by dy/dx (\xc2\xa7 77). \n\n\n\n144 DIFFERENTIAL CALCULUS. \n\ndu \'du \'du dy __ , . \n\nd^ ~ dx\'^dy~dx~ \xc2\xb0\' \xe2\x80\xa2 \xe2\x80\xa2 * \xe2\x80\xa2 ^^^ \n\nEXAMPLES. \n\n1. M = jj/3 \xe2\x80\x94 3jj/j;2 _j_ 2x^ = O. \n\ndx -^ dy ^-^ 3y\'-3x x-^y \n\n2. y"^ \xe2\x80\x94 2xy 4- a\'^ = O. dy/dx = y/{y \xe2\x80\x94 x). \n\ndy _ (2y -j- ^W \n\n\n\n3. y 4_ 3ay _ 4^2^^ _ ^2^2 _ o. \n\n\n\n^ji; 2y^ -\\- 3a^y \xe2\x80\x94 2a^x \n\n\n\n4. X - ax;/ -r ^^^^ ;/ - o. ^^ ^^^ _ ^^^.^ _^ ^^3 . \n\ndy 3x^ 4- Aax -\\- a} \n\n5. y-i _ x^ - 2ax^ - a}x = 0. -f - ^-^\xe2\x80\x94^ =?\xe2\x80\x94 -. \n\n\xe2\x80\xa2^ -^ dx 2y \n\n6. y^ \xe2\x80\x94 x\\i \xe2\x80\x94 ;f*) = O. dy/dx = {x \xe2\x80\x94 2x^)/y. \n\n\n\ndy/dx = 2x{x\'^ \xe2\x80\x94 a})f3ay{y -\\- a). \n\n8. {av - x\'y - (x - 4)5(x - 3)6 = o. \n\ndy^3^. 5(x - 4y{x - 3)8 3(x - 4)\xc2\xb0(x - 3)5 \ndx 4 8(47 - x\') 4(4:1/ - x^) \n\nDifferentiating (i), remembering that \n\ndx\\dx) dx^ dxdydx* \n\nd^ ldu\\ _ 9V 9^^// ^ \ndx\\dyl dydx 9y ^^\' \n\n\n\nIMPLICIT FUNCTIONS. 145 \n\nwe have \n\nd? ~ dx\'\' "^ dx\'dydx "^ \\dydx ^ 9/ dxjdx dy dx"" ~ \xc2\xb0\' \nd\'^u d\'^u dy \'d\'^ul dyV dudy _ \n\nHence, \n\n\n\ndx\' \n\n\n\nX^dx\' "^ Vjc- ay ^jc "^ a/ UW J / aj^\' \' * * ^^^ \n\nSubstituting expression for dy/dx from (2), and simplify- \ning, we have \n\ndx-" ~ l_dx\' \\dyl ^dx dy dx dy ^ a/ \\dxl ]l \\dy/ \' ^^^ \n\n(6) \n\n\n\nXi \n\n\ndx \n\n\n> V4; g \n\n\n"^^ ^^ \n\n\n\n\ndx"! \n\n\n\' \xc2\xa5 \n\n\n\n\n\n\n\n\nEXAMPLES. \n\n\n\n\nI. \n\n\nx^ \xe2\x80\x94 3\xc2\xab;r>/ \n\n\n+ ;.3^ \n\n\n0. \n\n\n\n\n\n\n\n\n\n\ndx \n\n\n= 3(x^ - \n\n\n- ay), \n\n\n-dy \n\n\n= 3(- \n\n\n\xe2\x96\xa0axJrf\\ \n\n\n\n\nav \n\n\n= 6x. \n\n\ndxdy \n\n\n= - \n\n\n3\xc2\xab, \n\n\n3\'\xc2\xab - 6^. \n\n9/ \n\n\n\n\n\n\n\n\n\xe2\x96\xa0 x^ d^y \nax dx\'\' \n\n\n= - \n\n\n2\xc2\xab3 \n\n\nxy \n\n\n\n\n{f- \n\n\n\xe2\x96\xa0axf \n\n\nIf \n\n\ndy/dx = \n\n\n/(6)giv \n\n\nes \n\n\n\n\n\n\n\n\n\n\n\n\n\n\ndx \n\n\n\n\n2X \n\n\n\n\n\n\n~ 3( \n\n\n- ax -\\-y \n\n\n^) " \n\n\nax \xe2\x80\x94 \n\n\nf \n\n\n\n14^ DIFFEKENriAL CALCULUS \n\n2. x^ -{-y^ \xe2\x80\x94 2dxy \xe2\x80\x94 a^ = o. \n\ndy/dx = {x - by)/{bx - y), \nd^y/dx"" = {b"" - i)ay{y - bxf. \n\n\n\nIf dy/dx = o, dy/dx"" = \xe2\x80\x94 i/aVi \xe2\x80\x94 b\\ \n\n3. ixy \xe2\x80\x94 y"" \xe2\x80\x94 a" =1 Q. \n\ndy/dx = y/{y \xe2\x80\x94 x), \ndy/dx"" = y{y \xe2\x80\x94 2x)/{y \xe2\x80\x94 x)\\ \n\n4. x^ + ^axy -\\- y^ = o. \n\ndy/dx z= \xe2\x80\x94 {x^ -\\- ay)/{y^ -f" ^\xe2\x80\xa2^)\xc2\xbb \ndy/dx"" = 2a^xy/{y"" -|- ax)^. \n\n5. y^ \xe2\x80\x94 x^/{2a \xe2\x80\x94 x) = o. \n\ndy/dx = \xc2\xb1 {^a \xe2\x80\x94 x) Vx/{2a \xe2\x80\x94 xy, \nd\'"y/dx"" = \xc2\xb1 3aV(2\xc2\xab - xf Vx. \n\nDifferentiating (3), we obtain \n\n~^ ^\\_ dxdy "^ 3/ ^^J ^^"^ "^ dy dx\' ~ ^\' \xe2\x80\xa2 \xe2\x80\xa2 v7 J \nSimilarly, it may be shown that \n\nd^~~d^ "^ \xe2\x80\xa2 \xe2\x80\xa2 \xe2\x80\xa2 "^ ay ^^\xc2\xab ~ \xc2\xb0\' \xe2\x80\xa2 \' ^^ \n\nin which the intermediate terms involve differential coeffi- \ncients of jF with respect to x of orders less than the n^^. \n112. Having any equation containing three variables \n\n\n\nDIFFERENTIAL EQUATIONS. 147 \n\nx^ y and z^ one must be an implicit function of the other \ntwo (\xc2\xa7 9). Each member may, therefore, be regarded as a \nfunction of but two independent variables. \n\nThe differential of each member is equal to the sum of \nits partial differentials; and since the partial differentials of \nthe two members are respectively equal to each other, the \ntotal differentials are equal. \n\nIt is not necessary to express the implicit function or \ndependent variable in terms of the others, but it is always \nimportant to distinguish it. \n\nIn a similar manner it may be shown that the total dif- \nferentials of the members of any equation are equal, re- \nmembering that the number of mdependent variables is one \nless than the number of variables. \n\nDIFFERENTIAL EQUATIONS. \n\nII3\xc2\xab An equation which contains differential coefficients \nis called a differential equation. \n\nA differential equation obtained by one differentiation \nmay, in general, be differentiated again, giving a differential \nequation of the j-^^:^;^^ order, and so on to differential equa- \ntions of the thirds etc., orders. \n\nThus, having y \xe2\x80\x94 2mxy -\\- x^ ^^ a^, regarding x as the \nindependent variable, differentiating, equating the results, \nand reducing, we have \n\ny dy \xe2\x80\x94 mx dy \xe2\x80\x94 my dx -\\- x dx ^^ o, \nor dy/dx = (my \xe2\x80\x94 x)/{y \xe2\x80\x94 ?nx). \n\nDifferentiating again,* we have \n\n* The importance of distinguishing between the independent and \ndependent variables becomes apparent at the second operation of \ndifferentiation \xe2\x80\x94 as in above dx is a constant, whereas dy = /{x). \n\n\n\n148 DIFFERENTIAL CALCULUS. \n\ndy^ -\\-y (Py \xe2\x80\x94 mx dy \xe2\x80\x94 2m dx dy -\\- dx^ = o, \n\nor, dividing by dx*, \n\n(y \xe2\x80\x94 mx) dy/dx^ -\\- dy^/dx"^ \xe2\x80\x94 2 m dy/dx -}- i = o. \n\nThe order of a differential equation is the same as that \nof the highest derivative it contains, and its degree is de- \nnoted by the greatest exponent of the derivative of the high- \nest order in any terra; provided that all such exponents are \nentire and positive. Thus, \n\n{dy/dxY \xe2\x80\x94 a/x =^ o is of the ist order and 2d degree. \n\ndy/dx"^ -{-ay =0 is of the 2d order and ist degree. \n\n. {dy)/dx\'\'y\'\' + M{d\'\'-y/dx\'\'-\'Y~\' + . . . = o is of the n^^ \norder and m^\'^ degree. \n\nEXAMPLES. \n\n1. y 4- JT^ \xe2\x80\x94 ^2^ dy/dx = \xe2\x80\x94 x/y, d^/dx^ = \xe2\x80\x94 ryy. \n\n2. / = 2j?x. dy/dx =p/y, d\'^y/dx\'^ = - p\'\'/y^. \n\n3 . a\'y\'\' + Vx"" = aH"". dy/dx = - ^ V/" V- ^ W-^-^\' = - ^*/aY. \n\n4. y - ^2\xe2\x80\x9e^ ^y^^ = \xc2\xab V31/2. d y/dx"" = - 2a*/gyK \n\n5.ax-y{x a). ________ __ _ ____^ _ . \n\n6, y = l>x\\ : dy/dx = \'ibx\'\'/2y. d\'\'y/dx\'\' = 3Pxy^y^. \n\n\n\n7. / = 2;>x + r^x^. \n\n\n\ndy^ _ i\xc2\xb1\xc2\xb1J^ d\'^y _ _f \ndx y \' dx\'^ y^\' \n\n\n\n8. xV=4\xc2\xab\'(2\xc2\xab\xe2\x80\x94 j). dy/dx = \xe2\x80\x94 2xy/{x\'^ -\\- 402), \n\nd \'^y/dx\'^ = 2i\'f 3,\'. \' - 4\xc2\xab\')/(x^ + 4\xc2\xab\')\'. \n\n\n\nDIFFERENTIAL EQUATIONS. 1 49 \n\n10. \\og{xy)-{-x\xe2\x80\x94y = a. {x \xe2\x80\x94 xy) \xc2\xa3^ -\\- y -\\- xy = o. \n\n12. )/^ \xe2\x80\x94 2;rj/-|-^^ = O. d\'^y/dx\'^ \xe2\x80\x94 y{y \xe2\x80\x94 2x)/(y \xe2\x80\x94 x)K \n\n13. ^^ + 3<T^-^/ +jJ^^ = O. d\'^y/dx\'^ = 2a^ ^yKy^ + ^\xe2\x80\xa2^)^\xc2\xab \n\n,4. .\'^ = 4\xc2\xab\'(3.->\'). \xc2\xa3=-^-^- \n\n15. j/3 = 2aji;\'^ - x^. d\'^y/dx^ = - Sa^/gx^\'\\2a - xf\\ \n\n16. x\'^ \xe2\x80\x94 xy-\\-J =0. d\'^y/dx^ = 2 (i + lA^). \n\n17. \\og(x+y)=x-y. d\'y/dx\' = 4{x-^y)/{x+y+iy. \n\n_ ^34. 5^2 _|_i2,r - 8 \n\n\n\n18, x\'^\xe2\x80\x942xy\xe2\x80\x942x\'^-8y=o. \n\n\n\ndx^ ^ (^^+4)\' \n\n\n\nIt is important to notice the difference between the suc- \ncessive differentials of an independent expression which \ncontains variables and those of the same expression limited \nto a constant value. Thus, suppose we nave (x"^ -\\-y^) un- \nlimited, and (x"^ +y) = \xc2\xab^ \n\nThen, \n\nd(x^-\\-y^) \xe2\x80\x942x dx-\\-2y dy, and d{x\'^-\\-f)=^2X dx-\\-2y dy^o, \n\nd\\x\'\'+f) = 2dx\'^2dy\\2iXidd\\x\'-\\-f) = 2dx\'\'+2d/^2yd\'y = o. \nd\\x\'-\\-y\')^o, and d\\x\'\'-^f) = A,dy d\'\'y^2dy d\'y^2y d\'y=o. \netc. etc. etc. \n\n\n\nI50 DIFFERENTIAL CALCULUS. \n\nIn the first case both variables are independent, but in \nthe second only one. The difference becomes apparent \nat the second differentiation. \n\nEquations derived by differentiating primitive equations \nor other differential equations are called immediate differ- \nential equations. \n\n114. Differential equations also arise by combining suc- \ncessive immediate differential equations with each other \nand the primitive equation in such a manner as to elimi- \nnate certain constants^ or particular functions which enter \nthe primitive equation. Thus, from \n\ny=^ ax -{\xe2\x80\xa2 b we obtain dy/dx = a. \nEliminating a^ dy/dx = [y \xe2\x80\x94 b)/xy \n\nwhich is independent of a. \n\nDifferentiating again, we have d\'^y/dx^ = o, which is in- \ndependent of both a and b. \n\nAs another example, take the equation of a circle \n\n(x - aY -\\-(y-by = J^\' (i) \n\nDifferentiating three times, we have \n\n2(x \xe2\x80\x94 a)dx + 2(y \xe2\x80\x94 b)dy = o. . . . (2) \n\n^^\'^^^f^(y-l,)^y=o (3) \n\n2dydy ~\\- dy dy -{- (y \xe2\x80\x94 b)d^y = o. . . (4) \nDividing (3) by "(4), member by member, we have \n\n(dx^ + dy\')/2idy d\'^y = d^/dy, which gives \n\n\n\nDIFFERENTIAL EQUATIONS, I5I \n\ndx\'^-df ld\'\'y _ dv_d^ldy_ \n\ndx" I dx\'~^dxdx\'/ dx\'\' ^^^ \n\ndx\'[\\dx) "^ \'J ^\\dxV dx ~ \xc2\xb0\' \n\nwhich is independent of a, b and R. \n\nAnd, in general, by differentiating an equation n times we \nobtain n differential equations between which and the prim- \nitive equation n constants or particular functions may be \neliminated, giving a differential equation of the \xc2\xab*^ order. \n\nEXAMPLES. \n\n1. Eliminate e^ and sin x from jj/ = ^* sin x. \n\ndy/dx = e^ sin x -\\- e^ cos x = jj/ -}- <?* cos x, \nd^y/dx^ = dy/dx -\\- ^* cos x \xe2\x80\x94 e\'\' sin x; \nand since e^ cos x = dy/dx \xe2\x80\x94 y, \n\nd\'^y/dx\'^ \xe2\x80\x94 2dy/dx \xe2\x80\x94 2y. \n\n2. y = a smx -\\- b sin 2x. \n\ndy/dx = a cos x -\\-\'2.b cos 2Jf, \nd^y/dx^ =. \xe2\x80\x94 a sin jr \xe2\x80\x94 4^ sin 2x, \nd^y/dx^ = \xe2\x80\x94 a cos x \xe2\x80\x94 83 cos 2x, \nd^y/dx^ = a sin x -|- i6(5 sin ix. \nHence, ^V^-=^\' + Sd^y/dx\'\' + 4^ = 0. \n\n3. j/3 \xe2\x80\x94 2^;^;\'^ = \xe2\x80\x94 a^. (^y\'^{dy/dxy \xe2\x80\x94 2^x^ dy/dx =. \xe2\x80\x94 ityx\'*, \n\n4. TV \xe2\x80\x94 nx\'^ \xe2\x80\x94 b = O. x^dy/dx\'^ \xe2\x80\x94 x dy/dx = Sjj/. \n?.(!-]- ;r^)(T + r\'^) = ax^ (t -1- x\'\')xr dy/dx = I -f-_j/^ \n\n\n\n152 DIFFERENTIAL CALCULUS, \n\nd. y z= a sin x \xe2\x80\x94 b cos x. d^y/dx^ \xe2\x96\xa0= \xe2\x80\x94 y. \ny. y = ax -\\- a \xe2\x80\x94 aK (-^ + i)dy/dx \xe2\x80\x94 (dy/dxY = ^\xe2\x80\xa2 \n\n8. _y = (a -j- ^jr ).?<\'\xc2\xbb\'. d\'^y/dx\'^ \xe2\x80\x94 icdy/dx = \xe2\x80\x94 c^y. \n\n9 jj/ = sin jf. {dy/dxY -\\- y^ = i. \n\nro. _)/ = (?^ cos jc. d\'iy/dx^ = 2dy/dx \xe2\x80\x94 2y. \n\n11. _;/ = sin log ^. ^t\'^^ ^y/dx"^ -\\- x dy/dx -f ^ = 0. \n\n12. y = log sin x, d^y/dx^ -\\- {dy/dxy = \xe2\x80\x94 i. \n\n13. f = ipx + r-x\\ yx^^ + "^t^)\'" \'^-^^ "^^^ " ""* \n\n14. y = a cos (^x -j- r). dy/dx"^ = \xe2\x80\x94 ^2^. \n\n15. y = sin~ijf. (l \xe2\x80\x94 x^)dy/dx^ \xe2\x80\x94 x dy/dx = o. \n\n\n\nlb y = ex \'\\- \\/i-\\-c^. y = xdy/dx + |/i + {dy/dx)*. \n\n17. ^ = (x 4" ^)^"*- ^/^^ \xe2\x80\x94 \xc2\xabj/ = e^^. \n\n18. ;j/=(r^-tan-ia;^tan-i ;p\xe2\x80\x94 I. (i -f- x\'^)dy/dx -\\- y = tan-*;ir. \n\n19. y = (ex -f- log X -j- l)~^. xdy/dx -\\- y = y^ log x. \n\n20. ^\'^ \xe2\x80\x94 2CX \xe2\x80\x94 c^ = o. y{dy/dxY -\\- 2xdy/dx = y, \n\nly \xe2\x80\x94 c cos mx-\\-c\' sin mx. ,\xe2\x80\x9e , , \xc2\xab , \xe2\x80\x9e \n\n21. K / , ,x dy/dx^ + ;\xc2\xabV = O. \n\n{ y =: C COS (iWJC + <r ). \n\n22. >/ = ;r log {{c -\\- c\'x)/x\\ x^d\'^y/dx^ + (/ \xe2\x80\x94 xdy/dxf = O. \n\n23. jj/ = JT sin nx/2n -\\- c cos nx -j- r\' sin ;2;c. \n\nd\'^y/dx\'^ 4" \'^l^ = c\xc2\xb0s fix \n\n24. jj/\'^ sin\'^ ^ + 2ay + ^2 = 0. {dy/dxf + "^y cot xdy/dx = y*, \n\n25. :j/ = (^3; + e-^)/{e\'^ \xe2\x80\x94 e-^). y"^ \xe2\x80\x94 \\ -\\- dy/dx = O. \n\n26. ^2j/_j_2ax^!/ -I- ^^^=0. {x" - i){dy/dxy = I. \n\n\n\nDIFFERENTIAL EQUATIONS. 153 \n\n27. jf/ = ae^^ -{- be-\'^^ . d\'^y/dx\'^ \xe2\x80\x94 c\'^y. \n\n28. y\xe2\x80\x94 ae^^ -\\r be-^^ -\\- ce^ . d^y/dx"^ \xe2\x80\x94 "jdy/dx = \xe2\x80\x94 6y. \n\n29. y={a-{-bx^x\'^/2)e\'\'-]-c. d^y/dx^ \xe2\x80\x94 2d\'^y/dx\'^ -\\- dy/dx =e^. \n\n30. y={a-\\-bx-\\-cx\'\')e^-\\-d. dy/dx*-2,d^y/dx^-\\-2d\'^y/dx\'\'-dy/dx=o. \n\n\n\n154 DIFFERENTIAL CALCULUS, \n\n\n\nCHAPTER VIIL \nCHANGE OF THE INDEPENDENT VARIABLE. \n\nII5* Having any expression or equation containing dif- \nferentials, or derivatives, of y regarded as a function of x^ \nit is sometimes desirable to obtain a corresponding expres- \nsion or equation, in which y^ or some other variable upon \nwhich y and x depend, is the independent one. This oper- \nation is called changing the independent variable. \n\nThe principle deduced in \xc2\xa7 73 enables us to make the \nchange in cases involving the first derivative only. \n\nWhen differentials of a higher order are involved it must \nbe remembered that we have written \n\n\n\nd_ \n\ndx \n\n\n\n(\xc2\xb1]-^:i and ^(^]~^ \n\\dx J~ dx\'\' dx \\dxy ~ dx\' \n\n\n\nin which x is the independent variable, and dx is a con- \nstant. \n\nRegarding both dy and dx as variable, we have \n\n\n\nf dy\\ __ dx dy \xe2\x80\x94 dy d\'^x , . \n\n\\d^l- ^? ~* .... (I) \n\n\n\nand \n\nd_ \ndx \n\n\n\nd d ldy\\ _ d (dxd\'^y \xe2\x80\x94 dyd\'x\\ \nix \' dx \\dxl ~ dx\\ dx\' J \n\n\n\n__{dx d\'y \xe2\x80\x94 dy d^x) dx-\\--^[dy d\'^x \xe2\x80\x94 dx dy) d\'x , . \n- dx~\' \' ^^^ \n\n\n\nCHANGE OF THE INDEPENDENT VARIABLE. 1 55 \n\nwhich should be substituted for d^y/dx^ and d^y/dx", re- \nspectively, in order that the results may be general, that is, \nin which neither x nor jv is independent. \n\nIf ttien y is made independent, we have dy \xe2\x96\xa0=. constant, \nd\'^y = o, dy = o, and (i) and (2) reduce to \n\ndy \xe2\x80\x94 dy d\'^x d\'^x I Idx V , \n\n^^ dx\' ^ ~~df I \\d^l \' \xe2\x80\xa2 \xe2\x80\xa2 \xe2\x80\xa2 V3) \n\nd^y _ ^dy(d^xy \xe2\x80\x94 dydxd\'^x \n\nr id\'\'x\\ d\'xdx-\\ /(dxV , . \n\nwhich may be used when the independent variable is \nchanged from x toy. \n\n\n\nEXAMPLES. \nChange the independent variable from x to^ in the following: \n\n\n\n\xe2\x96\xa0\' dx\' + \\dx) - ^\' \n\n\n\nd\'^x j.dx_ \n~df\'^\'d^~^\' \n\n\n\nd^y , IdyV^ \xe2\x80\x94 d\'^x , dx \n\n^\'d^ + \'Ad-xf =\xc2\xb0- -^;^ + ^-^^ = \xc2\xb0- \n\nd\'^y , dy^ , d\'^x dx^ dx \n\nd\'y , idyy dy d\'^x , Idx^ \n\n5. (^/ + dx\'^f/\'^ -\\-adx dy = 0. (I + dx^dy-^f^ - a d\'^x/dy\'\' = O. \n\nIf X or jl\' is given as a function of a third variable, 0, which we \nwish to make the independent one, we first transform the given ex- \npression, by means of (i) and (2), into its general form in which \nneither x hor y is independent, and then substitute for x, dx, d^x, \nd^x, or y, dy, d^y, dy, their values in terms of 6 and its differential. \n\n\n\n156 DIFFERENTIAL CALCULUS, \n\n6. Having t \xe2\x80\x94 x -\\- x"^, show that \n\n\xe2\x80\x94 -(4^+1)^ + 2^^. \n\nChange the independent variable from jr to in the following \nequations : \n\n7. -^ -^ A \xe2\x80\x94 - \xe2\x80\x94 o, when x = cos 9. \n\n^ dx"^ I- x^dx^ I - x"^ \n\n5r + "> = \'\'\xe2\x80\xa2 \n\n\n\n^\'y , I ^ , , 2 \n\n\n\n\n\n\n10. ^v\'* ^- + ax-^ -X- dy = o, when ;i; = ^ . \n\n^4(._.)| + ., = o. \n\n11. Having jc = r cos B, a.ndy = ?- sin 0, change the independent \nvariable from x to 9 in the equation \n\n^-v+d^l /d^ ; (^> \n\nFrom i) we have R = J ,, ^ -^^ ^ . ; \n\nand, since d\'^6 = o, \n\nf/jr = cos 6 dr \xe2\x80\x94 r sin dd, \n\ndy = s\\n B dr -\\- r cos 6 ^0, \n\nd\'^x = cos ^V \xe2\x80\x94 2 sin B dr dB \xe2\x80\x94 r cos a^0% \n\n</V = sin rtTV 4- 2 cos //r r/0 \xe2\x80\x94 r sin a^0^ \n\n\n\nCHAl^GE OF THE INDEPENDENT VARIABLE. 157 \n\ndx" -f dy"" = dr" + rV92, \nd^xdy \xe2\x80\x94 d\'^y dx = rd\'^rdQ - 2dr^dB \xe2\x80\x94 r^dB\\ \nSubstituting these values, we have \n\n12. Having x \xe2\x80\x94 a{<p \xe2\x80\x94 sin <p) and jj/ = <3(i \xe2\x80\x94 cos 0), change the \nindependent variable in eq. {a) from jc- to (J). \n\nAns. 7? = \xe2\x80\x94 4a sin (0/2). \n\n13. Having jr = r cos 6, and jj/ = r sin 6, transform \n\n2; =: (;f^ \xe2\x80\x94 ydx)/{ydy -\\- xdx) \nso that r will be independent. Ans. z = rdO/dr. \n\n14. Having dy/dx = 3(jr \xe2\x80\x94 2)\', show that d^x/dy* = \xe2\x80\x94 \'2-/<^{x \xe2\x80\x942) ^. \n\n15. \xe2\x80\x94 -i \xe2\x80\x94 r -^ = o, when ;i; = sm 9 \'; \n\n</ji;\' I \xe2\x80\x94 x^ dx \n\nd^y _ \n\n\n\ni6- :7:r5 + :xn-;:2 x = O\' ^^^^ ^r = a tan \n\n\n\n</;\xc2\xab:2 \' \xc2\xab\xc2\xab + ^\'^ dx \n\n\n\nd^y \n\n\n\nPART 11. \n\nANALYTIC APPLICATIONS. \n\n\n\nCHAPTER IX. \n\nLIMITS OF FUNCTIONS WHICH ASSUME INDETERMI- \nNATE FORMS. \n\nIi6. The symbols \n\nO/O, 00 /OO, O.OO, 00\xe2\x80\x9400, 0\xc2\xb0, 00\xc2\xb0, i" \n\nare indeterminate forms. \n\nWhen for a particular value of the variable a function of \nthe variable assumes any one of the above forms its corre- \nsponding value is indeterminate. \n\nThe limit of such a function, under the law that the \nvariable approaches the particular value, is determinate. \n\n(\xc2\xa739-) \n\nIn many cases this limit may be found by simple alge- \nbraic methods, otherwise a method of the Calculus is gen- \nerally used.* \n\n117. Form 0/0. \n\nLet fx/<px be a fraction such that both terms vanish \n\nT \xe2\x80\xa2 -J ^ J lii^it fx \n\nwhen X = a. It is required to nnd -\xe2\x80\x94 . \n\nX \xc2\xbb-> a (px \n\n* Any operation by which this limit is determined is getK rally \ncalled the evaluation of the corresponding indeterminate form. \n\n158 \n\n\n\nINDETEBMINATE FORMS. 159 \n\nFrom eq. (i), \xc2\xa7 62, we write \n\n, Ax^h)=fx-\\-hf\\x^eh\\ \n\n^{x -{-h)=cpx^ hct)\\x + d\'h). \n\nHence \n\n/{a + ^) ^ f{a + Oh) \n<p(a-\\-h) (p\\a^e\'h)\' \n\nfrom which, as h ^-\xc2\xbb o, we have \n\nfa/cpa = /\'a/(p\'a, \nor \n\nlimit f^ __ Y f\'^ _ f^ \nxM-^a (px (p\' X <p\'a \n\nSimilarly we may deduce \n\nlimit ffi _ ,. f!^_ _ ,. f^ _ f^a \n\n^, limit A f""^ \n\nZ**^ and 0**^^ representing, respectively, the derivatives of \nthe numerator and denominator of the same and lowest \norder, both of which do not vanish when x = a. \n\nIf /"^ and (p^a are both finite, the limit is finite. \n\n\\i f^a is zero, and <p"a is not zero, the limit is zero. \n\nli /""a is not zero, and 0"^ is, the limit is infinite. \n\nUfa and cp^\'a are both infinite, the limit is undeter-= \nmined. \n\nTherefore the required limit is the ratio, corresponding \nto the particular value of the variable, of the derivatives of \nthe numerator and denominator, of the sa7ne and lowest \norder, both of which do not vanish or become infinite. \n\n\n\nl6o DIFFERENTIAL CALCULUS. \n\nEach ratio as obtained should be carefully inspected and \nfactors common to both terms should be cancelled if pos- \nsible before proceeding to the next ratio. \n\nEXAMPLES. \n\n\n\n0, \'i(m>n. \n\n1, if ;;?=\xc2\xab \n00 , iim<.n. \n\n\n\n3. (sin x/x)^^^ = cos j;]^ = I.* \n\n4. (^--i)A]o = ^^]o=i. \n\n5. tan ^/^Jq = sec\'* xH = i. \n\n6. {x^ - a^)/ix\' - a^)\\ = 3^V2^]\xc2\xab = 3\xc2\xab/2. \n\n7. (^2 ~ ^2)/(^ _ ^f^^ - 2x/2{a - X)\\ = CO . \n\n8. {X - af\'/ia - x)y\\ = |(x - .)Vy|(;. - .)V4]^ = o. \n\n9. (i \xe2\x80\x94 sin x)/cos ^J^/2 = cos x/sin \xe2\x80\xa2^J^/g = o\xc2\xab \n\n10. (^^ - ^-^)/sin jc]^ = (^^ + ^-^)/cos jf]jj = 2. \n\n11. xVsin x]q = 2^/cos jt]^ = o. \n\nx^ I IX \'~\\ _ X cos ^ I _ cos ^ \xe2\x80\x94 ^ sin X I _i \nsin jr/ cos ;cJo 2 sin x Jo 2 cos x Jo~\'2\' \n\nX \xe2\x80\x94 sin x\\. _ I \xe2\x80\x94 cos jf I _ sin x~^ __ cos ^"1 i \n\nI \xe2\x80\x94 cos x\\ sin X I _ I \n\nsin\'* ^ _Jq 2 sin ;r cos jr_j0 2* \n\n* Hereafter, for abbreviation, the forms f{x)^^^ and fx~\\^ will fre- \n\nlimit , ^ ^ \nquently be used to express {fx). \n\n\n\nINDETERMINATE FORMS, \n\n\n\ni6i \n\n\n\n15. \n\n\n\n<\xc2\xbb*\xe2\x80\x94 2 sin X- \n\n\n\n;(? Jo I \xe2\x80\x94 cos X Jo \n\n-1 = \n\n\n\n^^-|-2 sin x\xe2\x80\x94e-=^ 1 \nsin jr Jo \n\n\n\ne^ -|- 2 cos jr + \n\n\n\n^mx _ ^ma n ^ ^^ mx "j _ ( 00 when r > I. \n\n^ \' {x - aj \\a ~ r(x \xe2\x80\x94 af^Ja " 1 o when r < 1. \n\nLimits of factors of the given or any derived ratio may be deter- \nmined separately (\xc2\xa7 36). \n\n\n\n\\/x tan X ~\\ k/^ "1 ^^" "^"l -^ \n\nThe given or any derived ratio may be separated into parts (\xc2\xa7 35). \n18. j/^- i/a-}-Vx-a ~] ^ / i_ ^ I \\ / -y "1 \n\n\n\n2x\\/x 2x\\/x \xe2\x80\x94 a. \n\n~ 2xr x-a Ja ~ ^Ta \ntan\xc2\xa3_\xe2\x80\x94 _sin^~| _ / sin x \\ /sec x \xe2\x80\x94 i\\~l _ sec ^ \xe2\x80\x94 il \n\nJo w A""^^^ yjo~ ^^ J( \n\n\n\nsec X tan x \n\n\n\nx^ I _ sec^x-|-tan\'*;r secjcj __i_ \nJo 2 Jo~2\' \n\ntan jf \xe2\x80\x94 sin or\\ /tan ^\\ /i \xe2\x80\x94 cos x\\~l i \xe2\x80\x94 cos or\\ \n"\xe2\x80\xa2 \xe2\x80\x94l?^ Jo = (\xe2\x80\x94 ) i--^-) J. = -7^ Jo \n__ sin x "1 cos X \n\n~(n- i)x\'\'-\'Jo ~(n- i)(n -2);r"-\'Jo "" "^ \' \n21. log (I +x)/x\']^ = 1/(1 + x)\\ = r. \n\nIn some cases it is advisable to transform the terms before apply- \ning the above rule. Thus \xe2\x80\x94 \n\nsin X "I 2 sin (x/2) cos (r/2)~l . . -, \n\n22. = ~- \xe2\x80\x94 -\xe2\x80\x94-- \xe2\x80\x94 = cot (x/2) L = 00. \n\nI \xe2\x80\x94 cos ^J 3 sin^ {x/2) ^ \' \'_[0 \n\n\n\n1 62 DIFFERENTIAL CALCULUS. \n\nii8. Form oo/oo , \n\nIf yiz = 00 = 0^, we may write \n\n\n\n~;r \xe2\x80\x94 ^r~r \xe2\x80\x94 ~-> and (\xc2\xa7117) \n\n0\xc2\xab <i>al fa o ^^ " \n\n\n\n0\xc2\xab 0^/ /< \nlimit r I / I "1 ,. r0\'^ //\'jt: "I \n\n^~.Ls/aJ="\'\xc2\xb0LwV7?J \n\n=-[(i)\'(g)]- \n\nand since limits of equimultiples of two variables with \nequal limits are equal, we have, multiplying by \n\n<t)xf\'xlfx (p\'x,^ \n\nlimit fx .. f\'x \n\nxM-^a 0x cp X \n\nSimilarly, we may deduce \n\nlimit /^^_ f\'\'x _ __ f\'x _ f^a \n\nThe method is therefore the same as in the preceding \narticle; but by \xc2\xa771 we have /\'\'^/0\'\'^ =: 00/co, when \nfa/cpa = co/00 for a finite value of a. Hence for finite \nvalues of a this method will fail to determine the limit un- \nless factors common to both terms of some derived ratio \nbecome apparent or the limit oi /""x/cp^\'x becomes otherwise \nknown. \n\n\n\nTaylor\'s Calculus, \xc2\xa779. \n\n\n\nINDETERMINATE FORMS, 163 \n\n\n\nEXAMPLES. \n\n\n\nlog sin 2x \\ _2. cot 2x _ /cos 2x\\/sin x\\/\\ \nlogsin;(;Jo cot x _Jo \\cosx /\\sin 2;i:/_Jo \n\n_ /2 cos^ JT \xe2\x80\x94 i\\ / sin X \\ sin ^cH \n\n\\ cos X /\\sin2x/Jo sin 2;ir_jo \n\n3 2J(rJo \n\n\n\ncos \n= 2- \n\n\n\nloor \n\n\n\n^ = - -4^ = ~ ^]o = O = ^ log ^"lo. \n\n\\Jx Jo lAUo -^ -^ \n\ni/x l _ _i/f!_~| _ sin^r"! _ \ncot ^Jo~ i/sin* ;rJo"~ ^^ Jo~ \n\nlog -^ "1 _ i/x "1 _ _ sin^ X [ \n\ncosecj;_Jo cot j; cosec x_| xcos^_Jo \n\n2 sin jr cos X I \n\n= -. =0. \n\ncos X \xe2\x80\x94 X sin ^_Jo \n\nlogjc I sin^ jv 1 . -1 \n\n5. \xe2\x96\xa0 = =: \xe2\x80\x94 2 sin ;\xc2\xab\xe2\x80\xa2 cos JT L = O. \n\ncot^Jo X Jo -J\xc2\xbb \n\nin which \xc2\xab is a positive integer. If n is fractional, the exponent of \nX ultimately becomes negative, giving the same result. \n7. Putting j^ = \\/xy whence y \xc2\xbb\xc2\xbb-> o as x ^-\xc2\xbb 00 , we have \n\nIn some cases we may with advantage place x \xe2\x80\xa2= a \xe2\x96\xa0\\- h and sub- \nsequently make ^ = o. Thus ^x \xe2\x80\x94 a/ \\/x^ \xe2\x80\x94 a^ reduces to 0/0, and \nthe ratios of all derivatives of both terms become 00/00 when x = a; \nbut putting jc = a -|- ^, we obtain \n\n\\/~^^^^a "I _ /^V3 n _ /?i/i^ ~] _ \n\n\n\nGO. \n\n\n\n164 DIFFERENTIAL CALCULUS, \n\n%. \n\n119. Form 0.00 . \n\nIf /^ = o and 0^ = 00 , we write \n\nwhich takes the form 0/0 or 00 /co when x = a^ and the \nlimit may be determined by the method of \xc2\xa7\xc2\xa7117, 118 \nwith the same limitations. \n\n\n\nEXAMPLES. \n\n-\'" ;r Jo \xe2\x80\x94 X Jo \n\n2. //^\'^ = \xe2\x80\x94 = = CO . \n\nJo x-^X 2 Jo \n\n3. .-VV],=;.V^V-],=0. \n\n4. e-^ log x]^ = log x/e\'\'\\ = (i A)A^^ = O. \n\n5. sin jir log cot x\\ = (sin jr/x) ;v log cot x\\ = log cot jf/(i/;i;)T \n\n= (x\'^/sin^^)(i/cot x)^^ \xe2\x80\x94 o. \n\n= I \xe2\x80\x94 \\j/{x^ + i)]^ = I, when tan-i(i/oo) = O. \n\n7. sec J? (:v sin x\xe2\x80\x94 7t/2)\'J^^^ = (;r sin x \xe2\x80\x94 7t /2)/cos \xe2\x96\xa0^J^/2 \n\n= (a- cos jc -{- sin .jf)/ \xe2\x80\x94 sin x \\, = \xe2\x80\x94 i. \n\n8, log (2 - i), tan ^]^ = log(a - ^) /cot ^]^ \n\n\xe2\x80\x94 \xe2\x80\x94 I / \xe2\x80\x94 7t/2a ~| _ \xc2\xa3 \n\n~~ a(2 \xe2\x80\x94 x/a)/ sin^(7rx/2\xc2\xab) J^ ~ it\' \n\n\n\nINDETERMINATE FORMS. 1 65 \n\n\n\n10. x\'^{\\ogxf\\^ \n\n\n\n\xe2\x80\x94 \xc2\xab/x(log xf\'^^_\\ \n\n\n\n.], \n\n\n\ntnx \ni/(log xfA( \n\nwhich remains indeterminate under the method; but placing x \xe2\x80\x94 e~^, \nwhence x^{\\og xf = (\xe2\x80\x94 ify\'/e"-^ and ^/m-^co as x-m-^o, we have \n(example 7, \xc2\xa7 iiS) o for the limit. \n\n120. Form 00 \xe2\x80\x94 00 . \n\n\\i fa \xe2\x80\x94 <pa= ^ \xe2\x80\x94 00 , we write \n\nwhich becomes 0/0 when x = a, and the rule of \xc2\xa7 117 will \napply after the given expression is placed under the form \nindicated in the second member. \n\nEXAMPLES. \n\n\n\nI. cot X \n\nX \n\n\n\nJo \\cot X II cot jrjo ^ ^\'^"^ ^ Jo \n\nsmx \xe2\x80\x94 xcosx _ X sin X _ sin jr-|-jtr cos ;tr _ \n\n;t: sin ;r Jo ~ sin X 4--^ cos X Jo 2cos jr\xe2\x80\x94 ;r sin jrJo~ \n\nIn some cases the desired form may be obtained by a more simple \ntransformation. \n\ni/~2 "I I \xe2\x80\x94 4/1 \xe2\x80\x94 a/or\\ ^ "] a \n\n2. X \xe2\x80\x94yx^ \xe2\x80\x94 ax I = *- L\xe2\x80\x94 = \xe2\x80\x94 \xe2\x96\xa0 -. = -. \n\n-^\xc2\xb0\xc2\xb0 lA Joe 2fl-aAj^ 2 \n\n3. tan ^ - sec x\\/^= (sin x-i)/cos x\']^^^ = -cos ^sin x\']^^^ =0. \n\n2 _ I ~| _ 2{x - I) - jx\'\' - i) "1 _ 2\xe2\x80\x94 .y-i n \nc\'- l~;\xc2\xabr-iji~\' (x-^-i)(^-l) Ji~ ^=^-1 Ji \n\n\n\n4 . \n\n\n\n\xe2\x80\x94 I \n\n2X \n\n\n\n5. X tan jf \xe2\x80\x94 ;r sec x/2^^, \xe2\x80\x94 {x sin x \xe2\x80\x94 ;r/2)/cos -^J^/j \n\n= (x cos x-\\- sin ;f)/\xe2\x80\x94 sin x^^j^ = \xe2\x80\x94 i. \n\n\n\n1 66 DIFFERENTIAL CALCULUS, \n\n121. Forms o*, oo*, i*. \n\nIf {fa)\'^\'\'.\xe2\x80\x94 o*^ or 00^ or i\xc2\xb0\xc2\xb0, we write \n\nlog (/x)*^ = (px log/x, whence (/xy^ = ^<^^io&/^^ \nand \n\n^^^^^ (/x)*^ = lim e\'^^ ^\xc2\xb0s -^-^ = (?\'^\'" [\'^\xe2\x96\xa0^ loff^-*^] \n\nin which (pa log fa = o.oo in each of the above cases, and \nthe method of \xc2\xa7 iig will apply after the given expression \nis placed in the form- indicated. \n\nEXAMPLES. \n\n3. x\'n =,10^-/-] =.iAi =1. \n\ntan\'^n tan \'^ log 6-^)"] ^ \n\n5. (2 - x/a)^\'^ Ja= ^ ^"^ J a ~ \'^\' \n\n6. cot ^sin ^J^ \xe2\x80\x94 ^sin ;\xc2\xab log cot ^ j^ _ j. \n\n\n\n^U^^ log a + 3\xc2\xbb" lo g ^ + . . \n^ a\xc2\xab^ + <5\xc2\xab^ + . . . \n\n\n\nJ^^^log(a^...\xc2\xab)^^3, . .\xc2\xab. \n\n\n\n122. Evaluation of Derivatives of Implicit Functions. \n\nDerivatives which for particular values of the variables as- \nsume indeterminate forms may be evaluated as in the pre- \nceding cases. \n\n\n\nINDETERMINATE FORMS, 1 67 \n\nEXAMPLES. \n\n1. y^ -\\- x^ = ax^. \n\nWhence ^1 ^ ?^^J=Jf!l = ^Hi^l . \n\n^\xe2\x80\xa2^Jo.o 3/ Jo.o ^y{dy/dx)_\\^,^ \n\nHence, {dy/dxf^ . ~ \xc2\xb0\xc2\xb0 \' ^"^ ^/^-^Jo . = ^ \xc2\xb0\xc2\xb0 \xe2\x80\xa2"\'"\' \n\n2. x^ -\\- y^ = 3axy. t/j//^x J^^ ^^ = o or 00 . \n\n3. r^ + 3\xc2\xab/ \xe2\x80\x94 20^:7 = ax"^. dy/dx~^^^ = i or \xe2\x80\x94 1/3. \n\n4. (^2 +y2)2 ^ ,,.(^2 _ y). ^^./^^J^ . = \xc2\xb1 I. \n\n5. w = ;V^ 4" 3\xc2\xabV^ \xe2\x80\x94 4^*^x7 \xe2\x80\x94 a\'^x^ = O. \n\n\'du/dx = \xe2\x80\x94 ^(i^y \xe2\x80\x94 id^x. \'du/dy \xe2\x80\x94 4jj/\' + 6a V "~ ^x. \n\n\n\nay \ndx \n\n\n\n~j _ ld^y-\\-d^x n _ 2a-{dyldx)-\\- d^ ~j \n\nJo.o ~ "^y\' + 3\xc2\xabV - 2a\'^Jo.o~ (^y-* + 3\xc2\xab\')(^j/^-^) - 2^2 Jo , \n\nn . Hence (#)-i^1 =i \n2 Jo.o \\dx I 3 ^^Jo.o 3 \n\n\n\n_ lidvldx) -|- \n~" 2,{dy/dx) \xe2\x80\x94 \n\nand ^v/^^]o.o=(2 \xc2\xb1 4/7) /3. \n\nt. u- ay - a\'^x^ -x\'^=o. dyldx\\ <, = \xc2\xb1 I. \n\n7. \xc2\xab = X* + rtxV \xe2\x80\x94 ay^ \xe2\x80\x94 o. 4j//^xJq = o or \xc2\xb1 I. \n\n8. \xc2\xab = oyS _ ^^2^ _[_ ^4 _ o. ^^\'/^Jfjo . = \xc2\xb0 \xc2\xb0\'\' \xc2\xb1 I^V^. \ng. M = jtr* \xe2\x80\x94 a2_;^y -f-y = 0\xc2\xab ^/\'^\'^Jo ~ \xc2\xb0 ^\'^ ^\xe2\x96\xa0\'\xe2\x80\xa2 \n\n10. M = JT* 4- >/3 -f \xc2\xab3 _ ^^j^y _ o. ^y/a\'x]^ ^ = 1- (I \xc2\xb1 4/^). \n\n11. \xc2\xab = ^/s \xe2\x80\x94 3axjj/ + jc* = o. </j//fltxJo .^ =r \xe2\x80\x94 o or 00 . \n\n12. \xc2\xab = ^* + \'2.ax\'^y \xe2\x80\x94 ay3 = o. dy/dx L ^ = o, or \xc2\xb1 |/2. \n\n* ^r/c/x 1^ ^ has two limiting values for the same value of the vari- \nable, hence it is discontinuous at the corresponding state. \n\n\n\nl68 DIFFERENTIAL CALCULUS. \n\n\n\nMISCELLANEOUS EXAMPLES. \n\n\n\n3 \n\nX \n\n\n\nlog x~\\ e^ \xe2\x80\x94 <?-^ \xe2\x80\x94 2x "1 \n\n-^\xe2\x80\x94 =0. 2. r =2. \n\nX _\\^ x-%\\nx Jo \n\nlog f 1 ^ J (^-i)tan\'^. y"] ^ ^ \n\niji \' -^^ Jo \n\n\n\nI \xe2\x80\x94 cos X \n\n\n\n1 I 1 \n\n= - , 6. sec jc \xe2\x80\x94 tan x \\ = o. \n\nJo 2 JV2 \n\n^ -\xe2\x96\xa0 "1 ^ - COS- Hi \xe2\x80\x94 ^n \n\n7- -*8^ = o. 8. \xe2\x96\xa0 \xe2\x80\x94 - = I. \n\nJ 00 ^2X-X^ Jo \n\n\n\n\n\n\na^ \xe2\x80\x94 b\'^ \n\n\n\n]10gU_-f-_\xc2\xab\xc2\xa3C)-l \n\n= \xc2\xab. 10. tf ^ = (f\xc2\xab. \n\n00 -^0 \n\n\xc2\xab ^1 /tan ;t:\\ V^n \xc2\xbb \n\n;^-iog.-l ^ ^ (L^zilI = _ i. \n\n^lA Jtt ^ tan^^ Jo 2 \n\n15. = O. 16. = CO. \n\nX Jo --^Jo \n\n19. (I + ax)^^ = ^. 20. log (^) /.]^ = 1. \n\n21. (I + ..)^J^= I. 22. ^^^^J, = log (7). \n\nlog^l 24 ^A^ n -^ \n\n23. -^ = o. 24. ^^ - 8 \xe2\x80\xa2 \n\n\xe2\x80\xa2^ Joo cot \n\n2 -"o \n\n\n\n= log -7 26. 2^ sin \xe2\x80\x94 = a, \n\nJo ^ -^ 2^J^ \n\n\n\n/ , cosec2 fA:~j " l?a COt X + COSec JT \xe2\x80\x94 iH \n\n27. (cos ax) \\ \xe2\x80\x94 e *\xe2\x80\xa2. 28. ; \xe2\x80\x94 = \\ \n\nJo cot^-cosecx+ij^/^ \n\n\n\n29. ^^"^^1 = I. \n\ngx ^sin x\\ \n\n31. : = I. \n\nX \xe2\x80\x94 sin jrjo \n\n= 3\xc2\xab. \n\n\\ ax \xe2\x80\x94 a _la. \n\n\n\nINDETERMINATE EORMS. \n\n30 \n\n32 \n\n\n\n169 \n\n\n\nI j:); \n\nlog X log x_ \n\n\n\n33 \n\n\n\n34. \n\n\n\n35- ^ \n\njf3 _ 2;t- \n\n\n\n^ ^"1 ^ L. \n\n\' X \xe2\x80\x94 \\ logx Ji 2 \n\n;f3 \xe2\x80\x94 3x +2 ~| _ \n\nx" - tx^ + 8;\xc2\xbb: - 3J1 ~~ "\'* \n\n^3\xe2\x80\x941 ~1 , tan \xc2\xabx \xe2\x80\x94 ft tan jf~l \n\n\xe2\x80\x94 ; = 3. 36. -. : = 2. \n\n2;tr^ -[- 2x \xe2\x80\x94 I w sin ^ \xe2\x80\x94 sin w^ j^^ \n\n37. tan Vlog (^ - V2)];r/2 = - \xc2\xab^ \xe2\x80\xa2 \n\n38. (jc \xe2\x80\x94 rt 4- V2^x - 2a^)/\\/x^ \xe2\x80\x94 af^^ \xe2\x80\x94 I. \n\n39. x/cO\\. X \xe2\x80\x94 7t/2 COS \xe2\x80\xa2^l;r/2 = \xe2\x80\x94 !\xe2\x80\xa2 \n\n40. (i \xe2\x80\x94 x) tan (;rx/2)l^ = 2/7t. \n\n41. (aA-2 - 2acx + ac\')l{bx\'\' - 2bcx + ^^2)^^ _ ^/^^ \n\n42. (i \xe2\x80\x94 sin j; + cos x)/(sin .a; -|- cos x \xe2\x80\x94 1)1^/2 = !\xe2\x80\xa2 \n\n43. (a - jr \xe2\x80\x94 a log a 4"^ log x)l {fl \xe2\x80\x94 ^2ax \xe2\x80\x94 ^^) J^ = \xe2\x80\x94 1, \n\n44. {x sin x - n/2)/cos -^J^/j = \xe2\x80\x94 i. \n\n45. {e^ - e-^ - 2) sec x/x\'^\'\\^ = 4. \n\n46. \\ ^2 \xe2\x80\x94 sxti X \xe2\x80\x94 COS ;c)/log sin x\\^,^ = \xe2\x80\x94 1/2 |/2. \n\n47. [(^ - ^-=^)\' - ^xXc^ + ^-^)]A\'\'], = - 2/3. \n\n48. [(jf \xe2\x80\x94 aY/{e\'\' \xe2\x80\x94 ^\xc2\xab)]^ = o or ^-\xc2\xab or 00 , according as \xc2\xab > I, \n\nn = 1, n < 1. \n\n49. 1/(1 + ^lA) 4- ^VV^(i + ^ia/]^) = o. \n\n50. 1/(1 + e-y^) \xe2\x80\x94 e-y^x{i + ^-iA)\']o = I. \n\nX + cos jr I I + cos x/x \n\n51. : = \xe2\x80\x94 \xe2\x96\xa0 \xe2\x80\x94 r 7- = I. \n\nX - sin xj^ I - sin x/^ J^ \n\n\n\n170 DIFFERENTIAL CALCULUS, \n\n\n\nCHAPTER X. \nDEVELOPMENTS. \n\n123. The development of a function is the operation of \ndetermining an equivalent finite or convergent-infinite series. \nWhen this can be done, the function will be the sum of the \nseries, which, in the case of a convergent-infinite series, is, \nalso, the limit of the sum of n terms as n increases without \nlimit. \n\nA convergent series having a given function as a limit \nmay be used to determine approximate values of the func- \ntion, the degree of approximation depending upon the ra- \npidity of convergence and the number of terms considered. \nA divergent series should not be employed in finding ap- \nproximate values of a function, or in the deduction of a \ngeneral principle or formula. \n\nLet S represent a function giving \n\n.S" = 2/, -f 2^2 + ^3 + \xe2\x80\xa2 \xe2\x80\xa2 \xe2\x80\xa2 + ^\xc2\xab + etc. \nDenote the sum of the first n terms by S^ , and the sum \nof the following terms by R^ called the remainder; then \n\nThe series is convergent if R is an infinitesimal as n in- \ncreases without limit, in which case S is the limit of S^. \n\nWhen S is the limit of S^ as n increases, it is also the \nlimit of Sn - 1 , and we have \nlimit \n\n\n\nSn \xe2\x80\x94 Sn-x = lim Un \xe2\x80\x94 C \n\n\n\nDEVEL OP MEN TS. 1 7 1 \n\nThat is, in a convergent series the \xc2\xab*** term is an infini- \ntesimal as n increases, but the converse is not necessa- \nrily true unless Sn has a finite limit under the law; for \nlim Un =^ o =^ lim [6\'\xe2\x80\x9e \xe2\x80\x94 \'^"\xc2\xab-i] may occur when 6" = oo. \nTherefore a series is not necessarily convergent when \nthe n}^ term is an infinitesimal as n increases. \n\n124. Taylor *s Formula has for its object the develop7?ient \nof a function of the sum of two variables into a series arranged \naccording to the ascending powers of one of the variables with \ncoefficients which are functions of the other. \n\nAssuming an expansion of the proposed form, we write \n\nf(x^r h) =^ X,^ XJi^ XJi\' -\\- . . . J^X^^^h^^R, {b) \n\nin which X,, X^^ etc., are functions of x to be determined, \nand R the remainder after n-\\- \\ terms. \n\n^ = o gives fx = ATj. \n\nPlacing x-\\- h^ s^ and differentiating, first with respect \nto X and then with respect to h^ we have \n\ndf{x + h)ldx = Ws/di) (ds/dx\\ (\xc2\xa7 77). \n\ndf(x + h)/dh = (dfs/ds)(ds/dh). \n\nBut ds/dx = ds/dhy hence \n\ndf{x + h)/dx = df{x + h)/dk \n\n\n\n172 \n\n\n\nDIFFER EN TIA L CAL CUL US. \n\n\n\nHence, differentiating the second member of {b) first \nwith respect to x and then with respect to h, we have \n\n\n\nax ax ax ax \n\n\n\n^^^\xe2\x80\xa2 + ^^^" + ^^- \n\n\n\n= X, + 2X,h + ^X.h\'^ + ^X,h\' + etc. + nXn+,h-^-V etc., \n\nwhich is an identical equation, and by the principle of in- \ndeterminate coefficients we have \n\n\n\ndX, _ \n\n\n\ndX. \n\n\n\nV ^^3 XT <dXn \xe2\x80\x9e \n\n2^3 > ~r: \xe2\x80\x94 3^4 > \xe2\x80\xa2 \xe2\x80\xa2 \xe2\x80\xa2 ~r~ =nXn+u etc. \n\n\n\ndx ^\'\'\' dx """ ^jc ^"4.--- ^^ \nSince X, =/r, dXjdx \xe2\x80\x94 fx^ .\'. X^ \xe2\x80\x94 fx. \n\n\n\nTherefore \n\n\n\n-\'=/"^ = 2X3, and X,^\\f"x, \n\n\n\ndX. \n\ndx \n\ndX, \n\ndx \n\netc. \n\ndX. I \n\n\n\n= i/\'"^ = 3^o and ^^ = -^/":^. \n\n2-3 \n\netc. \nf" x\xe2\x80\x94nXn^u and ^\xc2\xab+i =p/\xc2\xab^. \n\n\n\n^Jt: \\n \xe2\x80\x94 I \n\n\n\netc. \n\n\n\netc. \n\n\n\nSubstituting these expressions for X^ , X^ , ^3 , etc., in \n{b), we have Taylor*s formula* : \n\nf(x + h) = fx + fx h + f"x hV2 + f\'"x liy|3 + . . . \n\n+ f\xc2\xabxhV|n + R, . . . . W \n\nin which fx represents what the given function becomes \nwhen h = o, f\'x^ f"oc, etc., represent :he first, second. \n\n\n\n* Formula published in 171 5 by Dr. Brook Tavlor. \n\n\n\nD E VEL OF ME NTS. 1 73 \n\netc., derivatives of fx, and i? the sum of all of the terras \nafter the {n ^ t)^^ \n\nThe second member of {c) is also known as the develop- \nment of the second state of a function of a single variable \n\n(\xc2\xa7s)- \n\nDesignating /(^ + ^) by jv\', and/jc by j, we have \n\nwhich is another form of (c). \n\nTo apply Taylor\'s formula, cause the variable with refer- \nence to which the development is to be arranged^ to vanish. \nDifferentiate the result and its derivatives in succession until \none of the highest order desired is obtained^ and substitute \nthem^ respectively, for their corresponding symbols in the \nformula. \n\nThus, to develop (x -\\- y)\'*^, place 7 = 0, and differentiate \nx^y whence \n\nf(x) = x^, f\\x) = mx\'^-\\ etc. \n\ny\xc2\xab(x) = m{m \xe2\x80\x94 1) . . , , {m \xe2\x80\x94 n-\\- i)x^"^. \n\nSubstituting in (<:), we have \n\n1.2 \n\n+ -^ A -^\xe2\x80\x94 ^ x"^-\xc2\xaby\xc2\xab + r: \n\n\\n -^ \' \n\n125. Lagrange\'s Expression for the Remainder in \n\nTaylor\'s formula put x -\\- h =^ X, whence h =\xe2\x96\xa0 X \xe2\x80\x94 x, \ngiving \n\nfX=fx ^fx{X - x) -\\-f\'x{X - xy/2 + . . . \n\n+/M^-^)V)^+^. . . (i) \n\n\n\n174 DIFFERENTIAL CALCULUS, \n\nAssume R = F{X \xe2\x80\x94 x)" ^ VV ~^ ^j i^^ which F is an un- \nknown function of X and x, which will make (i) exact for \nall values of x and X, giving \n\nfX -fx -fx{X -x)-... \n\n-r\'x{X - xY/\\n_ - F{X - xY-^y \\n + i = o. . (2) \n\nSubstitute z for x (except in F)^ and let Fz represent \nthe result which in general will not be equal to zero, \ngiving \n\nFz =fX-fz -fz[X -z)- .., \n\n- rz{X - zYI\\n_ - F(X - zY + V I;. + I . (3) \n\nFrom (2) and (3) we see that Fx = o, and from (3) we \nhave FX =0. As s varies from x to Xy Fz increases and \nthen decreases or the reverse, and F\' z^ if continuous, must \nchange its sign by vanishing for some value of z between x \nandX (\xc2\xa716, \xc2\xa763.) \n\nDifferentiating (3) with respect to z and reducing, we \nhave, since the terms with the exception of the last two \ncancel in pairs, \n\nF\'z = - /" + ^^(AT - zYI\\>l-\\-F{X - zY/\\n. \n\nLet x-\\-(^n{X \xe2\x80\x94 x), in which 6*^ is a positive number less \nthan unity, represent the value of z for which F\'z = o. \nThen \n\nF=r^\\x-i- 0^(X -> x)) = /" ^^\\x + M), \n\n\n\nDEVELOPMENTS. 1/5 ^ \n\nand ^ R = f^^\\x + e nh)h^\'\'^^ /\\ n + i *, \n\nin which o </9^<i. \n\nThis expression for R enables us to determine the condi- \ntions of applicabihty of Taylor\'s formula. \n\nWhen as n-m-^oo , R is an infinitesimal, the formula gives \na finite or convergent-infinite series for the function. \n\nThis condition is fulfilled when as 7im-^co ^ fx and /"\'^jc \nremain continuous between states corresponding to all values \nof X from any assumed value of x to x -\\- h, for then \n/""^^{x-^-dnh) is always real and finite, and (\xc2\xa741) /^"+VU+i \napproaches zero as a limit. f \n\nWith any assumed value of x which fulfils the above \ncondition, h will frequently have limiting values. They are \nthe numerically least positive and negative values of h that \ncause yir or any of its derivatives to become discontinuous \nwhen X -\\- h \\s substituted for x. \n\nIf, for any assumed value of x, fx or any of its deriva- \ntives of a finite order, becomes imaginary or infinite, the \ncorresponding limiting value of h must be zero, and the \nformula is inapplicable. It may, however, develop the func- \ntion for other values of x. \n\nTo illustrate the use of R, take the example \n\n{x + yf^ = x^ + pix\'^-\'y -f \'-^^i^-^ x^\'-y -f . . . \n\n+ ^^"^ ~ ^^ \xe2\x80\xa2 \xe2\x80\xa2 , J^ ~ ^ \'^ "^ x-y- + R, \n\nin which /\'^jc = m{m \xe2\x80\x94 i) . . . . [m \xe2\x80\x94 n-\\- i)x*\'\'"^, \n7n(m \xe2\x80\x94 1) . . . (m \xe2\x80\x94 n) y"\'^^ \n\n\n\nand R = \n\n\n\n\\n^i [x^ 6,yY \n\n\n\n-m+V \n\n\n\n* Known as Lagrange\'s expression for the remainder, \nt If, as \xc2\xab\xc2\xbb\xc2\xbb-\xc2\xbb\xc2\xbb , /^jcm-^sa /a requires evaluation, \n\n\n\n1/6 DIFFERENTIAL CALCULUS. \n\nWhen m is fractio7ial or negative, n may increase without \nlimit, the series will have an unlimited number of terms, \nand m \xe2\x80\x94 n will become negative when n>m numerically, \ngiving \n\nf^x = m(m \xe2\x80\x94 i) , . . (m \xe2\x80\x94 n-\\- i)/x\'^~\'^. \n\nIf ^ <o, the development fails when the exponent n \xe2\x80\x94 m \nis a fraction with an even denominator. \n\nIf ^ = Ojfx ultimately becomes oo and the formula is \ninapplicable. \n\nIf jc > o, we have the ratio of R to the (\xc2\xab + 2)th term \n\nequal to \n\n{xl{x + 6\',;.))-\'\xc2\xbb+\', \n\nwhich vanishes as n 3\xc2\xa9-> oo ; and R, therefore, diminishes \nindefinitely when the successive terms in order likewise \ndecrease. \n\nThe ratio of the {n + i)th term to the nXh is \n\nm \xe2\x80\x94 n -\\- \\ y m/n \xe2\x80\x94 i + \\/n y \n\nn X 1 x^ \n\nthe limit of which, as n ^-> oo, is \xe2\x80\x94 (j/^v). \n\nHence, when x is numerically greater than y the succes- \nsive terms in order will ultimately decrease indefinitely. \n\nIn which case i? will be an infinitesimal; and we conclude \nthat when 7n is fractional or negative, the binomial formula \ndevelops (^H-jf)"" for all positive values of x numerically \ngreater than 7. \n\nEXAMPLES. \nI. Develop {x-\\-yf\'^. \n\nf{x) = |/^ f\'{x) = 1/2 V^, f"{x) = - 1/4 \\fj\\ etc. \n\n(^ + J)\'/^ = 4/^-1- y/2 i/x- //8 \\/7\' 4- R, \n\nwhich fails for ;tr = or < o. \n\n\n\nD E VEL OP MEN TS. 177 \n\n2. Develop cos {x -\\- y). \n\nf{x) = cos X, f\\x) = \xe2\x80\x94 sin x, f"{x) = \xe2\x80\x94 cos x, \n\nf"\\x) \xe2\x80\x94 sin X, etc. \ncos (x -\\- y^ = cos X \xe2\x80\x94 y sin X \xe2\x80\x94 y^ cos x/2 -\\- y^ sin \xe2\x96\xa0^/|3_ \n\n+ y* cos x/|4_-l- i?, \nwhich is true for all values of x and^. \nMaking x = o, we have \n\ncos ;>/ = I \xe2\x80\x94 y/2 +y/|4_\xe2\x80\x94 /l6_+ ^. \n\n3. sin {x -\\-y) = sin x -]- y cos x \xe2\x80\x94 y"^ sin x/2 \xe2\x80\x94 y cos x/\\$ \n\n-\\-y* sin x/|4^+y cos x/\\s_-j- R. \nWhence . sin ;/ = ;/ - y/|3 -|_y/|5__ //|7_+ ^. \n\n4. sin-l(jc+^) = sin-i;t: -j -} \n\n\n\nI + 2x\' f 3x(3 4- 2x^) y I ^ \n\nt/(i - xo^ 11 \\/{i - x\'y li \n\nwhich fails when jr = i or > i numterically. \nFor values of jr < 1 numerically, limiting value of y =^ i \xe2\x80\x94 x. \nMaking x = o, we have \n\nsin-i;/ = -j,4-y/|3_4. 3y/j^+ ^. \n\nwhich is true for all values of x andy. \n\nWhence ta.n-^y = y \xe2\x80\x94 y^/3 -\\-y^/s + \xe2\x96\xa0^\xc2\xab \n\n6. Develop loga(x -f- y)\' \n\nfx = logaX, f\'x = Ma/x, f\'x = \xe2\x80\x94 Ma/x"^, etC, \n\n/\xc2\xabx = (- iY-^Ma\\ n - I /x^ Hence, \n\niog.(. +^) = iog\xe2\x80\x9e. + i/\xe2\x80\x9e(z _ \xc2\xa3, + ^ _ . . . +(_,)\xe2\x80\x9e-.\xc2\xa3_)+,. \n\nIn which i? = \xc2\xb1 May^\'^\'^/in -f- i)(x -|- 6^^)\'*+^ is an infinitesimal, \nas wm-^ 00 when x = or > j numerically. \n\n\n\n17S DIFFERENTIAL CALCULUS. \n\n\\i X \xe2\x80\x94 Q>, the formula fails. If ;r = i, we have \n\nioga(i -^y) = Ma{y-fl2 + ...-!-(- i)\xc2\xab-{rV\xc2\xab) 4- R, \n\nlog (I +;^) = / - ;^V2 + y/3 - jV4 + ^. \n7. a^+jv = \xc2\xab^(i -^Xogay -\\- \\og^ay\'^/2 + . . . + \\og\'>\'ay\xc2\xbb/\\n) + R. \n\nR = a> ^ "^\' log ^ a_y /[wj-j \nis an infinitesimal as n n-^ 00 , since log a is a constant, while \n[y/i^ ~{~ i) I \'^"^ ^- Making ^ = o, we have \n\na^ = I -\\- log ay + log^ ay^/2 + . . . + log\xc2\xabfl!jj/Vt+ -^0. \n\n8. Develop (x^/^+/)^ \n\nThe variables are \\/x and y^. \n\nWhen the variables considered are not represented by the first \npowers of letters, substitute the first powers of other letters for the \nvariables, develop the result, and resubstitute the variables for the \nauxiliary letters. \n\nThus, placing x^^^ = r, and y"^ \xe2\x80\x94 s, we have \n\n(x\'/\' +/)\' = (^ + ^)\\ Ar) = r\\ f\\r) = Sr\\ f"{r) = 20r^, etc. ; \nHence, \n(y/2 -\\-y^y = {r + sf \xe2\x96\xa0= r^ -f sr*s + lor^^^^ + lorV + 5^^* -{- s^ \n\n= //^+ 5xv\' + lo/zy-f IOX/+ s^^^y +y^\'^\' \n\n9. V^ + x+y -= V^T^+ -7=-- -7=== 4- R, \nwhich fails when x = \xe2\x80\x94 a or < \xe2\x80\x94 a. \n\n10. (x-a +yf^ = {x- af^ + six-af\'y/2+is(x- a+Qyfy/S. \nx = a gives //\' = i50\'/y/V8, .\'. B = 64/225; \n\nbut the development would fail with more terms. \n\\i X < a, the formula fails. \n\n11. tan {x -\\-y) = tan x -j- sec\' xy-{- 2 sec\'* x tan jfj/y2 \n\n4- 2 sec\' Jf(i + 3 tan\' j^)//|3,+ -ff. \n\nInapplicable when x s tt/s. \n\n\n\nDEVELOPMENTS. 179 \n\n1/ ix^ \xe2\x80\x94 I y^ \n\n14. sec-i ix -\\-y) = sec-i X -| ; ^ h -^\xe2\x80\xa2 \n\n15. cos\'\' {x \xe2\x80\x94 y)z=. cos\'* ^ +JK sin 2x \xe2\x80\x94y^ cos 2x \xe2\x80\x94 2y^ sin 2j:/3 \n\n+JJ/* cos IX / 2, + 2^ sin 2x/i5 + i?. \n\n16. 2(;f + >\')\' - 3(^ + J)\' + I = ^\'(2x - 3) + I \n\n+ 6(jf2 _ x)y -\\- 2,{2x \xe2\x80\x94 i)y + 2JJ/3. \n\nt^;c\xe2\x80\x94 J t^x 34/-^* 9r-^ 274/ j^^" |3_ \n\n18. (\xc2\xab;. + a/ff = a\\x^ + 3^7/ + 3Vy + i//). \n\n19. log sin {x -\\-y) = log sin x \'-\\-y cot x \xe2\x80\x94 y"^ cosec\' x/2 \n\n-\\-y^ cos x/2) sin\' ;i: -f- \xe2\x96\xa0^. \n\n3 9 81 \n\n\n\n21 \n\n\n\n\n\n\n+ \n\n22. (- jt\' + x)-^ = y-^ -\\- 2XJ-3 + 3x2jj/-4 - 4;c3^-5 + R. \n\n23. sinh (;r +^) = sinh x{i +y/(2_+y/)4_+ i?) \n\n+ cosh^(;/4-y/|3_+^\'). \n\n24. cosh (;c 4-j}\') = cosh x{i -\\-y\'^/2 +y/|\xc2\xa3+ R) \n\n+ sinh ^(^ +y/|l+ ^V|5_+ ^\')- \n\n126. Stirling\'s Formula. \xe2\x80\x94 In Taylor\'s formula inter- \nchange the symbols x and h, and place h =^ o, giving \n\nfic = fo + f\'ox4-f"o^ + ...+ro-J\' + R, . (a) \n\n\n\nI80 DIFFERENTIAL CALCULUS. \n\nwhich is Stirling\'s * formula for developing a function of a \nsingle variable into a series arraiiged according to the asceiid- \ning powers of the variable^ with constant coefficients. \n\n/o, /\'o, etc., represent what the given function and its \nsuccessive derivatives respectively become when the vari\xc2\xab \nable vanishes. \n\nPlacing /x = Uy we write \n\n-1 , du\'\\ d\'u-] x\' . . d^u~] x^ . ^ \n\n^ = 4 + ^Jo^+^ Jo 7 + \xe2\x80\xa2 \xe2\x80\xa2 \xe2\x80\xa2 + ^ Jo |7 + ^- \n\nTo apply the formula, differentiate the function and its \nderivatives in succession imtil one of the highest order desired \nis obtained. In the function and its derivatives make the vari\' \nable equal to zero^ and substitute the results^ respectively ^ for \ntheir corresponding symbols in the fori7iula. \n\nThus, develop (i + xY". \n\nf{x) = (i + ^)^ f\\x) = m{i + x)^-\\ . . . \n\nf\'^ix) = m(m \xe2\x80\x94 i) . . . (m \xe2\x80\x94 n -\\- i){i -\\- x)^~\\ \n\n/(o) = I, /\'(o) = ^> /"(o) = ^(^ \xe2\x80\x94 i), etc., \n\nfn{o) = m(m \xe2\x80\x94 i) . . . (m \xe2\x80\x94 n -\\- i). \n\' Hence, \n\n(i + x)"^ = I + ^x + ^(^ \xe2\x80\x94 1)0^/2 + . . . \n\n+ m(m \xe2\x80\x94 i) . . , {m \xe2\x80\x94 n -\\- i)x\'^/^n + i?. \n\n127. From \xc2\xa7 125 we have, by interchanging x and h and \nmaking h = o, \n\n* This formula is generally known as Maclaurin\'s, but it was pub- \nlished by Stirling in 1717; and not by Maclaurin till 1742. It is a \nparticular case of Taylor\'s formula which was published in 171 5. \n\n\n\nDE VEL CEMENTS. 1 8 1 \n\nin which x^^\'^ /\\n +1 :^-^oas;2:^-^co (g 41). \n\nR is therefore an infinitesimal as n ^-> 00 , provided fx \nis continuous for all values of x from zero to the value \nassumed, \\i f\'^x\'^^^ as /2^->co, i? must be evaluated. \n\nThe limiting values of x in any case are the numerically \nleast negative and positive values of x that cause /a^ or any \nof its derivatives to become discontinuous. \n\nIt is important to note that if j^** o is imaginary or infinite \nfor any value of ;?, the formula is inapplicable for all values \nof X. \n\nTo illustrate the use of R, take the example in \xc2\xa7126, \nwhence \n\nR = -(---\')^(---) (. + .\xe2\x80\x9e.)^-\xc2\xbb->.\xc2\xab... \n\nJV/ien m is fractional or negative, and n is increased with- \nout limit, m \xe2\x80\x94 n will become negative, giving \n\n(i + xy-\'\' = 1/(1 + xy-^ ; also \n\n/-x = m{m- 1) . . . {m-n+ i)/(i + ^)\xe2\x80\x94 j\xc2\xab^\xc2\xab,= c^ ; \n\nand i?B-\xc2\xbbco . o. The evaluation of R becomes necessary in \nthis case. Being laborious it is omitted, but it will show \nthat R is infinitesimal provided x < 1 numerically, and the \nseries will be converging; otherwise not. \n\n\n\n1 82 DIFFERENTIAL CALCULUS:, \n\nEXAMPLES. \n\n.-. /(O) = O. \n\n.-. T\\o) = Ma. \n\n.\', /"(O) =-Ma. \n\n.\'. f"\'{0) = iMa, \n\n.-. /i^(0)= -\\^Ma, \n\netc. \n\n\\n \xe2\x80\x94 \\ Ma \nfn{x) = (- i)\xc2\xab-li^==^. /\xc2\xab(0) = (- l)\xc2\xab-l|\xc2\xab-j Ma, \n\n\n\nI. Develop loga \n\n\n(1+^). \n\n\n/[x) = \n\n\n]0ga (I + ^). \n\n\nfix) = \n\n\nMa \n\n\nr(x) = \n\n\nMa \n\n(i + ^r \n\n\nf"\'{x) = \n\n\n2Ma \n\n(i + xf \n\n\n/iv(x) = \n\n\n\\i_Ma \n\n\n(I + xf \' \n\n\netc \n\n\n;. \n\n\n\nHence, \n\n\n\nand \n\n\n\nl0ga(l+x)=Ma(x-\'i+-^. . . \xc2\xb1^W^, \n\n\\ 2 3 nj \n\n\n\nlog(i+x)=x-^ + | _... \xc2\xb11^+^?\', \n\n\n\nI x^+\'^ \n\nin which i?\' = (\xe2\x80\x94 i)^ \xe2\x80\x94 ; -, \xe2\x80\x94 ; \xe2\x80\x94 ;, \xe2\x80\x94 ;^ \xe2\x80\x94 - is an infinitesimal under \n\n^ \' \xc2\xab + I (l + dnX)n+l \n\nthe law that n increases without limit, provided x = or < i nu- \nmerically. \n\n2. (I - ^2)1/2 = I _ x\'^/2 - X4/8 - etc. + J?. \n\n3. sin ;r = X \xe2\x80\x94 x^/\\3_ + ^y |5_ \xe2\x80\x94 x\'/ 17_ + R, \n\nin which R = sin [(\xc2\xab + i)7r/2 + Q;^.r]-y^+^/| ^\' -j- i is an infinitesi- \nmal. \n\n4. cos ^ = I - xy|2 + x*/\\4_ - ^y|6_ + ^ \nin which R = cos ((\xc2\xab + i)7r/2 + e\xc2\xab^)jr\xc2\xab+y|?H-j. \n\n\n\nD E VEL OPMENTS. 1 8 3 \n\nSince cos x \'= a^sin ;r/^x, the development of cos j: may be \nobtained by differentiating that of sin x \n\nThe radian measure of i\' is 0.000291 \xe2\x80\x94 , for which value the develop- \nments of sin \\\' and cos i\' converge rapidly, giving their values with \ngreat accuracy. \n\nx\'^ x^ \n5. a^ = i-4-log a .x-\\- log"\'* a f- \xe2\x80\xa2 \xe2\x80\xa2 \xe2\x80\xa2 + log\xc2\xab\xc2\xab \\- J?, \n\nin which /^ = c^"^^ log a x / \\n 4- i diminishes as n increases \nwithout limit. \n\nPlacing a = g, we have \n\n\n\n2 \\n \\n -\\- 1 \n\nin which x = 1 gives \n\nand X =x 4/\xe2\x80\x94 i gives \n\n^\xc2\xbb\'-. = i + xi/-i--- -1\xe2\x80\x94 + -+etc. \n\n\n\n(-lV^-.)+(.-|\' + e..)y- \n\n\n\n= cos X -\\- 4/\xe2\x80\x94 I sin ;c {a) \n\nSubstituting \xe2\x80\x94 x for x, we have \n\n^-X -1 _ ^^g ^ _ ^_ J gjj^ j^ (J,\') \n\n\n\nHence, cos ^ = ( ^^^"^ + ^"^^"OA, \n\nsin X = ( ."\'^\'"1 - .-^"^-0/2 i/^ \n\n\n\n. . (0 \n\n\n\nwhich are known as Euler\'s expressions for the sine and cosine. \nFrom {c) we obtain \n\nxV\'~\\ -xV~\\ 2xV~l \n\ne \xe2\x80\x94 e e \xe2\x80\x94 I . . \n\n|/\xe2\x80\x94 itanx== \xe2\x80\x94 \xe2\x80\x94 = -;=- ~ "7^ \xe2\x80\xa2 \xe2\x80\xa2 \xe2\x80\xa2 V") \n\n\n\n1 84 DIFFERENTIAL CALCULUS, \n\nIn (a) and {b) put x = mx, then \n\n^mxV\xe2\x80\x94i- , ./ \xe2\x80\xa2 \n\ne = cos mx \xc2\xb1 y\xe2\x80\x94 i sin mx. \n\n\n\ndemx \n\nor, since e \n\n\n\ncos mx \xc2\xb1 |/\xe2\x80\x94 I sin mx = (cos ;c \xc2\xb1 |/\xe2\x80\x94 i sin ^)\'\xc2\xab, \n\nwhich is De Moivre\'s formula. \n\nExpanding the second member by the binomial formula, and equat- \ning separately the real and imaginary parts, we obtain, m being a \n^ positive integer, finite expansions for cos mx and sin mx in terms of \ncos X and sin x. \n\nThus, m =3 gives \n\ncos sx \xc2\xb1 |/\xe2\x80\x94 I sin 3J\xc2\xbbr = cos^;c +\xe2\x80\xa2 3 cos\'^JtrCi V\'\xe2\x80\x94 i sin jf) \n\n\xe2\x80\x94 3 cos jc sin^ ;c T 4/-- I sin\' Jp. \n\nHence, cos sx = cos^; \n\nsin 3;r = 3 C( \n\nIn (c) place \\/\xe2\x80\x94 i = /, giving \n\ncos X = (^^\xc2\xbb 4- <^ \' ^0/2 , sin X = ((f^* \xe2\x80\x94 e-^^)/2t. \n\nFrom which, putting xi for x, and multiplying both members of the \nsecond by t^, we have \n\ncos xi = (<?^ -|" ^~^)/2. sin jfi = i(e^ \xe2\x80\x94 ^~^)/2. \n\nIf /(\xe2\x80\x94 x) = \xe2\x80\x94 /(x), the development of /(x) will con- \ntain powers of x of an odd degree only. Such functions \nare called odd functions.* Entire functions all terms of \nwhich are of an odd degree with respect to the variable are \nexamples, also sin x, tan x^ cot x, cosec x, sin~^^, and \ntan~-^;ic. \n\nIf/(\xe2\x80\x94 x) =/(x), the development of /(x) will contain \npowers of x of an even degree only. Such functions are \n\n\n\nRice and Johnson\'s Calculus, \xc2\xa7 169. \n\n\n\nDE VEL OPMENTS. 1 85 \n\ncalled even functions, of which cos x, sec x, vers x, and all \nentire functions containing powers of the variable of an \neven degree only, are examples. \n\n6. Develop sec x. \n\nf\\x) = sec X tan x, ,., ^\'(o) _ ^ \n\nf\\x) = sec X (I + 2 tan^^), ... /"(q) = i. \n\nf"\\x) = sec ^ tan jr (5 -f- 6 tan^^p), ... y-\'/\'(o) = o. \n\np^{x) = sec ^ (5 + 28 tan^x + 24 tan*;\xc2\xbbr), .-. /i\'(o) = 5. \n\netc- etc. \nHence, sec x = i +^2/2 + 5vJ:*/24 + 6iJfy72o4-^. \n\n7. Develop cos^x. \n\n/\'(\xe2\x80\xa2^) = 3 sin X (sxn^x \xe2\x80\x94 i), .*. /\'(q) = o. \n\n/\'V) = 3 cos X (3 sin*;r \xe2\x80\x94 i), .\xe2\x80\xa2. /\'(o) =\xe2\x80\x94 3. \n\nf"\'{x) := 3 sin X (i \xe2\x80\x94 3 sin^jc -f- 6 cos\'x), .*. f"\\o) = o. \n\n/\'^^{x) = 3 cos X (1 \xe2\x80\x94 g sin\'^x \xe2\x80\x94 12 sin^x + 6 cos^jc), .*. /^"(o) = 21. \netc. etc \n\nHence, cos^^f = i \xe2\x80\x94 3^:2/2 + yx*/S \xe2\x80\x94 etc. \n\n8. Develop (i + e^Y^, \n\nf\\x) = \xc2\xab(i + ^^f - V, . \\ f\\o) = \xc2\xab2\xc2\xab-*. \n\n/"(x) = \xc2\xab(i + ^^)\xc2\xab-2 ^-[(i + ,^) + ^^(\xc2\xab - I)], \n\n.-. /"(o) = n(n+i)2\'\'-\\ \n\n3 r(i+^")\' + (\xc2\xab-i)(i + ^->- n \n\nL + 2(\xc2\xab - l)(l + ^^)^^ + ^2^(W - I) (\xc2\xab - 2) J \n\n... y-(o) = ;,2(\xc2\xab + 3)2\xc2\xab-^ \netc. etc. \n\nHence, \n(I + e-r = ."fi + ^ + \xc2\xbb(^1+J]fl + "\'(" + 3) f. + ,,,1. \n\n|_ 2 2^.2 2^ |3 \' J \n\n\n\n1 86 DIFFERENTIAL CALCULUS. \n\n\n\n9. Develop e^ ^\'^ -^ with respect to sin \'^x. \n\n^m sin- 1^ ^ j_^ ^ gj^_ 1 ^ _^ w2^sin-i;r)V2 + w3(sin-i;p)Vj3_+ i?. \n\n10. Develop <?\xc2\xbb\xc2\xab sin" at vvith reference to x. \n\nThe general rule may be applied, but the following method of \nUndetermined Coefficients* is perhaps simpler in this case. \n\nPlace \xc2\xab?\xc2\xbb\xc2\xab sin-i x^y^ then dy/dx = me^ sin-i jc/ 4/1 \xe2\x80\x94 ^2, \n\nand d^\'y/dx^ = ni^ e^ sin-i ^/(i _ ^2) _j_ ;^^^w sin-i ^(i \xe2\x80\x94 x^f /^ \n\nHence (i \xe2\x80\x94 x^)d\'^y/dx\'^ \xe2\x80\x94 x dy/dx = w\'^j/ (i) \n\nAssume >/ = A o -{- A ix -{\xe2\x80\xa2 A ^x^ -\\- A%x\'^ -\\- . . . -\\- AnX^ + R. \n\nThen ^/^;t: = ^, + 2^2;>: + . . . + \xc2\xab^\xc2\xab;tr\xc2\xab-l + R\\ \n\nd\'^y/dx"^ = 2^a + . . . + w(\xc2\xab ~ i)AnX\xc2\xbb-^ + ^". \n\nSubstituting in (i) and equating the coefficients of x\xc2\xab in the two \nmembers, we find \n\n^\xc2\xab+2= ^\xc2\xab(^2 +\xc2\xab\')/(\xc2\xab+!)(\xc2\xab + 2) (2) \n\nfrom which ^2, ^3, ^4, etc., may be determined in order when A^ \nand A I are known. \n\nAo = ^"^sin-l ^j^ = 1, ^^ \xe2\x80\x94 ^gmw^r^xl^^ _ ^aj^ _ ^^ \nHence (2), Ai = m^/\\2^ Aa = m(m^ + iVlS. etc., and \n\nComparing this result with that of example 9, we have, by equating \nthe coefficients of m, \n\nyZ o2y.5 \'22r2\xe2\x80\x9e7 \n\n11. sin-i. = .+|+^ + 3Ai + ^. \nSimilarly, by equating the coefficients of m^, \n\n* Todhunter\'s Calculus. \n\n\n\nDE VEL OP ME NTS, 1 8/ \n\n\'3.4 3-4-5.6 3.4. 5.6.7.8 \nEquating the coefficients of m^, we have \n\n13. (sin-i^)3 = ^3 + 3.^1 + I_yi^5 + 3. . 5,^1 4. E_ _^ ^)|^\' + ^\xc2\xb0 \n\nDividing both members of example 12 by 2 and differentiating, we \nhave \n\nsin-i^ , 2x^ . 2.4Jr* 2 . 4 . 6jr\' , ^ \n\n\\/i- x^ 3 "^3. 5 ^3-5.7 \n\nFrom which, multiplying both members by i \xe2\x80\x94x"^, we obtain \n\n/ 2x1/2 . , X^ 2 X^ 2 ^ X\'\' \n\n15. (i \xe2\x80\x94 x^) \' sm-i^f = X V- R^ \n\n^ ^ \xe2\x96\xa0\' \' 3 3 5 3 5 7^ \n\nin which, putting x = sin 9, we have \n\nsin^e 2 sin* 6 2 4 sin\xc2\xabe , ^ \n\n16. 9 cot = I h ^. \n\n3 3 5 3 5 7 \n\n17. When the determination of the successive derivatives of \na higher order is laborious, a simpler method of Undetermined \nCoefficients may be employed provided the development oi f\'{x) \nis known. \n\nThus, since sin-^ x is an odd function, which vanishes with x, we \nassume \n\nf{x) \xe2\x80\x94 sin-ix = Ax-\\- Bx^ + Cx^ -j- Dx\'^ + etc. \n\nDifferentiating, we have \n\n/\xc2\xbb=--r==^ = ^ + 3^^\' + 5C^*+ 7^^* + etc. , (i) \n\n4/1 \xe2\x80\x94 or^ \n\nDeveloping 1/ |/i \xe2\x80\x94 \xe2\x80\xa2^\'^> we have, provided \\ar < i, \n\n^ 2\'2.4\'2.4.6\' \n\n3.5.7...(2>^-i )\xc2\xa3!l^^^^_ . . (,) \n\n\' 2 . 4 . 6 . . . 2\xc2\xab \n\n\n\n88 DIFFERENTIAL CALCULUS, \n\n\n\n(i) and (2) are identical. Hence, \n\n\n\nand coefficient of ^ + i term = \n\n\n\n2.3 2.4.5 2.4.6.7 \n\n3 5 . 7 . . . (2\xc2\xab \xe2\x80\x94 l) \n\n\n\n2.4.6.. . in{in \xe2\x96\xa0\\- i)\' \n\n\n\nand sin-i x \xe2\x96\xa0=\xe2\x96\xa0 xA h - \xe2\x80\xa2 . \n\n^2.3 \'2.4.52.4.6.7\' \n\n, 3 \xe2\x96\xa0 5 \xe2\x80\xa2 7 . . \xe2\x80\xa2 (2/^ - i)x2"+i ^ \n\n2 . 4. 6 . . . 2W(2\xc2\xab + l) "*" \' \n\nwhich is convergent for jf < i, since each term is less than the cor- \nresponding term in the geometrical progression x -{\xe2\x96\xa0 x^ -\\- x^ -\\- etc. \n\nIt I T I "J \n\n^ = sin-1 - = - H ^ + ^ h :ff. \n\n6 2 2 2.3.8 2.4. 5.32 \n\nFrom which tt = 3.14159 . . . \n\n17-I. Develop tan~"^ x. \n\nSince tan~^.r is an odd function which vanishes with x, we assume \n\nf(x) \xe2\x80\x94 tan-i x\xe2\x80\x94Ax^ Bx^ + Or^ + Dx\'\' + etc. \n\nDifferentiating, we have \n\nf\\x) = 1/(1 + ^2) = ^ -f- 35^2 + 5 Cr* + 7Z>x\xc2\xab + etc. . {a) \n\nDeveloping 1/(1 -|- ;>:\'\'\xc2\xbb), we have, provided jc < i, \n\n(l+;*:\'^)-l = l-.r2 + ;\xc2\xbb:*-x\xc2\xab+ etc (3) \n\n(a) and (3) are identical. Hence, \n\nA = \\, B=-i/3, C=i/5, Z> = -i/7, etc. \n\nand tan-i JIT = .\xc2\xbb: 1 h etc.+ (- i)** \xe2\x96\xa0 1- R, \n\n3 5 7 \' 2\xc2\xab+i \' \' \n\nin which ^ is an infinitesimal as \xc2\xab 3\xc2\xae-^ qo provided x = or <i \nnumerically. \n\n\\i x=i \xe2\x80\x94 tan (7r/4), we have \n\n;r/4 = I - 1/3 -f 1/5 - 1/7 + R, \n\n\n\nD E VEL OP ME NTS. 1 89 \n\nTo obtain a development which converges more rapidly, let \n<f) = tan- 1(1/5), a"d 6 = J0 ihen 6 = 4 tan"! (1/5). Hence, \n\n4 tan \xe2\x80\x94 4 tan\' 120 \n\ntan e = ^ ^ ^ , 7^ = \xe2\x80\xa2 \n\nI \xe2\x80\x94 6 tan\'\' -\\- tan* 119 \n\nWe also have \n\ntan (0 - 45\xc2\xb0) = (tan - i)/(i + tan 0) = 1/239. \n\nTherefore \n\n\xe2\x80\x94 45^ = tan-i (1/239), and 45\xc2\xb0 = \xe2\x80\x94 tan-i (1/239), \n\nor ;r/4 = 4 tan-i (1/5) \xe2\x80\x94 tan-i (1/239).* \n\nDeveloping tan-i (1/5) and tan-i (1/239), and substituting in \nabove, \n\n\n\nTt _ ll I I I \\ \n\n~ \\239 ~ 3(239)^ 5(239)^ ~ ^ ^- j. \n\nwhich converges rapidly. Seven terms of the first set and three in \nthesecond give it \xe2\x80\x94 3- \'41592653589793 \xe2\x80\xa2 \xe2\x80\xa2 -t \n\n,8. ^-f-+log(i+.) = 2.-f +4-^-^+^. \n\n19. ^ cos ;c = I -r ^ - ix^l\\\'h_ - 4^V|4_- 5-^V|5_+ ^. \n\n20. ^sin a; = I -I- ;c + x^/^ \xe2\x80\x94 xV8 - x^ / \\t^ - JtrV240 + R, \n\n21. log {x + VTT^\') = ^ - -^713 + 3\'^y|5_+ R* \n\n22. ^*t2 = x2 H- jtr3 + xY|2 + ;cy|3 + R. \n\n23. (l + x)/(l \xe2\x80\x94 x) = I + 2X + 2X\'-^ + 2Jf3 -f- ^. \n\n24. log -^^ = -J- _ -^-^ + -^-L-^ + ^. \n\n\xe2\x80\x94 ^ _ \n\n* Known as Machin\'s Formula. \nf Haddon\'s Differential Calculus, \n\n\n\n190 DIFFERENTIAL CALCULUS. \n\n[Suggestion. \xe2\x80\x94 Put x/{x \xe2\x80\x94 i) = i \xe2\x80\x94 z, and in development of (i \xe2\x80\x94 2) \nput z \xe2\x80\x94 1/(1 \xe2\x80\x94 x).^ \n\n25. (i + 2;c + 3x2)-V2 \xe2\x80\x94 \\ \xe2\x80\x94 X ^ ix^ \xe2\x80\x94 7^V4 + 3^72 + R. \n\n26. COS-l \xc2\xbbr = 7r/2 \xe2\x80\x94 ^ \xe2\x80\x94 j:Y2 . 3 ~ 3Jr-y2 . 4 . 5 -j- -^. \n\n27. ^V(^^ \xe2\x80\x94 x) = ^2 _ ^4/2 \xe2\x80\x94 xy|3 + R. \n\n28. (i - ^2)-2 = I 4. 2x2 + 3^4 _^ 4^6 ^ ^^ \n\n29. ^Vcos >r= i+^4--^\'^ + 2jtV3 + \xe2\x80\xa2^V2 + 3^Vio + ^, \n\n30. ^^sinx \xe2\x80\x94 I _^ ^2 ^ ;p4/3 _j_ y^>_ \n\n\n\n31. V I + 4-^ + isjc\'-^ \xe2\x80\x94 \\ A^ 2x -\\- 4^2 -f- ^. \n\n32. e-V\'^^ = I - lA\' + i/2;t;* - i/6jr6 + i?. , \n\n33. g-x^ r^ I _ ^2 _|_ ^4/2 -f ^, \n\n34. (I + ^\xe2\x80\xa22)V^ = I + 5-^73 + 5-^V9 - 5^V8i + R. \n\n35. /an-i^ ^ I _p ;^ _}_ ^2/2 _ x^/6 - 7^724 + R, \n\n36. sin2 X = x2 \xe2\x80\x94 xV3 + 2x73\' .5 + ^. \n\n37. log sec X = x^/2 -{-x*/i2 -\\- jrV45 + R. \n\n38. {a\'^-{-dxy/^ = a-\\- dx/2a - b\'^x\'\'/8a^ ^ 3d^x^/4Sa^ -\\- R, \n39; \xe2\x96\xa0 ^^ log {i+x) = x-i- xV2 + 2x3/|3_ + 9^71 5_ + ^. \n\n40. tan Jtr = X + -^Vs + 2xVi5 -f: i7-^V3i5 + \'^\xe2\x80\xa2 \n\n41. log (i -\\- sin x) = X \xe2\x80\x94 x^/2 -j- x^/6 \xe2\x80\x94 xVi2 + ^\' \n\n42. log (i + e^) = log 2 + x/2 + x\'\'/2^ - x*/2^\\4_ + R. \n\n43. (^^ + ^-^)\xc2\xab = 2\xc2\xab(i + \xc2\xab;t:2/2 + n(2n \xe2\x80\x94 2)xYl4 ) + i?. \n\n44. cot X = i/x \xe2\x80\x94 x/3 \xe2\x80\x94 x^/f . 5 \xe2\x80\x94 2x^/2^ . 5 . 7 + i?. (By method \nof Undetermined Coefficients, assuming cotx = Ax\xe2\x80\x94^-\\- \nAix -{- A^\'^ 4" etc.) \n\n45. tan" X = X* -{- 4x6/3 + 6xV5 + ^\xe2\x80\xa2 \n\n46. By means of Taylor\'s and Stirling\'s formulas deduce the fol\' \n\nlowing : \n\nsin (x \xc2\xb1 j) = sin x cos y \xc2\xb1 cos x sin y, \ncos (x \xc2\xb1 jj\') = cos X cosy =F sin x sin jj/. \n\n47. sinh X = {e"" - e-^)/2 = x + ^y 3_ + xy|5_ + R. \n\n\n\nDEVELO PMEN TS. 1 9 1 \n\n48. cosh x = {e^-^ e-^)/2 = i + ^^2 + xy\\A_ + R. \n\n49. cosh\'^x = I + nx\'\'/2 + ii{2n \xe2\x80\x94 2)xy |4_ + ^. \n\n50. log {s\'m\\\\x/x) = x\'^/6 \xe2\x80\x94 xViSoH- A\'. \n\n51. ia.nh-\'^x = x-{-x^/^-{-x^/~,-\\^R, \n\n52. sinh-l (x/a) = x/a \xe2\x80\x94 x^/2 . 3a^ + 3^5/2 . 4 . 5a^ + i?o \n\n53. Given j/^ \xe2\x80\x94 ^y -^ x = o; develop y in terms of the ascending \npowers of x. \n\ny = fx, /b = o or \xc2\xb1 4/3. \n\n3;/^^ dy/dx \xe2\x80\x94 \'^dy/dx -f- i = o gives \n\ndy/dx\\ = - i/(3j/^ - 3)]^j = 1/3, for;/ = o. \n\n2y{dyjdxf -\\-y\\d^y/dx\'\') - dy/dx"" = o gives \n\nd\'y/dx-^]^ = - ty{dy/dxy/i3v\' - 3)]o = O. \n(dyY . (dy\\ dy , dv d-\'y , J^y d^y \n\n. . d\'^y/dx^]^ =2/27. \n\nHence, _)/ = x/3 -\\- x^/3* + ^\xc2\xab \n\n54. Given 2_;/^ \xe2\x80\x94 _:j/;c- \xe2\x80\x94 2 = o; shovir that \n\ny = I -\\- x/2 . 3 \xe2\x80\x94 x^/2^2>* + ^\xe2\x80\xa2 \n\n55. Given jv\'^ \xe2\x80\x94 S/y = 6x; show that \n\ny = 2 -{- X \xe2\x80\x94 x^/2\'^2 4- x*/2 . 3 . 4 + i?. \n\n56. Given j^x \xe2\x80\x94 Sy \xe2\x80\x94 Sx = o; show that \n\n;/ = - ^ - xV8 - 3;cV26 + R. \n\n57. Given ^y^x \xe2\x80\x94 y \xe2\x80\x94 4 \xe2\x80\x94 o; show that \n\ny= - 4-4^x- 3i4yx\' + i^. \n\n58. Given y\'^ \xe2\x80\x94 a\'jj/ + \'^J-^ \xe2\x80\x94 x^ = o; show that \n\n\n\n192 DIFFERENTIAL CALCULUS. \n\n59. Given sin = ;ir sin (a -|- 0); show that \n\n(p =z mt -\\- SAVi a . X -\\- sin 2a . x\'^/2 + 2 sin \xc2\xab . (3 \xe2\x80\x94 4 sin^ \xc2\xab)>^y|3_-f- 1^. \n\n^ ^ X e\'\' -\\- 1 ^ \n\n60. Develop .* \n\nBy Stirling\'s formula we write \n\nSince /vV =/(\xe2\x80\x94 ^), the development contains no power of x of an \nodd degree. \n\nWriting (^^ + i)/(^^ \xe2\x80\x94 i) = i + 2/(^^ \xe2\x80\x94 i), we have \n\nX ^^ + I X j; ^ jc^-^ \n\n/(^) = r .1^ \xe2\x80\x94 ; = r + ::^^ \xe2\x80\x94 ; = - + jr^- ^ + \n\n\n\n2<?^\xe2\x80\x94 I 2\'^*\xe2\x80\x94 I 2\' ^^\xe2\x80\x94 I* \n\nDifferentiating, we have \n^^(/(^)+/\'(x))=/\'(^) + ^ + \'-^^ (I) \n\n,-(/(^) + 2/\'(x)+/\'V))=/\'\'(x) + \'-3^i\xc2\xb1^^ (2) \n\ne- (A^) + 3/\'(^) + ^f\'\\^) + /\'"(^) ) = f\'iA + \'\xe2\x80\x94 V^- (3) \n\netc. etc. \n\nMaking jv = o in (i), (3), etc., we find \n\n/(o)=:i, /\'(o)-l/6, /iv(o)=._i/3o, /vi(o) = l/42, \n\n/viii(o) =. - 1/30, etc. \nHence, \n\n7<?^"^~ 6\'7 3^|4 42|6^ 3^15 \' \n\n* Edward\'s Differential Calculus. \n\n\n\nDE VEL OP MEN TS. 1 9 3 \n\nor, placing \n\n^ = ^\xe2\x80\x9e - = B,, -- = ^\xe2\x80\x9e -- = A, etc., \n6 . 30 42 30 \n\n\n\nAr\\ ^ /y\xc2\xab* -^^ \xe2\x80\xa2Y\'*\' \n\n2 \n\n\n\n\n\n\nThe coefficients represented by B^, B^, B^^ etc., are \nused in the higher branches of analysis, and are called \nBernoulli\'s numberS: \n\n128. Extension of Taylor\'s Formula to functions of two \nor more sums of two variables each. \n\nLet u\xe2\x80\x94f{x, 7), X and J being independent variables; and \nlet it be required to develop /(x + /^,ji/-l-^), in which h and \nk are variable increments of x andjK respectively. \n\nIf, in/(.r, j), X be increased by h, and/(:\\^ -\\-h,y) be de- \nveloped by Taylor\'s formula ; then if, in each term of the \nresult, _y be increased by /^ and developed in a similar man- \nner as a function oi y -\\- k, we shall havey"(jt: -\\- h, y -[- k) =\xe2\x96\xa0 \nsum of the latter developments. \n\nOtherwise, develop (p{t) = f{x -\\- hf, y -\\- kt) as a func- \ntion of /, by Stirling\'s formula, and in the result make \n/= I. \n\n<t>{f) = f(po -^ht,y-\\- kt), .-. 0(0) = f(x, y) = u. \n\nIn order to express conveniently the successive deriva- \ntives with respect to /, place x -\\- ht =^ w, and j^ + /^/ = .?, \ngiving 0(/) =/(\xc2\xab/, s). \n\nHence (\xc2\xa7 102), \n\nd(t>(t) ^ a0(/) -dw dc()(t) 9\xc2\xa3 \ndt \'dw dt \'ds df \n\n\n\n194 DIFFERENTIAL CALCULUS. \n\n\'dw dx /* 9j- dy \n\n\n\nBut \n\n\n\n. (Ex. 7, p. 71.} \n\n\n\nAlso dw/dt = A, \n\n\n\n\'ds/dt ~ k. \n\n\n\n\n\n\n^/90(/)\\ ^ 9V(/) 9^ j 9W) ?\xc2\xa3 ^ 5M^U 4- 5^^ \n^/V ^jt: / ^^ \'dw dt dx 9i- ^/ dx"^ dx dy \n\n\n\nk. \n\n\n\nHence, \n\n\n\n^ \'^ ^jt:\' ^ dxdy \' dy^ \n\n\n\netc. \n\n\n\netc. \n\n\n\n0\'^+^(/) = d\'\'^\'(P{t)h\'\'+^/dx^+\' + (^ + iW+\'(p{t)h\'\'k/dx^dy \n+ . . . + d\'\'+\'(p{t)k\'\'^\'/dy^^\\ \nTherefore \n\n0\'(o) = {du/dx)/i + (a?^/^jj;)^, \n\n,\xe2\x80\x9e, \\ S\'^^TS I S\'^^ 7 7 t 9\'^^72 \n\n(\xc2\xb0)=^^^+^<^^^^ + ^^. \n\n\n\netc. \n\n\n\netc. \n\n\n\n^^\xe2\x96\xa0<\xc2\xbb-\')=[^\'--+<\'+\')?^\'\xc2\xbb+\xc2\xab- \n\n\n\n\n\n\n^ = 0ni? \n\n\n\nDEVELOPMENTS. IQS \n\nSubstituting in Stirling\'s formula, and making / = i, we \nhave \n\n/(x + /,^ + .) = . + |. + |. \n\n+ bL^\'^" ^ ^\xc2\xab\'-^\' -^Z ^ ^\'^\xe2\x96\xa0^ -^Z ^ -tv\' J ^ {a) \n+ etc. etc. \n\n\n\nIt should be noticed that \n\n0\'^+\'(/) is, in general, a function of ^ + ^/.and jv + /^/ ; \n\n0\'\'+^(o) is the same function of x and j ; \n\n(p^\'^\'^(dnt) is the same function of ^ + hdj and_)^ + >^6\'\xe2\x80\x9e/ \xe2\x80\xa2 \n\n0^+^(6\'^) is the same function of x + ^\'w// and y -\\- 0,ik. \n\nTherefore the remainder term in {a), which is equivalent \n\nto \xe2\x80\x94 \xe2\x96\xa0 \xe2\x80\x94 0\'\'+\'(6\'^), is the same function of x -\\- d^^h and \n\n\n\nji-\\- J \n\n\n\ny + dnk that \xe2\x80\x94 ; \xe2\x80\x94 (p^^^{6) is of x and y. Hence, the re- \n\nn -\\- 1 \n\n\n\nmainder term, denoted by i?, may be written \n\n\n\ny=>\' 4-0/1 /& \n\n\n\nFrom \xc2\xa7 125 and \xc2\xa7 127 we see that formula (a) develops \nthe given function provided that, as n increases without \nlimit, 0"(o) is real and finite and R is an infinitesimal or. \n\n\n\n196 DIFFERENTIAL CALCULUS. \n\nwhat is equivalent, provided that u and all of its successive \npartial derivatives of the n^^ order are continuous between \nall states, corresponding to values of x and y from any- \nassumed values io X -\\- h and y -\\- k, under the same law. \n\nHaving u \xe2\x80\x94 f{x, y, z), we may, in a manner analogous \nto above, deduce \n\n^/ r 7 I ZL I 7\\ I 9^7 I 3^71 I 9^7 \n\n\n\n2\\_dx\' \' df \' ^s^ \n\n\\dx dy dxdz dy dz /J \n\n+ etc. etc. \n\ndx dy dz \n\n\n\n+ \n\n\n\nk^ + I \n\n\n\nAx-x^Qnh \n\n\n\ny\xe2\x80\x94y-\\-Qnk \nZ-Z-\\-Qnl \n\n\n\n(^) \n\n\n\nthe remainder term being indicated by the symbolic form \ndescribed in \xc2\xa7 109. \n\nIn a similar manner, a formula for the development of a \nfunction of any number of sums of two variables each may \nbe deduced. \n\n129. Extension of Stirling\'s Formula. \xe2\x80\x94 In {a), \xc2\xa7 128, \nput jc = o and J = o; then write x and y for h and k re- \nspectively, giving \n\n^=/(x,j)=/(o,o) + g. + |;,]^^^^ \n\n\n\n+ etc. \n\nd^ dy A{K^.Ky) \n\n\n\n, I ra , a i^^\' \n\n\n\nBEVEL PM EN TS. 1 97 \n\nwhich is a formula for the development of any function of \ntwo variables in which /(o, o), (du/d.x)o, (9W^^)o, etc., \ndenote constants resulting from making x = o and/ - \xe2\x80\xa2 o in \nu, du/dx, \'d\'^u/dy^^ etc., respectively. \n\nThe conditions of applicability, for any assumed values \nof X and y, are that u and all of its successive partial deriva- \ntives shall be continuous for all values of x andj\' from o to \nthose assumed. \n\nIn a similar manner we may deduce from {h), \xc2\xa7 128, and \nits extension, corresponding formulas for the development \nof any function of three or more variables. \n\nEXAMPLES. \n\n1. Develop {x + hy^iy + ky. \n\nu = /{x, y) = x\xc2\xbby^, \xe2\x80\x94\xe2\x80\x94\xe2\x96\xa0 = mx^~\'^y\'^, -\xe2\x80\x94- = nx**^y\'^-\'^y \nax ay \n\n-\xe2\x80\x94 ^ = m{m \xe2\x80\x94 iW\'^-Sy^, -\xe2\x80\x94 ^ = mnx^\'^-\'^y^-\'^, \xe2\x80\x94 ^ = n{n\xe2\x80\x94 i)x^yn-Z, \ndx^ \' \'^ dxdy ^ dy^ ^ \' -^ \n\netc. etc. etc. \n\nSubstituting in fornnula (\xc2\xab), \xc2\xa7 128, \n\n{x + hy^iy + ky = x^y\'*\' + rnx\'^^-\'^y^\'h + \xc2\xabx\'\xc2\xabjj/\xc2\xab-i/^ \n\n4- m{m \xe2\x80\x94 i);ir\xc2\xab\xc2\xab- V/iV2 + /?mx\'\xc2\xab-ij)\xc2\xab-M/& \n+ n{n - i)x^y^-^k\'^/2 -\\- R. \n\n2. Develop (;c + /^)2[(a-i-jj/) + y^]3. \n\nu=f{x,y)^x\\a+yf, \'^^ = 2x{a + y)\\ ^ = ^x^a -^ y)K \n\ng=.(.+.)3 |^=6.(.+,)i |-r-6.-^(.+v). \n\n9 \xc2\xab / , X 9 \xc2\xab 9 " \n\n,.>,\xe2\x96\xa0\xe2\x96\xa0 = I2(g + r). 1 ;=i2;r, 7-1-7^=12. \n\n\n\n198 DIFFERENTIAL CALCULUS. \n\nHence, substituting in formula {a), \xc2\xa7 128, \n(x + h)Ha + J + kf = x\\a ^yf + 2x{a -^ yfh + zx\\a -\\-yYk \n\n+ 3(\xc2\xab +^)V^2^ + (ix{a -\\-y)hk\'\' + ^tr^i^ \n\n3. Develop ?< = <?\xe2\x80\xa2* sin jj/. \n\n^. s _ f\'du\\ _ fd\'^uX _ _ f\'du\\ _ \n\n\\dx }q \\dx\'^Jo \' \' \\dy JQ \n\n\\dxdyjQ \\dy^ la \n\ni^\\=., f^Uo, f9!fl^=-r. \n\nXdx\'dyJQ \\dxdyy \\dy^ Id \n\netc. etc. \n\nx^v v^ x^v xv^ \nHence, <f^sin y = y ^ xy + -^ ~6"^6 6~\'~^* \n\n130. Theorems of Lagrange and Laplace. \n\nSuppose y = z -\\- x(p{y), (i) \n\nin which x and z are independent, and let it be required to \ndevelop /{y) according to the ascending powers of x with \ncoefficients which are functions of z. \n\nPlacing u =/{y), in which case u will also be a function \nof X and z, then developing by Stirling\'s formula, we have \n\n\\ax /x=o I \\dx /x=o 1.2 \n\n+(93 f+j,,. . . . (,; \n\nin which the coefficients of the different powers of x are \n\nfunction<^ of z. In order to determine their values, we \n\n9// d\'\'i^ , , , . \n\ntransform ~\xe2\x80\x94 , ,-7, etc., before making x = o. \nax cix^ \n\n\n\nDE VEL OPMENTS. 1 99 \n\nDifferentiating (i) with respect to x^ we obtain \n\n-dy/dx = 0(>-) + x\\dct)i^y)l^y\\{^y/dx\\ \nwhence \'dy/dx = 0(jJ^)/[i \xe2\x80\x94 ^90(^)/9y]. \n\nDifferentiating (i) with respect to z, we have \n\ndy/dz = i-{-x[d(Piy)/dy]{dy/dz), \n\ngiving dy/dz = i/[i \xe2\x80\x94 xd(piy)/dy]. \n\nHence dy/dx = <P{y){dy/dz). \n\n\n\nSince \nwe have \n\nObserving that \n\n\n\n82/ _ dii dy du _ du dy \ndx ~ dy dx\' dz ~ dy dz* \n\n\n\ndu/dx=cp{y){du/dz) (3) \n\n\n\nd_ \n\ndx \n\n\n\n(\xc2\xab-"!) =l-(*w|) \n\n\n\ndy dxdz ^"^^^Uzdx\' \' \' ^4) \n\n\n\nand that u =/{y) gives \n\ndu ^ d/jy) dy du ^ df(y) dy \ndz dy dz^ dx dy dx* \n\nwe have, differentiating (3), \n\ndx\' dxV^-^\'dzl dxV^-^\' dy dzl \n\n\n\n200 DIFFERENTIAL CALCULUS. \n\nHence, by (3), ^ = 1(^(7)\'^). \nAgain, \n\nHence, by (3), |^! = \xc2\xa3(^W\'^)- \nSimilarly, from \xe2\x80\x94 = -^;^^0(;,) _j, \n\nwe deduce ^-^. = _^0(_,) _j. \n\nwhich shows that the law of formation is general. \n^ = o gives JJ^ = ^, u=/(z), and -- = \xe2\x80\x94^. \n\nXT 9^1 ^^ \\9/W \n\n\n\netc. etc. \n\n\n\nSubstituting these expressions in (2), we have \n, x" 9\xc2\xab-\' /\'v^\xc2\xab3/(\xc2\xab)\\ , \xe2\x80\x9e \n\n\n\nwhich is called Lagrange\'s Theoretn. \n\n\n\nDEVELOPMENTS. 20I \n\nSuppose / \xe2\x80\x94 F{z + x(p{t)). \nPlacing/ \xe2\x80\x94 z-{- x(p{t), we have \n\n/ = F(y), u =f(t) =/(i^(j)), and y = z + x<p{F{y)), \n\nto which Lagrange\'s theorem is applicable provided we \nwrite f{F) for /, and <P{F) in place of 0; therefore we \nhave \n\n. I ^" 9""\' r-7^77i- 9/(^(^)) 1 \n\n+ \xe2\x80\xa2 \xe2\x80\xa2 \xe2\x80\xa2 + \\n_ dz"-^ \\J>K.n^)) dz J \n\n+ ..., \n\nwhich is called Laplace\'s Theorem. \n\nSince the theorems of Lagrange and Laplace depend upon \nthat of Stirling\'s, they hold only when x is small enough to \nmake the developments convergent. \n\nEXAMPLES. \n\nI. Develop ;j/ = z -\\- xe** \n\nIn this case /( r) = y, f{z) \xe2\x80\x94 z, \n\n(p(y) = e^ , and (p{z) = /. \n\nHence, from Lagrange\'s theorem, we obtain \n\n\n\n2 \n\n\n\n13. \' \' |\xc2\xab \n\n\n\n202 DIFFERENTIAL CALCULUS. \n\n2. Given log^ = ;rv, develop^. \n\nWe may write y = e=>^y, and putting xy^=:y\\ we have y = xe^ \\ \nwhich may be developed by making 2 = and y =y in example i\xc2\xab \ngiving \n\nReplacing y by xy and dividing by x, we have \n\n3. Develop y = z -{- xy^. \n\nHere <J){y) =jj/\xc2\xab, 0(2) = 2\xc2\xab. \n\nHence, \n\nz -\\- xy^ = z -{- z^x -\\- 2\xc2\xabs2m-i 1_ 3^(3\xe2\x80\x9e _ i)^n~z j. j^^ \n\n4. Develop ^ = 2 + (f sin jj/. \n\nHere x = e, (p{y) = s\\ny, 0(2) = sin\xc2\xab, \n\n\n\n1M = \n\n\n\n2 sm 2 cos z = sm 23;. \n\n\n\n|^(0W\') = \n\n\n\n6 sin cos2 2 \xe2\x80\x94 3 sin^ 2 = (3/4)(3 sin 32 \xe2\x80\x94 sin z), \n\netc. etc. \n\nHence, \n\ny = z -{- e sin z -\\ sin 22 + -^(3 -siri 33 \xe2\x80\x94 sin 2) + i?, \n\n5. Having y = z -\\- e sm y, develop sin y. \nHere /(jj/) = sin y, f{z) = (p{z) = sin z, \n\n\'Q/{z)/dz = cos 2, jf = ,?, \n0(2) Q/{z)/dz = sin 2 cos 2 = sin 22/2. \n\n\n\n"^ V^^^^ ~^ / ^ ^^^^"\' ^ ^\xc2\xb0^ ^^ ^ ^3 ^^" 32 - sin z)/4. \n\n\n\nD E VEL OP MEN TS. 203 \n\nHence, \n\nsin jj\' = sin z "1 sia 20 + yCs sin 32 \xe2\x80\x94 sin z) + R. \n\n6. Having jj/ = z + <f sin/, develop sin 2y. \n\nHere /(jj/) = sin 2y, f{z) = sin 2z, 0(2) = sin a, \n\n\'^f{z)/dz = 2 cos 22, X =\xe2\x96\xa0 e. \n\nHence, \n\n(p{z)\'^f{z) / dz = sin 2 2 cos 22 = sin 32 \xe2\x80\x94 sin z. \n\n^ .?* \n\nsin 2;/ = sin 22 -|- (sin 32 \xe2\x80\x94 sin z)e -f- \xe2\x80\x94 (sin\'\' 2 2 cos 22) 1- R, \n\n7. Similarly, develop sin 3/. \n\nIn this case f{y) \xe2\x80\x94 sin 3jj/, f{z) \xe2\x80\x94 sin 32, 0(2) = sin z, \n\'^f{z)/dz = 3 cos 32, X = ^. \n\nHence, \n\nd . ^\' \nsin 3/ = sin 32 + <? sin 2 3 cos 32 + "t^^^"\'\' ^ 3 cos 32) f- i?. \n\nQf3 2 \n\n8. Similarly, develop cos J. \n\nHere /(>/) = cosj, /(z) = cos 2, 0(2) = sin 2, \n\n^/{z)/dz = \xe2\x80\x94 sin z, X = e. \nHence, \n\ncos / = cos z \xe2\x80\x94 e sin\'\' 2 \xe2\x80\x94 3 sin\' 2 cos 2 1- R. \n\n9. Having \xc2\xab< = ;// + (? sin \xc2\xab, develop u, sin \xc2\xab, sin 22<, sin 3\xc2\xab, and \ncos u in terms of t and ^. By comparison with examples 4, 5. 6, 7, \nand 8 we have \n\nu = nt-\\- (sin nt)e+ sin 2\xc2\xab^\xe2\x80\x94 ^ + (3 sin 3^/ \xe2\x80\x94 sin nt)-- + i?. \n\n<? <?" \n\nsin \xc2\xab = sin nt + sin 2\xc2\xab/ 1- (3 sin 3\xc2\xab/ \xe2\x80\x94 sin ni)\xe2\x80\x94 -\\- R. \n\n2 o \n\nsin 2M = sin 2ni -\\- (sin 3\xc2\xab^ \xe2\x80\x94 sin nt)e -\\- R. \n\nsin 2^ = sin 3\xc2\xab^ + i?. \n\ng /J \n\ncos \xc2\xab = cos W/ \xe2\x80\x94 (l \xe2\x80\x94 cos 2/2/) (- (3 COS 3W/ -- 3 COS \xc2\xab/)\xe2\x80\x94 + ^. \n\n2 O \n\n\n\n204 DIFFERENTIAL CALCULUS, ^ \n\nlo. Kepler\'s Problem.* \nHaving nt =: u \xe2\x80\x94 6 s\'m u, (i) \n\n\n\n_ / iH-e W/^_ u \n\n2 ~ \\I - 6/ \n\n\n\n^^" T = \\Tzn.) t^" 2 \' <2) \n\n\n\nf = a(i \xe2\x80\x94 e COS?/), ^3) \n\nfind 9 and r in terms of /. \n\nFirst develop 6 in terms of ?/. In (2) put I 1 = m^ and since \n\nfrom {d), Ex. 5, \xc2\xa7 127, \n\nwe have \xe2\x80\x94 =z = \xe2\x80\x94 ^^ \xe2\x80\x94 \xe2\x80\x94 \\ \n\n-y\xe2\x80\x94 1 (^ + 1)^" ~^ \xe2\x80\x94 {m \xe2\x80\x94 i) \nHence, e = i7=^\' \n\nPlacing \xe2\x80\x94 j \xe2\x80\x94 = A, and |/\xe2\x80\x94 i = ?", we have \nw -f- I \n\nI \xe2\x80\x94 A^\xc2\xbb\xc2\xbb \nTaking the logarithms of both members, \nQi = ui-\\- log (I \xe2\x80\x94 Xe-^i) - log (i \xe2\x80\x94 A^\xc2\xab0 \n\n\\g-ui _j g-iui _| ^-3\xc2\xabt + . . .1 \n\n/ A2 A3 \\ \n\nA\'\' A3 \n\n= Ui + A((?\xc2\xab\xc2\xbb \xe2\x80\x94 ^-\xc2\xab\xc2\xbb") H (^2m\xc2\xbb _ ^-2ki) _{ (^giui _ ^-3m/^ -f- . , . \n\n2 3 \n\nA" . . A3 . \n\n= 2/z 4- 2X1 sin \xc2\xab + 2 \xe2\x80\x94 i sin 2w + 2 \xe2\x80\x94 i sin 32/ + . . . \n2 3 \n\n* Price\'s Calculus, Vol. HI. pp. 561-564, \n\n\n\nDEVELOPMENTS, 20 5 \n\nHence, \n\n^ \xe2\x96\xa0=u-\\- i\\X sin u-\\- \xe2\x80\x94 sin 2u -\\ sin 3m + . . .1, . (4) \n\n\n\nin which A. = \n\n\n\n3 \n\xe2\x80\x9e^-^ ^ (i + e)V^-(i-6)V^ \n\nI _ (I _ e^)t/2 e , \n\n\n\n-F +. \n\n\n\nIn (4) and (3) substitute for u, sin u, sin 2m, sin 3M, and cos u their \nrespective values in terms of / from example 9, replace A by its value \nin terms of e, omit terms involving powers of e higher than the third, \nand we have \n\n6 = \xc2\xab/ + 26 sin nt ^ sin int -\\ (13 sin 3\xc2\xab/ \xe2\x80\x94 3 sin nt) +. . ., \n\n4 12 \n\nCe^ 3e\' \\ \n\nI \xe2\x80\x94 e cos \xc2\xab/ -| (i \xe2\x80\x94 cos2\xc2\xab/) \xe2\x80\x94 (cos3\xc2\xab/ \xe2\x80\x94 cos\xc2\xab/)4-. . . J, \n\nwhich are important equations in Astronomy, \n\n\n\n2g6 diI\'FErential calculus. \n\n\n\nCHAPTER XL \n\nMAXIMUM AND MINIMUM STATES. \n\nFUNCTIONS OF A SINGLE VARIABLE. \n\n131. A Maximum state of a continuous function of a \nsingle variable is a state greater than adjacent states which \nprecede or follow it. Thus, fx has a maximum state \ncorresponding to x ^=- a, provided that as h vanishes we \nhave ultimately and continuously /(^ > /(^ \xc2\xb1 h). \n\nA maximum state is, therefore, a state through which, as \nthe variable increases conti?2uously, the function changes \nfrom an increasing to a decreasing function, and its first \ndifferential coefficient changes its sign from plus to 77iinus \n\n(\xc2\xa763).^ . \n\nA Minimum state is one less than adjacent states which \nprecede or follow it. Thus, fa is a minimum provided \nthat, as h vanishes, we have ultimately and continuously \nfa<f{a\xc2\xb1h). \n\nA minimum state is, therefore, a state through which, \nas the variable ificr eases continuously, the function changes \nfrom a decreasing to an increasing function, and its first \ndifferential coefficient changes its sign from minus to plus. \n\nA maximum is not necessarily the greatest, nor a mini- \nmum the least, state of a function. \n\nA function may have several maximum and minimum \nstates. \n\n\n\nMAXIMUM AND MINIMUM STATES. \n\n\n\n207 \n\n\n\nTo illustrate, let ABCD-IJ be the graph of a func- \ntion (\xc2\xa7 20). \n\n\n\n\nThe ordinates PA, RC, \xe2\x80\x94 TE, and fF/ represent maxima \nstates of tlie function, and QB, \xe2\x80\x94 SB, \xe2\x80\x94 ZF, and \xe2\x80\x94 VH \nrepresent minima states. \n\nQB > IIV illustrates the fact that a minimum state may \nbe greater than a maximum. \n\nThe ordinate at ^represents a zero maximum, and the \nordinate at y^ a zero minimum. \n\nThe point /at which two branches of a curve terminate \nwith separate tangents is called a salient point, and the \npoints C and F at which two branches of a curve terminate \nwith a common tangent are called cusps. \n\n132. A continuous function must have at least one \nmaximum or minimum state between any two equal states; \nfor if, in passing through any state, the function is increasing \nit must change to decreasing, and if decreasing it must \nchange to increasing, at least once before it can again \narrive at that state. The maximum ordinate PA, between \nthe equal ordinates ML and M\' L\' , illustrates the principle. \n\nSimilarly, it may be shown that a continuous function \nJias at least owe minimum state between any two maxima, \n\n\n\n208 \n\n\n\nD IFFEREN TE4 L CAL CUL US. \n\n\n\nand one or more maxima between any two minima. That \nis, as the variable increases, maxima and minima of a con- \ntinuous function occur alternately. \n\nI33\xc2\xbb The general definition given for maxima and \nminima assumes that the function is continuous, and \nthat as the variable increases adjacent states precede \nand follow those considered. Some exceptional cases arise \nwhich are illustrated in the following figure. \nB \n\n\n\n\n\n\nAs X increases, the function represented by the ordinate \nof the curve ABC in passing through PA has no adjacent \npreceding states, fx does not change its sign, and is neither \nzero nor infinite; yet as PA is smaller than adjacent states \nit is generally considered a minimum. \n\nAt Q the positive ordinate is an asymptote to both \nbranches of the curve; and although the unlimited value of \nthe ordinate does not represent a possible value of the \nfunction, yet fx changes its sign; therefore the ordinate \nQB \\^ said to be an infinite maximum. \n\nAt R the positive ordinate is an asymptote to one branch \n\n\n\nMAXIMUM AND MINIMUM STATES. 209 \n\nof the curve, and the negative ordinate to the other \nfx does not change its sign , therefore neither of the \nordinates \xc2\xb1 RC^ respectively, is considered as a maximum \nor a minimum. \n\nE is called a terminating point, and the corresponding \nordinate SE is generally considered as a maximum althougli \nfx does not change its sign. \n\nMETHODS OF DETERMINING MAXIMA AND MINIMA. \n\nI34* Any particular state oi fx, sls fa, may be examined \ndirectly by determining whether, as k vanishes from any \ndefinite value, we have ultimately and continuously \n\nfa > f(a \xc2\xb1 h) or fa < f{a \xc2\xb1 h). \n\nThus, let fx\xe2\x80\x94 c^{x \xe2\x80\x94 a)\\ \n\nX = a gives fa = c, and f(a \xc2\xb1 h) = c -\\- /z^ \n\nHence, /<3; <f{a -k h) 2iS h vanishes from any value, and \nfa \xe2\x80\x94 ^ is a minimum. \n\nAgain, let fx = {x \xe2\x80\x94 i){x \xe2\x80\x94 2)^ \n\nX = 2 gives /2 \xe2\x80\x94 o, and /(2 \xc2\xb1 /;) = (i \xc2\xb1 Ji)fi. \n\nHence, /2 < /(2 \xc2\xb1 /^) as /^ vanishes, and f2 \xe2\x80\x94 o is a ^ero \nminimum. \n^:=4/3 gives /(4/3) = 4/27, and /(4/3\xc2\xb1/0 = \xc2\xb1/^\'-/z\' + 4/27. \n\nHence, /(4/3) > /(4/3 \xc2\xb1 ^0 ^s h vanishes, \nand /(4/3) = 4/27 is a maximum. \n\nLet fx \xe2\x80\x94 sin x, \nX \xe2\x80\x94 n/2 gives /{jt/i) = I, and f{7r/2 \xc2\xb1 h) \xe2\x80\x94 sin (tt/z \xc2\xb1 /.). \n\n\n\n2IO DIFFERENTIAL CALCULUS. \n\nHence, /(^/s) > /(7r/2 \xc2\xb1 h) as h vanishes, \nand f{\'^/^) = I is maximum. \n\nJt \xe2\x80\x94 o gives /o = o, and /(o \xc2\xb1 /^) = sin (\xc2\xb1 ^). \n\nHence, /o < /(o + /^), and /o > /(o \xe2\x80\x94 /^), as h vanishes; \ntherefore /o \xe2\x80\x94 o is neither a maximum nor a minimum. \n\nI35\xc2\xbb In general, maxima and minima are determined by \nfinding those values of x corresponding to which, as the \nvariable increases, f\'x changes its sign. \n\nAssuming that x increases continuously, that/ir is continu- \nous, and that every state considered has adjacent preceding \nand following states, a maxiiiium state is characterized by a \nchange of sign from plus to minus in the first differential co- \nefficient, and a minimum state by a corresponding change from \nminus to plus. \n\nConversely, if, in passing through fa, f\'x changes fro?n plus \nto minus, fa is a maximum, and if f\'x changes from minus to \nplus, fa is a minimum. \n\nf\'x, if disconti?iuous, may change its sign by passing \nthrough a double value, one positive and the other negative, \nas illustrated by a salient point or by passing through infin- \nity as illustrated by a cusp with common tangent perpendic- \nular to the axis of X. \n\nf\'x, if continuous, can change its sign only by passing \nthrough zero. \n\nA maximum or minimum oi fx corresponding to a salient \npoint is an exceptional case distinguished by a double value \nfor f\'x. \n\nHence, in general, values of x corresponding to which \nf\'x changes its sign as x increases, are real roots of one or \nthe other of the two equations \n\nfx \xe2\x80\x94 o, . . . (i) and /\'^=co. ... (2) \n\n\n\nMAXIMUM AND MINIMUM STATES. \n\n\n\n211 \n\n\n\nThe figure, p. 207, illustrates the fact indicated by (i) and \n(2), that, in general, the tangent corresponding to a maxi- \nmum or minimum ordinate of a curve is parallel or per- \npendicular to the axis of abscissas. \n\nfx does not necessarily change its sign as x passes \nthrough roots of (i) and (2), as may be seen in the cases \nrepresented by particular ordinates of the following curves. \n\nY \n\n\n\n\nAt the points A and B^ where the tangents are parallel to \nX,f\'x = o; and at C and D, where the tangents are per- \npendicular to X, fx = 00, but/\':r does not change its sign \nas/{x) passes through the corresponding states. \n\nThe points A, B, C, and D are called /<?/;//^ of inflexion. \n\nThe real roots of (i) and (2) are, therefore, called critical \nvalues^ and the next step is to determine which of them \ncorrespond to states at which fx changes its sign as x in- \ncreases^ and, therefore, correspond to maxima or minima of \nfx. \n\nThe general method, for any critical value, as a^ is to \ndetermine whether, as h vanishes, f\'{a \xe2\x80\x94 h) andy\'(\xc2\xab -\\- h) \nultimately have and retain different signs. If so, a cor- \nresponds to a maximum or a minimum, according as the \nsign of f(a \xe2\x80\x94 h) is plus or minus. \n\n\n\n212 DIFFERENTIAL CALCULUS. \n\n\n\nEXAMPLES. \n\n1. Let fx = b^-{x - a)V\\ \n\nThenf\'x = 2/3(x \xe2\x80\x94 a)i/3 = oo gives the critical value a. \n\nAs k vanishes, /\'(a \xe2\x80\x94 /z) is negative and /\'{a -\\- h) is positive. \n\nHence, /a = <5 is a minimum. \n\n2. fx = 6x -|- 3x^ \xe2\x80\x94 4x^. \n\nf\'x = 6(i -|- X -- 2x^) = o gives the critical values i and \xe2\x80\x94 1/2. \nf\\\\ \xe2\x80\x94 h) =\xe2\x96\xa0 6/z(3 \xe2\x80\x94 2h), which is positive when k < 3/2. \n/\'(i -f- /^) = 6/z(\xe2\x80\x94 3 \xe2\x80\x94 2h), which is negative when h > o. \n\nHence, /"i = 5 is a maximum. \nf\'{\xe2\x80\x94 1/2 \xe2\x80\x94 h) is negative when h > o. \n/\'{\xe2\x80\x94 1/2 -f- -^) is positive when k < 2/3. \n\nHence, /(\xe2\x80\x94 1/2) = \xe2\x80\x94 7/4 is a minimum. \n\n3. /r = a -f^ (x - 3)1/3. \n\n/\'jf = i/3{x \xe2\x80\x94 b)^/^ = 00 gives x = b, \nf\'{b "^ h) are both positive for all values of h. \nHence, fib) = \xc2\xab is neither a maximum nor a minimum. \n\n4. /x = (x - i)v + 2)^: \n\nf\'x = (x \xe2\x80\x94 i)^(x + 2)\\nx + 5) = o gives \n\nX \xe2\x80\x94 \\, X = \xe2\x80\x94 2, X = \xe2\x80\x94 5/7. \n/i = o is a minimum. \n\n/(\xe2\x80\x94 2) = o is neither a maximum nor a minimum. \n/(\xe2\x80\x94 5/7) = 124.93/77 is a maximum. \n\n5. fx = {x + 2)V(x - 3)\xc2\xab. \n\nf\'x = (x + 2)\'^(x \xe2\x80\x94 i3)/(x \xe2\x80\x94 3)3 = o and 00 gives \n\nX = \xe2\x80\x94 2, X = 13, X = 3. \nyi3 := 135/4 is a minimum. \ny3 \xe2\x80\x94 00 is a maximum. \n\n6, fx \xe2\x80\x94 {a \xe2\x80\x94 xy/{a \xe2\x80\x94 2x). f{a/^), min. \n\n\n\nMAXIMUM AND MINIMUM STATES. ^13 \n\n-7. fx = \xe2\x80\x94 \xe2\x96\xa0 . /(\xe2\x80\x94 ), max. / \xe2\x80\x94 , min. \n\n8. fx = xy^(2a \xe2\x80\x94 x)y^. /o, min. /(4\xc2\xab/3), max. \n\n136. The preceding method, or that indicated in \xc2\xa7 134, \nmust be employed for testing critical values from the equa- \ntion f\'x = 00 ; but when/\'^,/"^, /\'"^, etc., are contmu- \nous for values of x adjacent to critical values, those derived \nfrom the equation fx = o may be examined by another \nmethod. \n\nIn Taylor\'s formula (\xc2\xa7 124) put x = a, and \\/rite \xc2\xb1 A \nfor k; then, since /\'<2 = o, we have \n\n+ (\xc2\xb1 ky + y^+^a \xc2\xb1 tiji)/\\n + I, (i) \n\nwhich form is exact for continuous values of /i from zero to \ncertain limits, provided /a,/\'a^ etc., to include /\'\'+^^, are \nreal and finite. \n\nIn order thsit fa may be a maximum, we must have ulti- \nmately, as k vanishes, \n\n/a>f(a\xc2\xb1/i); \n\nand fa a minimum requires, under the same law, \n\n/a</ia\xc2\xb1/i). \n\nThat is, fa a maximum requires that, as // vanishes, \n\nAa\xc2\xb1A)-/a, \n\nand therefore that the second member of (i) shall ulti- \nmately become and remain negative ; and/^ a minimum re- \nquires that the second member of (i) shall, under the same \nlaw, ultimately become and remain positive. \n\n\n\n214 \'^ DIFFERENTIAL CALCULUS. \n\nAs h vanishes, the sign of the second member of (i) will \nultimately become and remain the same as that of H\'\'f"al2- \nand since h^ /2 is always positive, \n\nfa a maximum requires/"^ < o, and \nfa a minimum requires /"<3; > o. \n\nIn the exceptional case when f"a = o, the sign of the \nsecond member of (i) will, under the law, ultimately depend \nupon that of \xc2\xb1 H^f"\'a/W, which changes with that of }i. fa \ncannot, therefore, be a maximum or a minimum unless \n\xc2\xb1 Iif\'"a/\\T^ = o, which requires/"\'^ = o. \n\nIf also f "\'a = o, the sign of the second member of (i) \nwill, under the law, ultimately depend upon that of \nAy^a/U; and since /^Vk is always positive, \n\nfa a maximum requires /^""^ < o, and \nfa a minimum requires /^^dJ > o. \n\nBy continuing the same method of reasoning it may be \nshown that, ii f^a is the first derivative in order which does \nnot reduce to o, fa is neither a maximum nor a minimum if \n72 is odd, and that it is a maximum or a minimum if n is even, \naccording sisf^\'a is negative or positive. Hence, we have \nthe following rule: \n\nHaving fa = o, substitute a for x in the successive deriva- \ntives of fx in order ^ until a result other than o is obtained. If \nthe correspondiftg derivative is of an odd order ^ fa is neither a \nmaximum nor a minimum j but if it is of an even order, fa is \na maximum or a minimum according as the residt is negative \nor positive. If a result oo is obtained., the 7?iethod of \xc2\xa7 135 \nshould be employed. \n\nThe relations l)etween the corresponding states of /jc,/\':^, \n\n\n\nMAXIMUM AND MINIMUM STATES. \n\n\n\n215 \n\n\n\nand f\'x^ in a case where ABCDE is the graph of fx, are \nshown graphically in the following figure.* \n\n\n\n\nEXAMPLES. \n\nFind the values of the variable which correspond to maxima \nor minima of the followijtg functions: \n\n1. fx =x^\xe2\x80\x94 5Jc4 + 5;c=*+ I. \n\nf\'x = 5x* \xe2\x80\x94 2o;i:^ + I S^"^ = o gives the critical values o, I, 3. \n\nf"x = 20^^ \xe2\x80\x94 6ox^ -\\- 30X. \n\n/"o =0. f"\\ \xe2\x80\x94 \xe2\x80\x94 10. /"3 = 90. \nHence, \n\n/i = 2, maximum, /3 = \xe2\x80\x94 26, minimum. \n\n\n\n/\'"x = 60^2 \n\n\n\n:20j:-|- 30, \n\n\n\n/\'"o = 30, and \n\n\n\nfo == I is neither a maximum nor a minimum. \n2. /x = jr^ \xe2\x80\x94 f^x"^ -\\- 24X \xe2\x80\x94 7. \n\nf\'x = ^ix"^ \xe2\x80\x94 6x-\\-S) = o gives x= 2 or x = 4. \n/";c = 3(2;c \xe2\x80\x94 6), .\'. /" 2 < o, /"4>0. \nTherefore /2, maximum, f^, minimum. \n\n\n\nCalcul, par P. Haag, page 71. \n\n\n\n2l6 DIFFERENTIAL CALCULUS. \n\n^. fx = sin^x cos X. \n\nf X =3 sin^jT cos^x \xe2\x80\x94 sin\'x = o, gives x = 60\xc2\xb0, etc. \nf"x =r \xe2\x80\x94 10 sin^x cos ;c -|- 6 sin ;r cos^;r. \nSince sin 60\xc2\xb0= Vs/^* ^"d cos 60\xc2\xb0 = 1/2, \n\ny" 60\xc2\xb0 = \xe2\x80\x94 3 \'^3/2, .\xe2\x80\xa2. /6o\xc2\xb0 = 34/3/16, maximum, \n\n4. jf* \xe2\x80\x94 Sx^ 4- 22^2 \xe2\x80\x94 24JC-+ 12. -^ = 3, min. \n\n:t = 2, max. \nx = J, min. \n\n5. ^^ \xe2\x80\x94 4x + 9, X = 2, min. \n\n6. JfV3 + fl!^\'^ \xe2\x80\x94 3a\'jp. X = a, min. \n\n:r = \xe2\x80\x94 3d!, max, \n\nX = 3, min. \n\nX = 1, max. \n\njf = P/2,a, min. \n\njf = \xe2\x80\x94 ^V3a, max. \n\nX = \xe2\x80\x94 I, max. \njr = I, min. \n\n^ = I, max. \nX = \xe2\x80\x94 I, min. \n\n^ = <f = 2.71828 . . . max. \n\nX = 7t/4, min. \nX = 57r/4, max. \n\n13. /x = ^^ +<?-\xe2\x80\xa2* + 2 cos X, \n\ny^x = ^^ \xe2\x80\x94 ^-^ \xe2\x80\x94 2 sin X, f"x \xe2\x96\xa0=e^-\\- e-^ \xe2\x80\x94 2 cos Jf, \n/\'";c = ^^ \xe2\x80\x94 e-^ + 2 sin x, /i^ x = <f^ -f <f-^ + 2 cos x, \n\nf\'o z=/"0=/"\'0 = 0, /ivo = 4. \n\nHence, /o = 4 is a minimum. \n\n\n\n7. \n\n\n^3 _ 5^2 + gx -p 10. \n\n\n8. \n\n\n3^2^8 _ 34^ _j_ ^6, \n\n\n\n\n.;c\xc2\xab-^+i \n\n\n9- \n\n\nx^-j-x-^l \n\n\n\n\nX \n\n\n10. \n\n\n1+x-\' \n\n\nII. \n\n\nxV^. \n\n\n12. \n\n\nsec X -\\- cosec jr \n\n\n\n14- \n\n\n\nI -^ X tan X \n\n\n\njv = cos X, max. \n\n\n\nMAXIMUM AND MINIMUM STATES. 217 \n\n137. The following principles frequently facilitate the \ndetermination of maxima and minima: \n\nIf F\\_f{x)\\ is an increasing function of /(jc), F\'\\^f{a)\\ is \na maximum or a minimum of ^[/(^)] according as/(^) is \na maximum or a minimum oi f[x). \n\nIf i^[/(^)] is a decreasing function of /(^), F\\^f{a)\\ is a \nmaximum or a minimum according as f{a) is a minimum \nor a maximum. Hence \xe2\x80\x94 \n\n1\xc2\xb0. Cf{a)^ in which C is a positive constant, is a maxi- \nmum or a minimum of Cf{x) according as/(^) is a maximum \nor a minimum of /(^). \n\n2^. C + /W is a maximum or a minimum according as \nf{a) is a maximum or a minimum. \n\n3\xc2\xb0. C \xe2\x80\x94 f{a) is a minimum or a maximum according as \n/(^) is a maximum or a minimum. \n\n4\xc2\xb0. The base being greater than unity, log f{a) is a \nmaximum or a minimum according as f{a) is a maximum \nor a minimum ; also any value of x that makes ^^\'^^ a maxi- \nmum or a minimum makes /(jjc) a maximum or a minimum. \n\n5\xc2\xb0. i/f{a) is a maximum or a minimum according as \nf{a) is a minimum or a maximum. \n\n6\xc2\xb0. ;\xc2\xab being any positive odd integer, [/(^)]" is a maxi- \nmum or a minimum according as f{a) is a maximum or a \nminimum. \n\n;2 being any positive even integer, \\_f{a)\'Y is a maximum \nor a minimum according as f{a) positive is a maximum or \na minimum or/(^) negative is a minimum or a maximum. \n\n[/(^)]\'* may, however, be a maximum or a minimum \nwhen /(^) is neither. Thus, {a^ \xe2\x80\x94 x^)^ is a minimum, \nwhereas (^^ \xe2\x80\x94 x^)a is neither a minimum nor a maximum. \n\nA radical sign which affects the entire function may \ntherefore be omitted, provided critical values which corre- \n\n\n\n2l8 DIFFERENTIAL CALCULUS. \n\nspond to maximum and minimum states of the power only \nare rejected. \n\nThus, having f{x) \xe2\x80\x94 \xc2\xb1 \\/ 2ax^ \xe2\x80\x94 x\\ we write \n\n0(j,) = \\_f{x)\\ = 2ax\' - x\\ \n\nThen <P\\x) = 4ax \xe2\x80\x94 ^x"^ = o gives x = o and 4a/^. \n\n(p(o) is a minimum and 0(4^/3) is a maximum of ^{x). \n\nBut/\'(^) = \xc2\xb1 (4a \xe2\x80\x94 ^x)/2V2a -- X = o gives x = 4^/3 \nonly, and /(o) is neither a maximum nor a minimum of \n/(x). \n\nSimilarly, {x \xe2\x80\x94 2ay/{x^ \xe2\x80\x94 c^) is a minimum when ^ = 2^, \nbut (^ \xe2\x80\x94 2a)/ ^x^ \xe2\x80\x94 d^ is not. \n\nTo illustrate the use of the foregoing principles ; \n\n\n\nLet f(x) = 1/(5 + log |/4^V - 2l^x\'), \n\n\n\nBy 5\xc2\xb0 we take 5 + log 1/4^ V \xe2\x80\x94 2l>x\\ \n\n\n\nBy 2\xc2\xb0 and 4\xc2\xb0 we take \\^4px^ \xe2\x80\x94 2bx^. \nBy 6\xc2\xb0 and 1\xc2\xb0 we take 2bx^ \xe2\x80\x94 x^ = (p(x). \n\n<p\\x) = 4bx \xe2\x80\x94 ^x"^ = o gives jjc \xe2\x80\x94 o, ^ = 4V3\xc2\xab \nV"(o) = 4^, 0"(4V3) = - 4\'^- \n\nHence, \n\n/(o) is a maximum and /{4b/ t,) is a minimum. \n\n138. In certain cases it is not necessary to determine the \nsecond derivative. \n\n1\xc2\xb0. In case of one critical value only, and it is known \nthat the function has a maximum .state or that it has a \nminimum state. \n\n\n\nMAXIMUM AND MINIMUM STATES. 219 \n\n2\xc2\xb0. If a is the only critical value, /(^) is a maximum or a \nminimum, provided f{a) is greater than or less than both \nf(a \xc2\xb1 h) for any assumed value of h. \n\n3\xc2\xb0. When /\'(a) is composed of two or more factors, one \nof which reduces to o, for x = a,/\'\\a) may be determined \nwithout using /"(jc). \n\nThus, let /\'(x) = tp{x)(p{x). \nThen /\'\\x) = iix)(p\'(x) + (p(x)tp\\x). \n\nSupposing that tp{a) = 0, we have \n/"(a) = 4>(<\')>P\'(a). \n\nHence, to obtain f"{a), multiply the differential coefficient \nof that factor of f\\x) which reduces to o by the other factors, \nand substitute a. \n\nTo illustrate, let \n\nf{x) = (x- \xc2\xab) V. \n\nf\'(pc) = 2x(x \xe2\x80\x94 a)(2x \xe2\x80\x94 a) =^ o gives \n\nX ^ o, X = a, X = a/2. \n\ny"(o) = 2(x \xe2\x80\x94 a)(2x \xe2\x80\x94 a)o = 2^^ indicating a minimum. \n\nf"(a) = 2jc(2ji; \xe2\x80\x94 cL)a = 2a^, indicating a minimum. \n\nf"(a/2) = /\\x{x \xe2\x80\x94 a)^,^ = \xe2\x80\x94 d^, indicating a maximum. \n\nEXAMPLES. \n\nJ^ind the values of the variable which correspond to maxima \nor minima of the following functions : \n\n(x -f- 3)^ X = o, min. \n\n\' (jc + 2)* \' X ^ \xe2\x80\x94 2, max. \n\n{y \xe2\x80\x94 \\f J = I, min. \n\nQ/ -f- 1)=^* Jf\' = 5, max. \n\n\n\n220 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\n3, 1/^2^2 _ ^4. \n\n\n\n4. \n\n\n\n2r;rj[:\' \xe2\x80\x94 TtX^ \n\n\n\n2.r \xe2\x80\x94 ^3. \n\n\n\n5. a*x \xe2\x80\x94 X \n\n6. aj^\' \xe2\x80\x94 ;\xc2\xab:*, \n\n\n\n^ = o, min. of power. \n\xe2\x80\xa2\xc2\xab\xe2\x96\xa0 = \xc2\xb1 <^/V\'2, max., min. \n^ = o, min. \n^ = 4r/3, max. \n\nX = a/4/3, max. \nX = 3rt;/4, max. \n\n\n\n7. x-" - X \n\n\n\n5/2. \n\n\n\n8. \n\n\n\nX^ - 2^" \n\n\n\n2^" + 8 \n9. x\'^. \n\n10 (2^x< + a^bx)/a^. \nb \n\n\n\n^ 1 \n\n\n\n{c - xf \n\n\n\nb a \n\n\n\nX = o, mm. \n\nX = 16/25, max. \n\nX = o, max. \njf = 1. 19, min. \nX = i/e, min. \n\nX = \xe2\x80\x94 a/2, min. \n\n^ ya \n\nX \xe2\x80\x94 \xe2\x80\x94 Y= 3"^^\xc2\xbb min. \n\nj/^ _|_ |/^ \n\n6 = tan-i |/3/dz, min. \n\n\n\nsin cos \n\n13. fx = x/2 \xe2\x80\x94 x^ sin {i/x)f2. \n\nfx \xe2\x80\x94 1/2 + cos (i/x)/2 \xe2\x80\x94 X sin (i/x). \n\n1,1 i-i I .1 I \n1 cos \xe2\x80\x94 = cos\' \xe2\x80\x94 X sin \xe2\x80\x94 = 2jr sm \xe2\x80\x94 . cos \xe2\x80\x94 . \n\n2 2 ;f 2X X 2X 2X \n\nHence, fx = cos \xe2\x80\x94 . ( cos 2x sin 1 = o gives \n\n2X \\ 2;\xc2\xbbr 2x1 \n\n\n\nM \n\n\n\n\n\n2X \n\n\n\\ 2X \n\n\n2XJ \n\n\n\n\nX \n\n\n= 1/^, \n\n\nJT = 00. \n\n\nf\\i/n) = \n\n\n- 4/^^ \n\n\nindicating a maximum. \n\n\na-^x \n\n\n\n\n\n\nX \xe2\x96\xa0= \xe2\x80\x94 a, min. \n\n\n{a - xf \xe2\x96\xa0 \n\n\nX = ay max.. \n\n\nab \n\n\n._./\xc2\xab= + \'\'\xe2\x96\xa0\' \n\n\n\nX Va" -^ b"" - x"" \n\n\n\nMAXIMUM AND MINIMUM STATES. 22 \n\n\n\nPROBLEMS. \n\n1. Divide a number a into two such parts that the product of the \nmS^ power of one and the w*^ power of the other shall be a maximum. \n\nfx = x^^{a \xe2\x80\x94 xY, f\'x \xe2\x80\x94 x"^-\\a \xe2\x80\x94 xy^-\'^[?na \xe2\x80\x94 (w + n)x] = o \n\ngives X ^ o, X ^ a, x = ma /{in -\\- n), \n\n/"ma/{m -]-\xc2\xab)=: \xe2\x80\x94 {m -{- n)c, indicating a maximum. \n\n2. Divide a number a into two such factors that the sum of their \nsquares shall be a minimum. \n\nfx=:x\'^-\\- a\'^/x^, X =^ \xc2\xb1 ^a, minimum. \n\n3. Into how many equal parts must a number a be divided that \ntheir continued product may be a maximum? \n\nLet X = the number of equal parts, then \n\nfx \xe2\x80\x94 {a/xY, . *. log fx = x (log a \xe2\x80\x94 log x). \n\nf\'x = fx{\xe2\x80\x94 I -j- log a \xe2\x80\x94 log x) = o gives x = a/e, \nf"x{a/e) = W^(\xe2\x80\x94 e/a), indicating a maximum. \n\n4. Let A be the hypothenuse of a right triangle; find the lengths of \nthe other sides when the area is a maximum. \n\n\n\nLet X = one side, then \\/h^ \xe2\x80\x94 x^ = the other. \n\n\n\n/x = area = x r/i\'^ \xe2\x80\x94 x\'^/2. f\'x = o gives /^^ \xe2\x80\x94 ix\'^ = o, \nwhence x = hj V2, f\'\\h/ V2) \xe2\x80\x94 \xe2\x80\x94 ^/iK \n\n5. What fraction exceeds its \xc2\xab*^ power by the greatest number \npossible ? \n\nLet X = fraction, then fx=x \xe2\x80\x94 x^. \n\nf\'x = I \xe2\x80\x94 \xc2\xabjc\xc2\xab-i = o gives X = 1/ \\n. \n\n\xe2\x80\x9e( r-h-\\ I \xe2\x80\x94 \n\nf \\\\/ \\n)^^\xe2\x80\x94n{n\xe2\x80\x94\\)l \xc2\xab\xc2\xab~^ , indicating a maximum. \n\n6. Of all isoperimetrical rectangles which has the greatest area ? \n\n\n\n222 DIFFERENTIAL CALCULUS. \n\n7. On the right line A c B joining \n\nthe two lights A and B, find the point between the lights of least \nillumination. \n\' Let c = number of miles from A 10 B. \n\nLet X = number of miles from A to required point. \n\nLet a = intensity of the light A at i mile from A, \n\nLet d = intensity of the light ^ at i mile from B. \n\nThen \n\na b \n\nfx \xe2\x80\x94 -^-\\-7\xe2\x80\x94^ \xe2\x80\x94 <A \xe2\x80\x94 intensity of light at point required. \n\nc ya \n\n\n\ngives X \n\n\n\n|/a -f /\\/b \n\n\n\n8. Find the relation between the radius of the base and the alti- \ntude of a cylinder, open at the top, which shall just hold a given \nquantity of water and have its surface a minimum. \n\nLet X = radius of base, and j = altitude. Then \n\nTtyx"^ = volume = v. .\'. y = v/itx\'^, \n\nTtx\'^ -f- 2Ttxy = surface = /(x). \nHence, \n\nfx = TtX^ + ITCXv/nx\'^ = Ttx\'^ 4- 2v/x. \n\nf X = 2TIX \xe2\x80\x94 2vlx* = o gives x = \'^vjn and y = \'[/v/tc. \n\n/"( I^z^/tt) = 67r, indicating a minimum. \n\ng. Find the maximum rectangle that can be inscribed in a given \ncircle. \n\nTake the origin at the centre, and let 2x and 2y respectively equal \nthe sides of the required rectangle. Then xy = 1/4 area of rectangle. \n\nx^ -{-y^ = R"^, in which R represents the radius of the circle, gives \n\n\n\ny= \xc2\xb1 V^\'\' \n\n\n\nHence, fx = \xc2\xb1x\\/R-^ - x"^- \n\n/\'jf = \xc2\xb1(2^\'^x \xe2\x80\x94 4Jr^) = o gives x = o and x^^R/ \\2. \n/"{r/ ^2) = \xe2\x80\x94 4^^, indicating a maximum. \n\n\n\nMAXIMUM AND MINIMUM STATES. \n\n\n\n223 . . \n\n\n\n10. Determine the maxitrum rectangle which can be inscribed in \na triangle of given base and altitude. \n\n11. Show that the difference between the sine and the versed sine \nis a maximum when the angle is 45\xc2\xb0. \n\n12. The base and ihe vertical angle of a triangle being given, show \nthat its area is a maximum when it is isosceles. \n\n13. Show that of all triangles of a given perimeter (a) and given \nbase {b), the isosceles has the greatest area. \n\n14. Divide a triangle whose sides are a, b, and c, respectively, into \ntwo equal parts by the minimum right line. \n\n\n\n\nAns. ^{c \xe2\x80\x94 a-^b){c-\\-a \xe2\x80\x94 b)/2, \n\n15. Through a given point {a, b) draw \nthe shortest straight line terminating in \n^and Y. \n\nLet 9 = angle required line CB makes \nwith X. \nThen \nCB=CF-{-FB = a/ cos Q + ^sin 0, \n\nwhich is a minimum when Q tan-i \\^b/a, giving CB ={a^^^ -{- ^^^Y\'"^ \nSimilarly, show that \n\nOB + C>C7is a minimum when = tan-i ^bja\'y \n\nOB X OCis a minimum when = tan-i(V\xc2\xab); \n\n0B-\\- OC-^ CB is a minimum when = tan-i ^ + ^^ ; \n\na -f- ^lab \n\nOB X OC X C!i5 is a minimum when \n\nla tan^ \xe2\x80\x94 5 tan\'^ + a tan \xe2\x80\x94 23 = o. \n\n16. Determine the maximum right cone which can be inscribed in \na sphere whose radius is A\'. \n\n\n\n224 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\n\nLet X = AF, and y = PB. \n\nThen \n\ntcv^x \nvol. = z/ = ~ \xe2\x80\x94 , but y^ = 2Fx \xe2\x80\x94 x\\ \n\nTherefore v = {2F7tx^ \xe2\x80\x94 7tx^)/3. \ndv/dx = nx{^R \xe2\x80\x94 3-^)/3 = o \ngives X = o, X = 4R/3. \n\nd^v/dx\'^ = 7r(4/? - 6^)/3 = - 4^V3]^=4;?/3. \n\n17. Find the radius of a circle such that the segment corresponding \nto an arc of a given length shall be a maximum. \n\nLet 20! = length of arc, and r = radius. a. \n\nDraw CD bisecting the arc, then \n\nZDCA = 6/2 = a/r, and = 2a/r. \n\nSegment = sector BCAD \xe2\x80\x94 aBCA \n\n= r\'9/2 - r"" sin 6/2 \n\nz=i ra \xe2\x80\x94 r^ sin {2a/r)/2, \n\nwhich is a maximum when r = 2a/7t, and \nthe segment is a semi-circle. \n\n18. With a given perimeter find the radius which makes the cor- \nresponding circular sector a maximum. Ans. radius = 1/4 perimeter. \n\n19. Find the maximum right cylinder which can be inscribed in a \ngiven right cone. \n\nLet VA = a, BA = b, AC = x, \nCD =y, CV = a \xe2\x80\x94 x. \n\nHence, vol. cylinder = z/ = Tty^x, \nVA -.ABwVC: DC, .\'. y= b{a - x)/a. \nTherefore v = nl)\'^{a \xe2\x80\x94 xfx/a^. \nOmitting nb\'^/a^, we have \n\nJ{x) = a^x - 2ax\'\' + x^. \n\n\n\n\n\nMAXIMUM AND MINIAiUM STATES. \n\n\n\n22 C \n\n\n\nf\\x) = a^ \xe2\x80\x94 ^ax -j- 3Jt^ = o gives x \xe2\x96\xa0= a or (7/3. \n/"[a/\'i) = \xe2\x80\x94 la, therefore v = /\\TTaP/2-i is a maximum. \n\n20. Circumscribe the minimum isos- \nceles triangle about the parabola j^=4\xc2\xabx. \n\nLet X = C/\' = ^ (9, 7 = PM, h = OD. \nThen \n\n^ _ A. \n\nBD\xe2\x80\x94{h + x)y/2x = {h-\\- x)\\/ax/x, \n\nand area A \n\n= {k -{-xf \'i/ax/x, X = h/2, min. \n\n21. Determine the minimum right \ncone circumscribing a given sphere. \n\nLet X = alt. = AD, y = radius of base, \nR = radius of sphere. \n\nThen V = vol. of cone = ny^x/^. \n\n\n\n\n\ny\\R:\\ s/x\'- -f / \\x - R. \nFrom vi^hich y"^ = R\'^x/{x \xe2\x80\x94 2R), \n\nand V = 7rR^x^/3{x - 2R). \nX = 4R, min. \n\n\\B \n\n\n\n22. Find the maximum parabola that \ncan be cut from a given right cone. \nLet AC = a, AB \xe2\x80\x94b, DC = x. \n\n\n\nThen AD = a \xe2\x80\x94x, DE= Via \xe2\x80\x94 x)x. \nAlso, a :x: : d : DG. .\' . DG\xe2\x80\x94 bx/a. \n\n\n\nParabola = \\b \\/ax\'\'^ \xe2\x80\x94 x\'^/^a, \nX = 3\xc2\xab/4, max. \n\n\n\n23. Find the maximum isosceles triangle inscribed in a given \ncircle. \n\n\n\n\n226 \n\n\n\nD IFFEREN TIA L CA L CUL US. \n\n\n\n\n]Let r \xe2\x80\x94 radius CA, AB \xe2\x80\x94 AE \xe2\x80\x94 x, BE = 2y. \nThen area A\xe2\x80\x94 u =^y ^ x^ \xe2\x80\x94 y^. \nAlso, u \xe2\x80\x94 \n\n\n\nAB y^ AE y^ BE \n\n\n\n4r \n\n\n\nx\'^y \n\n\n\nx\\/4) \n\n\n\nAnd u \n\n\n\ni/Ar^- X\' \n\n\n\n. X = ^4/3, max. \n\n\n\n2r ir \n\n24. Find the maximum cylmder that can be inscribed in a given \nprolate spheroid. \n\nLet Q.X \xe2\x80\x94 axis, and y \xe2\x80\x94 radius of \nbase, of required cylinder. \nThen, vol. of cylinder \n\n= z/ = 2iiy\'^x \xe2\x80\x94 27txb\'^{a\'* \xe2\x80\x94 x\'^)/a^, \nwhich is a maximum for x \xe2\x80\x94 (2/ 4/3. \n\n25. Find the minimum isosceles triangle circumscribing a given \ncircle. \n\nLet y = radius CE, x \xe2\x80\x94 BF^ \n\n2y - AD. \n\nThen area A = xy. \n\nSimilar triangles, BCE, BED, give \n\ny : y : ^ x\'\' -\\- y^ : x - r. \n\n\n\n\n\nHence, y - r\\/x/{x - 2r), and \n\n\n\narea ^ - xr \\/x/{x - 2r), \nD which is a minimum for x = 3^. \n\n26. " A boatman 3 mi from shore goes to a point 5 \nmi. down the shore in the shortest lime. He rows 4 B_ \nmi. and walks 5 mi an hour, Where did he land? \n\nLet B be the boat 3 mi. from S, vvhich is 5 mi. \nhorn the point /*. Let IV be the landing-place, and \n\nThen, number of hours \n\nr- i ^ 1/9+^74 -i- (5 - x)/s, \nwhich is a minimum for jc = 4. \n\n\n\n*Todhunters Difl. Calc, p. 213. \n\n\n\nMAXIMUM AND MINIMUM STATES. \n\n\n\n227 \n\n\n\n27. Through a given point P within an angle \nBAC draw a right line so that the triangle \nlormed shall be a minimum. D \n\nDraw PD parallel to AC, and let AD = a. \n\nPD = h, AX - X. ,\'. DX =: X \xe2\x80\x94 a. \nThen x \xe2\x80\x94 a\\b \\\\ x : AR. \n\nAXR \xe2\x80\x94 X AR-sAw A J 2., which is a minimum for x \xe2\x80\x94 2a. \n\n\n\n\nXP=PR. \n\n\n\n\n28. The volume of a cylinder being constant, find its form vyhen the \nsurface is a mmimum. Ans. Altitude = diameter of base. \n\n29. Find the height of a light A above the \nstraight line OB when its intensity at ^ is a \nmaximum. \n\nLet a = intensity of light at i foot from the \nlight. \n\nB OA - y, OB-b, I OB A - 6. \n\nThe intensity varies directly as sin 9, and mversely as BA . \nIntensity at B = ay/{b^ -|- y^f^ , which is a maximum when \n\ny = d \\/2 1 2. \n\n30, Having y =^ x tan a \xe2\x80\x94 x\'^/^h cos\'^ a . 1\xc2\xb0. Find the maximum \nvalue of y. 2*. Considering/ = o and a as varying, find maximum \nvalue of X. \n\n1st. y \xe2\x80\x94 h sin- a, a maximum, x = h sin 2a. \n\n\n\nAns. \n\n\n\n2d. X \xe2\x80\x94 \xe2\x96\xa0 2h, a maximum, a = 45\' \n\n\n\n31. On the right line CC\\ -- a joining the centres of two spheres \n(radii A, r) find point (rom which the maximum spherical surface Is \nvisible. \n\n\n\n\n22S DIFFERENTIAL CALCULUS. \n\nLet CP = X. .\'. PCx - a - X. \n\nArea zone ASHB = inR X HS. \n\nX- Rv. R: CS. .-. CS := R^x and HS-R- R^/x. \nHence, zone ASBH = 2;rA\'(A\' - R\'\'/x). \nSimilarly, zone DEL \xe2\x80\x94 2ni(r \xe2\x80\x94 r\'^/{a \xe2\x80\x94 x)\\. \n\nVisible surface --^ 27t\\^R\'\' + r^ - (^R^x + rV(\xc2\xab - ^))], which is \na maximum for x = aR^^\'^/^R^^^^ + r^^% \n\n32. Find the path of a ray of light from \na point /i in one medium to a point B in \nanother medium, such that a minimum \ntime will be required for light to pass \nfrom A to B; the velocity of light in the \nfirst medium being V, and in the second \nv. [Fermat\'s Problem.] \n\nIt is assumed that the required path is \nin a plane through A and B perpendicu- \nlar to the plane separating the media. \n\nLet ACB be the required path. Through A, C, and B draw per- \npendiculars to DE. \n\nLet a - AE, b = DB, d- DE, \n\nThen AC = \xc2\xab/cos 0, BC = b/cos (p\' , \n\nCE = a tan 0, CD \xe2\x80\x94 b tan (p\' . \n\na tan (p -\\- b tan 0\' \xe2\x96\xa0= d. .\'. d<p\'/d(p = \xe2\x80\x94 a cos^ <P\'/b cos\'\' 0, \n\nh \nTime =. i, from A to B = \n\n\n\n\nV cos V cos 0\' \n\n3 \n\n\n\na\'/ _ \\ F cos 0/ 1 F\' cos 07 id_0\'\\ \n\n70 ~~ d\'0 + d0\' \\d0)\' ^^ ^^^\' \n\n\n\n</0 \n\ndf \n\nd0 \n\n\n\nsin \n\n\n\nsin \nF\'" \n\n\n\nsin = T^sin 0\', \n\n\n\nd0\' \n\n- o gives \n\nsin _ \nsin \n\n\n\nV_ \n\n\n\n(a) \n\n\n\nfor t a minimum. \n\n\n\nMAXIMUM AND MINIMUM STATES. \n\n\n\n229 \n\n\n\nEquation {a) expresses an important law, which is known as Snell\'s \nlaw of refraction. \n\n33. To inscribe in a given sphere a right cone with a maximum \nconvex surface. ^B \n\nLet R = radius AC, x = AP, \n\n\n\ny \xe2\x80\x94 V2RX - x^= TB, \ns = 27tTBAB/2 \n\xe2\x80\x94 convex surface. \n\nThen AB = M^Rx, and \n\n\n\n= It V4R\' \n\n\n\n2Rx^, \n\n\n\n\nwhich is a maximum for x = 4/^/3. D \n\n34. From two points A and B draw two right lines to a point F \n\nin a given right line OX, so that \nAF 4" BF shall be a minimum. \n\nLet X\xe2\x80\x94 OF, a = OC, b\xe2\x80\x94 CA, \na\' = OD, b\' = DB. \nAns. /LAFC-DFB, or = 0\'. \nP D In some cases the general \n\nmethod ( \xc2\xa7135) apparently fails when it is obvious that maxima and \nminima stales exist. \n\n35. Find the maximum and minimum distances from a given ex- \nternal point to a given circumference. \n\n\n\n\n\nLet A be the point, r = radius, a \nThen \n\n\n\nFA = Vr^ + a-" \n\n\n\n2ax, \n\n\n\n= AO, OB =x. \n\nand dFA/dx = \xe2\x80\x94 2a. \n\n\n\nIt is obvious, however, that AM is a minimum and AN a maxi- \nmum. The method of \xc2\xa7 135 depends upon the assumption that the \nvariable increases continuously ; whereas x in the above expression \n\n\n\n230 DIFFERENTIAL CALCULUS, \n\nchanges from increasing to decreasing, or vice versa, as PA passes \nthrough AM 2.n& AN. \n\nLet 6 = angle AOP. Then x = r cos 0, and \n\n\n\nPA = Vr^ -\\- d^ \xe2\x80\x94 2ar cos \ndPAjdb = 2ar sin 6=0. \n\n\n\no, min. \n\n11. max. \n\n\n\nOtherwise PA may be expressed in terms of y and the problem \nsolved. \n\n36. Find those conjugate diameters in an ellipse whose sum is a \nmaximum or a minimum. \n\nLet X and y be any two semi-conjugate diameters, and let j=j;\'-f->\'\' \nand a=^ + <5\' = f^ Then [Anal. Geom.]y^+y2 = t^ \n\n\n\n. \xe2\x80\xa2. / = V^2 _ ^\'2 and J = y + V^^ \xe2\x80\x94 x"^. \n.*. ds/dx\' = I \xe2\x80\x94 x\' I Vc^ \xe2\x80\x94 x"^ = o gives x\'^ \xe2\x80\x94c\'^/i =y2. \n\nThat is, equal conjugate diameters are those whose sum is a maxi- \nmum. \n\nExpressing x\' and y in terms of the inclination of x to the trans- \nverse axis, denoted by 6, we have ds/dB = {ds/dx\')(dx\'/dQ). \n\na and b are, respectively, maximum and minimum states of x\\ \ngiving dx\' /dB \xe2\x80\x94 o, and therefore ds/dB = o. Hence the sum of the \naxes is a minimum. \n\n37. " A rectangular hall 80 feet long, 40 feet wide, and 12 feet high \nhas a spider in one corner of the ceiling. How long will it take the \nspider to crawl to the opposite corner o n the floo^ if he crawls a foot \nin one second on the wall and two feet in a second on the floor?"* \n\nAns. 55.4754 seconds, minimum. \n\n1 39* ^^ find the maximum and minimum distances from \na given plafie curve to a given point in its plane. \n\nLet y = fx be the equation of any plane curve, (a, b) \nthe coordinates of any point in its plane, and i? the dis- \ntance from [a, b) to the point {x\\y\') on the curve. \n\n* Problem proposed by Professor H. C. Whitaker in American \n\nMathematical Monthly, Vol. L No. 8. \n\n\n\nMAXIMUM AND MINIMUM STATES. 23 1 \n\n-If {x\' ^y\') move along the curve, R in general becomes a \nvarying distance measured on the radius vector joining (^, b) \nwith the moving point {x\' ^y\') and i?" =(a:\' \xe2\x80\x94 of + (jv\' \xe2\x80\x94 ^)^ \nWe wish ,to find the maximum and minimum values of R. \n\nPlacing the first derivative of {x\' \xe2\x80\x94 ay -\\- (y\' \xe2\x80\x94 by equal \nto zero, we have \n\n^\' _ ^ _l- (/ _ b){dy/dx\') = o. . . . (1) \n\nThe equation of the normal to y ^= fx at {x\',y\') is \ny \xe2\x80\x94 y\' = \xe2\x80\x94 {dx\'/dy\')(x \xe2\x80\x94 x\'),^i^S. Hence (i) expresses \nthe condition that (x\',y\') is on the normal through {a, b). \n\nThe required value of R is therefore estimated along the \nnormal through {a, b), and is a maximum or a minimum \naccording as the second derivative (dropping the primes), \n\nI + (Jy/dxY + (/ - i){dy/dx\'), \n\nis negative or positive, and, in general, is neither a maxi- \nmum nor a minimum when the second deiivative reduces \nto zero (\xc2\xa7 136). \n\nAs {a, b) may be any point upon any normal, we con- \nclude that the radial distance of each point of a normal \nfrom the curve is, in general, a maximum or a minimum \nwhen measured upon the normal. (See figure, page 232.) \n\nThus, let BAM be a normal to the curve NMO at M. \n^With A and B as centres, and with the radii AM a.nd BM \nrespectively, describe the circumferences rMr and RMR. \nThe figure shows that the radial distance of A from NMO \nis a minimum when measured upon the normal AM, and \nthat the corresponding distance of the point ^ is a maxi- \nmum. This is evident from the fact that the circumference \nvMr^ in the vicinity of and on both sides of M, lies within \nthe curve NMO, while the corresponding part of the cir- \ncumference RMR lies without. \n\n\n\n232 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nConsider the point [a^ b) to move upon the normal, and \nlet yoc^yj be its variable coordinates. When the normal \n\n\n\n\ndistance of \\Xyy ) from the curve is neither a maximum nor \na minimum, we have \n\nI + {dy/dxY + (y --y){d-^y/dx\') = o, \n\nwhence by combination with (i) we obtain \n\n\n\ndx\' \n\n\n\n\\ dx^ jdx \' dx \n\n\n\n(2) \n\n\n\nfor the coordinates of a point on the normal whose distance \nfrom the curve measured along the normal is, in general, \nneither a maximum nor a minimum. Representing this dis- \ntance by p, we have \n\np^ = {x-Tf + {y-y)\\ \n\nwhich combined with (2) gives \n\n\n\np= 1 + \n\n\n\ndx\'\' \n\n\n\n\'\xe2\x96\xa0 l<\xc2\xa3l \n\nI dx\' \n\n\n\n(3) \n\n\n\nMAXIMUM AND MINIMUM STATES. 233 \n\nIn the figure, page 232, \\x^y ) lies somewhere between A \nand B. It separates those points of the normal each of \nwhich has a minimum radial distance from the curve lying \non the normal, from those points of the normal each of \nwhich has a corresponding maximum distance on the same \nline. \n\nIt is important to observe that a circumference described \nwith \\x, J\' ) as a centre and with a radius equal to p will, \nin general, intersect the curve NMO at M. \n\nThis circle is important in the discussion of curves, and \nequations (2) and (3) will be referred to hereafter. \n\nIMPLICIT FUNCTIONS. \n\n140. Having J given as an implicit function of x^ by an \nequation f{^x^y) = o not readily solved with respect toj, \nwe may differentiate as indicated in \xc2\xa7 no and obtain an \nexpression for dy/dx. Placing it equal to o, we may com- \nbine the resulting equation with the given, and find critical \nvalues of x. \n\nOtherwise, let u=f{x^y) = o (i) \n\nThen (i), (\xc2\xa7111), \n\ndu/dx = du/dx -f (di^/dy) (dy/dx) = o. . (2) \n\nMaxima and minima values of y in general require \ndy/dx = o. Hence, \n\n^ \xe2\x80\x94f(^,y) \xe2\x80\x94 o, combined with \'du/dx = o, \\ \n\ngives critical values of x. \nEq. (6) {\xc2\xa7 in) gives \n\nd\'^y/dx^ = - (dW^^l/idu/dy), \n\nwhich, if not zero or infinity, is positive for a minimum and \nnegative for a maximum oi y. \n\n\n\n234 DIFFERENTIAL CALCULUS. \n\nHaving y \xe2\x80\x94 fZj z = cpx, \n\ndy/dx = (dy/dz) X (dz/dx) = o, \nwill give critical values of x. \n\nEXAMPLES. \nI. u = x^ -\\- y^ \xe2\x80\x94 2^^^ = o\xc2\xab \n\ndu/dx = sx^ \xe2\x80\x94 Sa"^ = o. .\'. X = \xc2\xb1 a. \nSubstituting in given equation, we havejj/ = \xc2\xb1 a ^2. \n\'d^ujdx^ = tx, \'du/\'dy = s^y". \n\n^\'Vn -6a 3/-. \n\n:74 \\x=a = , 3/- > \xe2\x80\xa2 \xe2\x80\xa2 ji\' = \xc2\xab y2 IS a maximum. \n\n\n\n^vn 6a 8/\xe2\x80\x94. . . \n\n\xe2\x80\x94 ^n, . . jj/ = \xe2\x80\x94 a y 2 IS a mmimum. \n\n\n\n\'1 - - \xe2\x80\x94 \n\n2. Jf^ \xe2\x80\x94 saxy -j-jj/^ = o. jr = o, jj/ = o, is a minimum. \n\nX = a 4/2, y \xe2\x80\x94 a I/4, is a maximum \n\n\n\n3. jc" -j-^\'\' \xe2\x80\x94 23xy \xe2\x80\x94 a\' = o. \na3 \n\n\n\n^ = , IS a maximum. \n\n\n\n4. 4XJJ\' \xe2\x80\x94 ;j/4 \xe2\x80\x94 jc* = 2. X = \xc2\xb1 I, >\' = \xc2\xb1 I, no max. or min. \n\n5. jv\'^ \xe2\x80\x94 3 = \xe2\x80\x94 2Jr(jj/x + 2). X = \xe2\x80\x94 1/2, J = 2, a maximum. \n\n6. jj/ = Tt^zjuy z r= {k"^ -\\- x\'^)/x, X z= k makes ^ a minimum. \n\n141. Having v given as an implicit function of x^ by two \nequations v = (p[x^y) and ti ^= f{x,y) = o, from which _y \nis not readily eliminated, we may proceed as follows : \n\ndv _\'dv dv dy \ndx dx \'dy dx \' \n\n\n\ndy \'du fdu ( \\ ro X \n\n\n\ndx dxl \'dy \n\n\n\nMAXIMUM AND MINIMUM STATES. 235 \n\n\' dx ^ dx dy dx I Qy \' \n\ndv . dv 9^/ dv du , . \n\nand ^=0 gives -\xe2\x80\x94 _---=o, . . (i) \n\nwhich combined with u = /(x, y) = o gives critical values \nof X. The sign of the corresponding value of d\'^v/dx^ will, \nin general, determine whether z/ is a maximum or a minimum. \n\nEXAMPLES. \n\nI. 2/ = x\xc2\xab+/, (;\xc2\xbb: - \xc2\xab)\' -\\-{y- bf - c^ = 0. \n\ndv/dx \xe2\x80\x94 2x, dv/dy = 2y, \n\ndu/dx = 2{x \xe2\x80\x94 a), du/dy \xe2\x80\x94 2{y \xe2\x80\x94 5). \n\nSubstituting in (i), we have ay z=dx; which combined with m = o \ngives \n\n\n\nX = a \xc2\xb1ac/ Vd" -h b"". \n\nThe positive sign gives a maximum, and the negative a minimum, \nfor V. \n\n2. Find the points in the circumference of a given circle which are \nat a maximum or minimum distance from a given point. \n\n3 Given the four sides of a quadrilateral, to find when its area is a \nmaximum. \n\nLet a, b, c, d be the lengths of the sides, (p the angle between a and \nb, rj) that between c and d. \n\nThen area r=. v = ab sin 0/2 -|- cd sin ^/2, \n\nand a" _|_ ^2 _ 2^(5 cos (p = c^ ^ d"^ \xe2\x80\x94 2cd cos ^, \n\neach member being the square of the same diagonal. \n\ndv ab , dv cd \n\n-\xe2\x80\x94 = \xe2\x80\x94 cos 0, \xe2\x80\x94 \xe2\x80\x94 = \xe2\x80\x94 cos tp, \n\nd(p 2 ^\' a^ 2 \n\n-7- = 2\xc2\xab<5 sm 0, \xe2\x80\x94 = \xe2\x80\x94 2cd sm z^. \n\n^0 d^i^ \n\n\n\n236 \n\n\n\nDIFFERENTIA L CA L CUL US. \n\n\n\nSubstituting in (i), we have \n\ntan (p \xe2\x80\x94 \xe2\x80\x94 tan ^. . ". (j) \xe2\x80\x94 180\xc2\xb0 \xe2\x80\x94 ^. \nThat is, the quadrilateral is inscribable in a circle. \ndv ab ^ , cd , d^ \n\n\xe2\x80\x94 - = cos (p \\ cos Tp\xe2\x80\x94- \xe2\x80\x94 O, \n\nd(p 2 \'2 d(f) \n\nab sin (f)=^ cd sin \'^{dip/d(p) ; \nfrom which dTp/d(p = ab sin cp/cd sin ip. \n\nSubstituting in above, we have \n\ndv/d<p = ff^ sin {<p -\\- ip)/2 sin ip. \nd\'^v _ ab cos {(p -(- ip) \n\n\n\nd(p\'\' \n\n\n\nd\'^v\' \n\n\n\n\xe2\x80\x94 ab i , ab\\ . ,. \n\n\xe2\x80\x94 : \xe2\x80\x94 -, U H ;l> indicating \n\n2 sin ip\\ cdr \n\n\n\na maximum. \n\n, . , 2 sin \xe2\x96\xa0ib\\ \' ccti " \n\n142. Having w given as an implicit function of x, by- \nthree equations \nw = F{x, y, z), V = (p{x, y, z) = o, and u = /{x, y, z) = o, \n\nand placing dw/dx = o for a maximum or a minimum, we \nwrite \n\n\n\ndw 9w 9w dy \'dw dz \n\ndx dx "dy dx \'dz dx \n\n\n\ndv \ndx \n\n\ndv dv dy dv dz \n~~dx \'^ dy dx\'^dz ^ ~ \xc2\xb0 \n\n\ndu \n\ndx \n\n\ndu du- dy du dz \ndx dy dx dz dx \n\n\n\n(l) \n\n\n\nEliminating dy/dx and dz/dx, we have a single equation \nwhich, combined with 7V = F{x,y, z), v = o, and u \xe2\x80\x94 o, \ngives critical values of x and the corresponding values oi y^ \nz, and w. \n\nBy differentiating equations (i), and eliminating \n\ndy/dx, dz/dx, d\'^y/dx\', d\'^z/dx\'^, \n^n expression for d\'^w/dx^ may be determined. \n\n\n\nMAXIMUM AND MINIMUM STATES. 237 \n\nExample. * A Norman window consists of a rectangle \nsurmounted by a semicircle. With a given perimeter, find \nthe height and width of the window when its area is a \nmaximum. \n\nLet y = height, 2X \xe2\x80\x94 width, w = area, J^ = perimeter. \n\nTtx"^ \nThen w = j- 2xy, v = 2{x -\\-y) \xe2\x96\xa0\\- nx \xe2\x80\x94 P =-0. \n\ndw/dx = Ttx + 2 V + 2xdy/dx = o, ) \n\n[.. . (i) \n\n2 -{- TV -\\- 2dy/dx = o. ) \nEliminating dy/dx^ and combining result with z^ = o, \n\nDifferentiating equation (i), we have \n\nd\'^w/dx\' = TT -{- 4dy/dx + 2xdy/dx^f \n2d^y/dx^ \xe2\x80\x94 o. \nHence, d^w/dx\'^j^^y = \xe2\x80\x94 tt \xe2\x80\x94 4, indicating a maximum. \n\nFUNCTIONS OF TWO OR MORE VARIABLES. \n\n143. A Maximum state of a continuous function of two \nindependent variables is one greater than any adjacent \nstate. Thus, z=/{x,y) is a maximum corresponding to \nX = a, y = d, provided as /i and k vanish from any values, \nwe have ultimately and continuously /(^, b)> fia^h^ d\xc2\xb1k). \n\nA maximum state is, therefore, one through which, as \neither or both variables increase continuously, the function \nchanges from an increasing to a decreasing function, and its \npartial differential coefficients of the first order change their \nsigns from plus to minus. (\xc2\xa771.) \n\n*Todhunter*s Diff. Calc, p. 214. \n\n\n\n238 DIFFERENTIAL CALCULUS. \n\nA Minimum state is one less than adjacent states. Thus, \n/{a, b) is a minimum, provided as h and k vanish from any \nvalues, we have ultimately and continuously \n\nf{a, b) <f(a\xc2\xb1h,b\xc2\xb1 k). \n\nA minimum state is, therefore, one through which, as \neither or both variables increase continuously^ the function \nchanges from a decreasing to an increasing function, and iis \npartial differential coefficie7its of the first order change their \nsigns from ?ninus to plus. (\xc2\xa7 71.) \n\nAny particular state of z =/(x, jf), as/(^, ^), may be ex- \namined directly by determining whether as h and k vanish \nfrom any assumed values, we have ultimately \n\nf{a,b)> f{a\xc2\xb1h,b\xc2\xb1k), or f{^a, b)<f(a\xc2\xb1h, b\xc2\xb1k). \n\n144. In general, however, maxima and minima are deter- \nmined by testing those values of the variables correspond- \ning to which, as the variables increase, both partial \nderivatives of the first order change their signs from plus \nto minus, or minus to plus. \n\nSets of roots of the equations \n\n\n\n\'dz/dx = o, \'dz/dy = 0, . . . . (i) \n\n( dz/dx \xe2\x80\x94 00 , dz/dy = co \n\'Exceptional \n\n. cases. I 1 S"^/-^^ = "\' ^\'/\'^^ = ~ (s) \n\n\n\n(3) \n\n[ dz/dy \xe2\x80\x94 00 , \'dz/dx =: o, . . . . (4) \n\n\n\nare, therefore, critical, and may be tested as indicated in \n\n\xc2\xa7 143- \n\nA maximum or minimum state of a function of two va- \nriables is illustrated by an ordinate of a surface which is \neither greater or less than all adjacent ordinates. The \n\n\n\nMA XIMUM A ND MINIM UM S TA TES. 239 \n\nconditions dz/dx = o and dz/dy \xe2\x80\x94 o indicate, in general, \nthat the corresponding tangent plane is parallel to XY. \n\n145. Lagrange^s Condition.\xe2\x80\x94 When the successive partial \ndifferential coefficients to include those of the n + 1 order \nare real and finite for a set of critical values, as {a, b), de- \nrived from equations (i), \xc2\xa7 144, a condition for a maximum \nor a minimum may be deduced from (^), \xc2\xa7 128. \n\nPlacing \n\nm =A (^ =B {^y\\ ^C \n\n\\dx^l(a,b) \' \\dxdy)^a,V) \' \\dy\'\'}(a,b) * \n\nwe have \n\n/{a \xc2\xb1h,b\xc2\xb1k) -f{a, b) = {Ah\' \xc2\xb1 2Bhk + C/^\'^)/2 + R. (i) \n\nIn general, as h and >^ vanish, the sign of the second mem- \nber will ultimately depend upon that of A?^ \xc2\xb1 2Bhk -\\- Ck^ ^ \nand when A ^ o \\\\. may be written \n\n\\i^Ah \xc2\xb1 Bky + {AC - B\')k\'\']/A. \n\nA maximum or a. minimum state will then require that the \nsign of this expression shall ultimately become fixed and \nremain so, while h and k change their signs by passing \nthrough zero. \n\nIf {AC \xe2\x80\x94 B^) < o, the numerator of the above expression \nwill be positive when k = o, and it will be negative for \nvalues of /i and ^ other than zero which make A/i \xc2\xb1 Bk =0. \nHence, a condition for a maximum or a minimum state is \n\n{AC- B\')> o, or AC> B\'; \nthat is, p^ Xp:] >(-f^)l ...(.) \n\n\n\n240 DIFFERENTIAL CALCULUS \n\nThis condition being satisfied, /(^, b) is a maximum or a \nminimum according as A and C are both negative or both \npositive. \n\nIf AC <. B^y there is neither a maximum nor a minimum. \n\nIf ^i = o and B :^ o, we have the same result, for the \nsign of 2Bhk + Ck^^ in the second member of eq (i), varies \nfor a fixed value oi k d.-s, h passes through the value \n- Cki\'2B. \n\nli A = B =^ o, then \n\nAC - B\' =^ o, and A/i\' \xc2\xb1 2BM + Ck\' = Ck\\ \n\nwhich vanishes when k \xe2\x96\xa0= o, for all values of h leaving \n\nthe question in doubt. The same result follows when \n\nA ^ B =: C ^ o. \n\nEXAMPLES. \n\nFind sets of values of the variables which correspond to \nmaxima or minima of the following functions : \n\n1. z = x^ -\\- xy -\\- y^ -\\- a^/x -\\- a^/y. \n\n\'dz/dx = 2x -]- y \xe2\x80\x94 a^/x"^ = o, \'dz/dy = 2y -\\- x \xe2\x80\x94 a^/y^ = O. \nFrom which we obtain x\xe2\x80\x94 y \xe2\x80\x94 aj 1/3, \n8V^-^\' = 2 + 2aVx^ a^V^r\' = 2 + 2aVy, \'d^z/dxdy = I. \n\nIn which substituting values of x and y, condition (2) is satisfied and \n\xc2\xab is a minimum. \n\n2.2 = cos X cos a -{- sin X sin a cos (y \xe2\x80\x94 f3). \n\n^z/dx = \xe2\x80\x94 sin X cos a -\\- cos x sin a cos (y \xe2\x80\x94 /3) = o, \nC)z/dy = \xe2\x80\x94 sin a sin x sin (jj/ \xe2\x80\x94 ^) = o, \ngive X = a, y \xe2\x80\x94 ft- \n\nf9!iU-,, (9!i) = _ sin\'., f^)=o. \n\nV O\'^^ / (a, ^) \\ ^\'^ / ia, /3) V^ ^1\'/ (a, ^) \n\n\n\nMAXIMUM AND MINIMUM STATES, 24I \n\nHence 3 is a maximum. \n\n\n\nX = a/2. \n\n\n\n\\y \n\n\n= c^/3, \n\n\nIX. \n\n\n\n\nX \n\n\n= y = a, : \n\n\nmin. \n\n\n\n\nX \n\n\n= y = o, \n\n\nmax. \n\n\n\n\nX \n\n\n= y= \xc2\xb1 \n\n\ni, min. \n\n\n\n\nX \n\n\n= \xc2\xb1 W3 \n\n\n>y=\'f \n\n\nil/3, min, \n\n\nX \n\ny \n\n\n= 2\', "^^^\xe2\x80\xa2 \n\n\n\n\n\n\nX \n\n\n= \xc2\xb1 V2, \n\n\n\n\n\n\ny \n\n\n= TV2, \n\n\nmin. \n\n\n\n\nX \n\n\n= 0=y, \n\n\nmax. \n\n\n\n\nX \n\ny \n\n\n~ i\' max, \n\n\n\n\n\n\nX \n\n\n= y = 0, \n\n\nmin. \n\n\n\n\n\n3. xy^{a \xe2\x80\x94 X \xe2\x80\x94 y). \n\n4. x^ -\\- y^ \xe2\x80\x94 2>axy. \n\n5 . x^ -\\- yi \xe2\x80\x94 x^ -\\- xy ^ y^. \n\n\n\n6. jf^y(6 \xe2\x80\x94 X ^y\\ \\ \n\n7. X* -\\- y^ \xe2\x80\x94 2x^ -j- 4xy \xe2\x80\x94 2y^. \xe2\x96\xa0< \n\n8. {2ax - x"") {2by - y""). ] \n\n9. g-\'^^-y\\ax\'^ + by""). \n\nX =\xe2\x96\xa0 o, y ^=- \xc2\xb1 \\, a <. b, max. \nX =\xe2\x96\xa0 \xc2\xb1 I, J = o, ^7 > b, max. \n\n10. sin X -\\-s\\ny -\\- cos {x -\\- y)- x = y = 37r/2, min, \n\nX = y = 7t/6, max. \n\n11. Divide a number a into three parts, such that the ni^^ power of \nihe first, by the \xc2\xab"* power of the second, by the r^^ power of the third \nshall be a maximum. \n\nAns. ma, na, and ra, each divided by (w + \xc2\xab + \'\')\xe2\x80\xa2 \n\n12. Find the minimum distance from a given point to a given plane. \n\n13. The volume of a rectangular parallelopipedon being given, find \nits edges when the surface is a minimum. Each edge = l^vol. \n\n14. An open tank to contain a given volume of water is to be con- \nstructed in the form of a rectangular parallelopipedon. Determine its \nedges so that the surface to be limed shall be a minimum. \n\nEach edge of base = i/2 vol. ; altitude = ^^2 vol. /2. \n\n15. Determine the maximum rectangular parallelopipedon which \ncan be inscribed in a given sphere. Each edge = 2i?/ 1/3. \n\n16. Of all isoperimetrical plane triangles which has the maxi- \nmum area ? \n\n\n\n242 DIFFERENIIAL CALCULUS, \n\n146. Functions of Three Variables.\xe2\x80\x94 Let 2/ =/(^,j, s). \nReasoning as in the preceding cases, it may be shown that \nsets of roots of the equations \n\n\'dtc/dx = o, \'du/dy \xe2\x96\xa0= o, \'du/dz = 0, . . (i) \n\nand du/dx = 00, du/dy = 00 , du/dz = 00, . . (2) \n\nare critical. \n\nDenoting a set of critical values from (i) by a, d, and r, \nwe have, {d), \xc2\xa7 128, \n\n/{a\xc2\xb1/i,d\xc2\xb1J^,c\xc2\xb1 I) -/{a, b, c) = [A A" + Bk\' + CP]/2 \n\n\xc2\xb1Dhk\xc2\xb1Ehl\xc2\xb1Fkl-\\-R, \n\nin which A^ B^ C, Z>, E^ F, represent the values of \n\na^ 3^ 8^ _a^ _a^ ^ \n\ndx\'\'\' dy" dz\'\' dxdy\' dxdz\' dy dz\' \n\nrespectively, when x =^ a, y =^ b, z ^^ c. \n\nIn order that/(^, ^, c) may be a maximum or a minimum, \n\n^{AH\' + ^/^^ + O") \xc2\xb1 Dhk \xc2\xb1 Ehl \xc2\xb1 Fhl, \n\nif not zero, should be either always negative or always posi- \ntive, as h^ k, and / vary through zero between certain positive \nand negative limits. \n\n147. Functions of n Variables. \xe2\x80\x94 By extending the above \nmethod of reasoning, it maybe shown that the sets of roots \nof the equations formed by placing the partial derivatives \nof the first order separately equal to zero are critical. \nEach set of critical values when substituted in the corre- \nsponding expansion should render the quadratic function of \nh^ k, /, etc., always negative for a maximum, or always posi- \ntive for a minimum, as h^ k^ /, etc., vary through zero \nbetween certain limits. \n\n\n\nPART III. \n\nGEOMETRIC APPLICATIONS- \n\n\n\nCHAPTER XIL \n\nTANGENTS AND NORMALS. \nRECTANGULAR COORDINATES. \n\n148. Equations of a Tangent and NormaL\xe2\x80\x94 The equa- \ntion of a straight line passing through (jc\', y\') on the curve \ny =^ fx is (Anal. Geom.) y \xe2\x80\x94 y\' = 7n{x \xe2\x80\x94 x\'). \n\nPlacing ifi \xe2\x80\x94f\'x\' \xe2\x80\x94 dy / dx\\ we have for the tangent line \nat(y,/)(\xc2\xa77i) \n\ny-y\'^-Wldx\'){x-x\') (i) \n\nand for the corresponding normal \n\ny -/ = - {dxldy\'){x - x\'). ... (2) \n\nThus, having y = ^x^ then f\'x = 9/27, /\'4 = 3/4. \nHence, ^^ \xe2\x80\x94 6 = (3/4) (x \xe2\x80\x94 4) is the tangent, and \n\ny \xe2\x80\x94 6= \xe2\x80\x94 (4/3)(j\\: \xe2\x80\x94 4) is the normal at (4., 6). \n\nIf the equation of a line is in the form u \xe2\x80\x94 <p[x^y) = o, \nwe have (\xc2\xa7 in) \n\n/\'x\' = dy\'/dx\' = - {du/dx\')/{du/dy). \n\n243 \n\n\n\n244 DIFFERENTIAL CALCULUS. \n\nSubstituting in above, we liave \n{bu/dy\'){y \xe2\x80\x94 y) = \xe2\x80\x94 i(du/dx\'){x \xe2\x80\x94 x\') for the tangent, \n{^u/dx\')[y \xe2\x80\x94 /) =: {^^n,/dy\'){x \xe2\x80\x94 x) for the normal. \nThus, having u ^= y"^ \xe2\x80\x94 gx = o, \n\n(du/dy) = 2j, {du/dx) = \xe2\x80\x94 9, \nand at the point (4, 6) \n\n(du/dy\'\') = 12, (du/dx^) = \xe2\x80\x94 9. \nTherefore, we have \n\ni2(y \xe2\x80\x94 6) = g{x \xe2\x80\x94 4) for the tangent, \nand -- 9(^ \xe2\x80\x94 6) =12(^^ \xe2\x80\x94 4) for the normal. \n\nEXAMPLES. \n\nFind the equations of the tangent and normal at the point (x\', y) \non each of the following curves: \n\njj/ \xe2\x80\x94 J)/\' = \xe2\x80\x94 (Px\'/a^y\'){x \xe2\x80\x94 x\'), \n\ny \xe2\x80\x94 y ^ {a^y /b\'^x){x \xe2\x80\x94 x). \n\n\n\nI. ^y + b\'^x\'^ = a^\\ \n\n\n\n3. ^2+y = i?2. \n\n\n\n5. y = a\\og sec (x/a). \n\n\n\n1 \n\n\\ y -y \xe2\x80\x94 -{y\' h \n\n\\ \n\n\n\ny-y = -{y \' lP){x - x\'). \n\ny\'y -f- x\'x = R"^, \n\ny = W /x\')X\' \n\n4. y"^ = 2^*?^ \xe2\x80\x94 ;\xc2\xabr\xc2\xab. y \xe2\x80\x94 y z=: [R \xe2\x80\x94 x\'){x \xe2\x80\x94 x\')/y . \ny \xe2\x80\x94 y =z \xe2\x80\x94 cot {x /a){x \xe2\x80\x94 x\'), \ny \xe2\x80\x94y\' =z tan (xycr)(jr \xe2\x80\x94 x\'). \n\n6. ;rV3 +J1/2/3 = ^2/3. J r= - ^ ^jV^ + f ^"^V \xe2\x80\xa2 \n\n{V = Qx/2 \xe2\x80\x94 \'^/2, \n>\' = - 2^9 + 29/9. \n\n5. X == r vers- ^ \\/2ry\xe2\x80\x94y^. y\xe2\x80\x94y\' = \xe2\x80\x94 (jf \xe2\x80\x94 y). \n\n\n\n1 \n\n\n\nTANGENTS AND NORMALS. 245 \n\n9. y^ = -^^. y-y=\xc2\xb1 ^^(3^ - ^\'\\ ^ - ^\'), \n\n^/ 2a- X \xe2\x96\xa0" \xe2\x96\xa0" (2a - y)3/2 \n\n\n\n10. tan -1- = k log |/x2 -j- y"^. y \xe2\x80\x94 y\' = \xe2\x80\x94, f^,(x \xe2\x80\x94 x\'). \n\n\n\n\\\\. y \xe2\x80\x94 ae=^l<=. y \xe2\x80\x94 y \xe2\x80\x94 [y\' /c){x \xe2\x80\x94 x\'). \n\n12. a;f^ + 2bxy -\\- cy\'^ + 2i/x -|- 2^?)/ -}- /= o. \n\nax\'-\\-h\' + ^ , \n\ny\xe2\x80\x94y= \xe2\x80\x94 T-r-\\ r-r-{x \xe2\x80\x94 x), \n\nbx Ar cy -\\- e \n\n13. ey \xe2\x80\x94 x. y \xe2\x80\x94 y \xe2\x80\x94ip^ \xe2\x80\x94 x\')[x\' . \n\n14. e"^ = sin X. y ~ y\' \xe2\x80\x94 cot x(x \xe2\x80\x94 x\'). \n\nIC). xy = a. y \xe2\x80\x94y\' = \xe2\x80\x94 (y /x\')(x \xe2\x80\x94 x\'). \n\n16. y = 2/^;^ + r^\'X^\' y ~y\' = \xe2\x80\x94 r-=-(x \xe2\x80\x94 ;f\').- \n\n\\2px\' -\\- r^x"^ \n\n8^3 ( ^-1-2;/ = 4a. \n\n4a^+j;^ \\ y - 2x = - \'ia. \n\nax\'\' \\y= \xc2\xb13 1/3-^/8 - a/8. \n\n\'^^ + \xc2\xab- ( ^ = T 8x/3 ^3 + 4i\xc2\xab/36. \n\nig. y{x\xe2\x80\x94i){x \xe2\x80\x94 2) = x \xe2\x80\x943. {x\' = 3\xc2\xb1V2) \n\ny= 1^2/(4 + 31/2), x = o/o. \n\n20. 3/ + j;2 = 5. (jc\' = l) jJ\' = T .29X \xc2\xb1 1.44. \n\n2 ^ (J^ = [3(^ + ^0 -5]//. \n\n21. jj/ == 6jr \xe2\x80\x94 5. -j \n\n(jj/= -y\'ix-x\')/3-\\-y. \ny=\xc2\xb1 (^+i)/2. \n\n\n\n22. y\'^ = 2x\'^ \xe2\x80\x94 x^. (x\' = I \n\n\' (y = T 2x \xc2\xb1 3. \n\n23. jj/ = (W^ + e-^/^y/2. V \xe2\x80\x94 / = {(="\'!\' \xe2\x80\x94 <f- V^)(jc \xe2\x80\x94 jtr\')/2. \n\n24. X = ^ log [(a +4/;^^:r/)/y] -4/^^ -ji\'2. \n\ny-y = - W/{^\' -y0V2](^ - ^\'). \n\n25. Find the angle at which x"^ =jj/^ -[- 5 intersects 8-r^ -{- i8jj/^ = I44\xe2\x80\x9e \n\nAns. 7C/2. \n\n26. Find the equation of the tangent to the curve x\'^{x ~[- y) = \na\'^{x \xe2\x80\x94 y) at the origin. Ans. y = x. \n\n\n\n246 DIFFERENTIAL CALCULUS. \n\n27. Find the angle of intersection made by the two curves \n\nx^ -\\- y^ \xe2\x80\x94 {laf, and j\'^ \xe2\x80\x94 x^ \xe2\x80\x94 \xe2\x80\x94 a^ . Ans. tan-i|/i5. \n\n28. Find the angle at which the curve {y"^ -|- x"^)"^ = 2a\'^{x^ \xe2\x80\x94 y^) \ncuts X. Ans. 7t/2. \n\n29. Find that point on an ellipse at which the angle between the \nnormal and diameter is a maximum. Ans. {a/\\/2, b/ ^2). \n\n30. Find the point on a parabola where the angle between a straight \nline to the vertex and the curve is a maximum. Ans. x = p. \n\nTo find the equation of the tangent to y =/(^) which makes \na given angle, 0, with X: \n\nLet a \xe2\x80\x94 tan 0; then dy\' /dx\' = a, which combined with \ny = /(x) w^ill determine {x\\ y\') for the point of tangency. \n\nThus, to find the equation of the tangent to y = 6x \nwhich makes an angle of 45\xc2\xb0 with X: \n\ndy\'/dx\' = s/y = I ; .\'\xe2\x80\xa2 7\' = 3- \nSubstituting in^ = 6x, we have x\' = 3/2, and the required \nequation is j \xe2\x80\x94 3 = x \xe2\x80\x94 3/2. \n\nEXAMPLES. \n\n1. Find the equation of the tangent to y = a -\\- (c \xe2\x80\x94 xy which is \nparallel to X. Ans. y = a, x = 0/0. \n\n2. Find the equation of the tangent to x^ -\\-y^ = f^ which is \nparallel to y =: 2x -{- 7. Ans. y \xc2\xb1 r/4/5 = 2{x T 2^/1/5). \n\n3. Find the equation of the normal to y\'^/4 + ^Vq \xe2\x80\x94 ^ which is \nparallel to 2x \xe2\x80\x94 y = 3. Ans. y = 2x T 2. \n\n4. Find the equation of a tangent to iSy"^ -\\- Sx"^ = 72 which is \nperpendicular to the right line pafsinof through the positive ends of \nthe axes. Ans. 2y = 2^ ^ 1^97- \n\n5. Fmd the equations of the tangents to gy"^ \xe2\x80\x94 2\'^x\'^ = \xe2\x80\x94 225 \nwhich makes an angle of 60\xc2\xb0 with the transverse a.^is. \n\n6. Find the points on y = x^ \xe2\x80\x94 2^^ \xe2\x80\x94 ^^x -\\- 85 where the tan- \ngents are parallel to X. Ans. (4, 5) and (\xe2\x80\x94 2, 113). \n\n7. Find the equation of the perpendicular through the focus of a \nparabola to a tangent at {x\' , y\'). Ans. y = \xe2\x80\x94 y\'{x \xe2\x80\x94 //2)//. \n\n\n\nTANGENTS AND NORMALS. \n\n\n\n247 \n\n\n\n8. y^ = 4ax + 4^^i find the locus of the points of intersection of \ntangents, and perpendiculars upon them from the focus. \n\nAns. X = \xe2\x80\x94 a; y = 0/0. \ng. y \xe2\x80\x94 2 = (x \xe2\x80\x94 i)^x \xe2\x80\x94 2; find the angle at which the curve \ncuts X, and the point where the tangent is perpendicular to X. \n\nAns. Tan-i 2; (2, 2). \n10. Show that normal to y^ = 4ax is tangent to y\'^= 4{x \xe2\x80\x94 2afJ{2\']a), \n\nWith oblique axes, inclined at an angle y5, dy\' /dx\' repre- \nsents the ratio of the sines of the angles which the tangent \nand curve at {x\\ y\') make with the axes. The equation of \nthe tangent remains unchanged in form, while that of the \nnormal becomes \n\n\n\ny \xe2\x80\x94y =. \n\n\n\n(,+,os/5|;)(.-y)]/[cos/j+g]; \n\n149. Let TM, PM, and NM, respectively, be the tan- \ngent, ordinate, and nor- \nmal at any point M \nwhose coordinates are \nW.y\'Y \n\nbe perpendiculars from j \nthe origin to the tan- \ngent and normal respectively. \n\nLet XTM = (f). Then, since dy\' /dx\' \xe2\x80\x94 tan 0, we have \nSubtangent = PT = / cot =/dx\'/d/. \nSubnormal = PN \xe2\x80\x94 y\' tan =-y\'dy\' /dx\' . \n\nNormal = NM \xe2\x80\x94 y\' sec cfy \xe2\x80\x94 y\' \\^i -\\- tan\' cp \n\n\n\n\n=/ Vi + {d//dx\'y \n\nTangent = TM = / cosec = / Vi + cot\' \n\n\n\n= y Vi + {dx\'/dyy. \n\n\n\n* Todhunter\'s Calculus, page 2S3; C Smith\'s Conies, \xc2\xa7 42= \n\n\n\n248 DIFFERENTIAL CALCULUS, \n\nAlso, from figure, or equations (i) and (2) (\xc2\xa7 148), \nOT =^ Intercept of tangent on X = x\' \xe2\x80\x94 y\'dx^ Jdy\', \nOS = Intercept of tangent on Y = y\' \xe2\x80\x94 x\'dy\' /dx\\ \nON = Intercept of normal on X = ^\' \xe2\x80\xa2\\- y\' dy\' / dx\' , \nHence, \n\np \xe2\x80\x94 OQ = Perpendicular to tangent = OS cos \n\n_ ^n I _ y \xe2\x80\x94 x\'dy\' /dx\' __ y\' dx\' \xe2\x80\x94 x\' dv[ \n\n~ V 1 4-tan=^ (f)~ Vi^ {d//d^~ Vdx" -h dy^\'\' \nq = OR = Perpendicular to normal = ON cos \n_ x\' -\\-y\'dy\'/dx\' __ x\'dx\' ^y\'dy\' \n~ \\\\^{dy\'/dx\'y ~ Vdx\'-\' + ^/^* \nTo apply these formulas, obtain general expression for \ndy\' Jdx\' from the equation of the curve, and substitute \nvalues of x* and f. \n\nEXAMPLES. \n\n1. //^^ + x\'Ja\' =1. / = aHyib\'x"\' + \xc2\xaby 2)1/2. \nSubt = - ayy{Px\') = {x"" - a})lx. Subn = - Px\'/d\'. \nIf a ^= b, we have for the circle r ^ a, p =: a, Subt = \xe2\x80\x94 y"\'/x\' , \n\nSubn = \xe2\x80\x94 x\' . \n\n2. y-\'/b"" - x\'^Jd\' = - I. / = - aUyib^x"\' + \xc2\xab*y\'2)V2. \nSubt = ay V(^*x\') = {x\'\'^ + a\'V^\'. Subn = b\'^x\'/d\'. \n\n\\ia =zb, we have for an equilateral hyperbola ;> = d^/i/x"^ +y^ \nSubt = y\'^/x\'y Subn = x\'. \nZ. y = 2px + r^x\'. \n\nSubt = {2px\' + rV2)/(/ 4- rV); Subn = / + ^-V, \n\n\n\nTa\xe2\x80\x9e = /... + .V=H-(^^fl\xc2\xb1Z.-)\'. \n\n\n\nNor = \\ipx\' + rV^ + (/ + r\'x\'Y \n\n\n\nIf r2 = o, we have, for the parabola referred to axis and tangent at \nvertex, \n\n\n\nSubt = 2x\', Subn = p, Tan = y\'^/p\'\' -^ y"^/p, \n\n\n\nTANGENTS AND NORMALS. \n\n\n\n249 \n\n\n\nNor = yy\'\'+/^ Perp. to tan=>/\'2/24/y\'^-|-/^ q=y\\x\'^-p)l ^^^p. \n\n4. xy = m. \n\nSubt = \xe2\x80\x94 y, \xe2\x96\xa0 Subn = \xe2\x80\x94 yyy= ~ y^/m, \n\nNor =y|/x\'2 -f yy^\', Tan = V/M^". \n\n5. j^/ = fz*. Subt = i/log a = i^a. \n5. X = r vtxs-\'^iy/r) \xe2\x80\x94 ^2ry \xe2\x80\x94 y , is the eq. of a cycloid.* \n\n\n\nSubn = ^ iry\' \xe2\x80\x94 y\'^ \n\n\n\nSubt = y-\'l^/iry\' - y\'\\ \n\nTan \xe2\x96\xa0= y ^ 2ry\' I ^ iry\' \xe2\x80\x94 y\'^, Noi = j/ary\' \nY \n\n\n\n\nP N A X \n\nSince the subnormal PN = MD, the normal at any point passes \nthrough the foot of the vertical diameter of the corresponding posi- \ntion of the generating circle, and the tangent passes through the \nother extremity. \n\nThe tangent and normal at any point of a cycloid are therefore \nreadily constructed when the corresponding position of the generating \ncircle is drawn. \n\nOtherwise, when the circle AB, upon the greatest ordinate as a \ndiameter, is drawn, through the given point i^draw MC parallel to \nthe base 0X\\ from C where it cuts the circle draw the chords CB \nand CA to the ends of its vertical diameter; through M draw the \ntangent ME parallel to CB, and the normal J/TV parallel to CA. \n\nTo construct a tangent parallel to any right line as RS, draw the \nrhord BC parallel to it, through C draw CM parallel to OX, and \nthrough M draw ME parallel to RS. \n\n7. y"^ = a\'^-\'^x. Subn \xe2\x80\x94y"^/nx\', Subt = nx\' . \n\n\'\xe2\x96\xa0"\' Prith described by a point on circum, of a circle rolling upon \n\na fixed rii/ht line. \n\n\n\n250 DIFFERENTIAL CALCULUS, \n\n8. f\' = ax" + x\\ \n\nIntercept of tan on y = \\x\' [{a + ^\')f \'^(^/s)- \n^9. ;>/\' = IX. y = 8, Tan = 4 |/i7. \n\n10. X = sec 2jf/./ = [{x^ - 1)1/2 sec-i^f - i]/[i + 4;t:2(;tr\xc2\xab - i)^^^]. \n\n11. y^ = x^/(2a \xe2\x80\x94 x). \n\nSubt = x\'{2a \xe2\x80\x94 x\')/{2a \xe2\x80\x94 x\'), Subn = ^(sa \xe2\x80\x94 x\')/{2a \xe2\x80\x94 x\')^o \n\nNor =/\'Vc-, Subn = c{e^=^\'/c) - e-^^\'/c)/^. \n\n\n\nTan = j"V l/j\'* \xe2\x80\x94 tf", Subt = cy\'/\\^y\'^ \xe2\x80\x94 c^. \n\n\n\n13. ^* +JJ/* = <r*. / = c^/^x\'^ + /\xc2\xab. \n\n14. ;c^ \xe2\x80\x94 "^axy -\\-y^ = o. \n\n\xc2\xa3q. of tan jj/ = \xe2\x80\x94 ;\xc2\xbb;(x\'2 \xe2\x80\x94 \xc2\xab;\'\')/(/\' \xe2\x80\x94 a^\') + (\xc2\xab^y)/(y\' - ax\'\\ \nSubt = (_j/\'3 \xe2\x80\x94 axy\')/{ay\' \xe2\x80\x94 x\'^). \n\nEq. of tan y = -x ^{y\'/x) + f ^. \n\nIntercept on X = |/aV, on F = f^a^^ . \nLength of tan between axes = a. \n\n\n\nArea between axes and tan = \\/a^x\'y\' /2. \n\np = \\ ax\'y\' . \n\\^ y - ce^f\'\'. Subt = a, Subn = f^/a, \n\n17, y = 2^^^ log x. Subn = \xc2\xabVj;\'. \n\n18. V = a2(a - ;c). Subt = \xe2\x80\x94 2{ax\' - x"^)/a. \nig. 3^!^ -|- a-\'\' = 2x^. Subn = x\'Y^. \n\nx\' = 4, Subt \n\nSubt = 2x\'/2y Subn = 2>(^x"^/2. \n\n\n\nTANGENTS AND NORMALS, 2$ I \n\nPOLAR COORDINATES. \n\n150. Polar Tangent, Subtangent, Normal, and Sub- \nnormal. \n\nLet PM =. r, corresponding to th\'fe point of tangency M \nof any tangent as TM. From the pole P draw PT per- \n\n\n\n\npendicular to PM. Draw PQ and MR perpendicular to \nTM, \n\nPT^ the part of the perpendicular to r, from P to its \nintersection with the corresponding tangent 7W, is the \npolar subtangent corresponding to M, PR is the polar \nsubnormal, and RM is the normal. \n\nLet = PMT, the angle made by r with TM, \n\nThen (\xc2\xa7 55, \xc2\xa7 70) \n\nFrom Trigonometry, \nsin = tan 0/ Vi -p tan^ 0, cos = Vi/(i + tan^ 0). \n\n\n\nTherefore, since tVr\' + rV<9^ = ^^, (\xc2\xa7 92,) \nsin = rdO/ds, cos = dr/ds, cot = dr/rdQ. \n\n\n\n252 DIFFERENTIAL CALCULUS, \n\nHence, \n\nPT=. Subt = r tan (p = r\'dO/dr, \n\n\n\nTM = Tan - ;Vi + {rdid/drY = rds/dr, \nPR = Subn = r cot = ^r/^6\', \n\n\n\nPM = Nor = l/r\' + {dr/ddf = ^V^iy, \n/\'(2 =/ = perpendicular to tangent \n\n= r sin = rdS/ds \n\n\n\nV(dr/dOy -f r" \n\nSince ^j/^9 is assumed to be positive, positive values only of/ are \nconsidered. \n\ni// = i/r\' + (dr/dey/r\\ \nPuttiT^g j/r = Uy from which dr"^ = r^dji^ we liave \n\ni// = ^^^ + (^2//^6\')\' (i) \n\nPN =^ q = perpendicular to normal = r cos \n\n\n\n= rdr/ds = =r = V/^ \xe2\x80\x94 /. \n\nj/i + {rdO/drf \n\n\n\nSince </^ =^ dr/{dr/ds) = rdr/r cos = rdr/\'q^ \nds/dr = r/q = r/ S/r" -/; /, ^j = rdr/ \\f7 \nBut / = rV^M, .*. ^i- = r\'dB/p. \n\nTherefore r\'dd \xe2\x80\x94 prdrl ^ r" \xe2\x80\x94 p\\ \n\n\n\nTANGENTS AND NORMALS. 253 \n\nEXAMPLES. \n\n1. r = a0; find the angle between r and the tangent, the sub- \ntangent, the subnormal, the normal, and/^. \n\n(p = tan-19, Subt = r\'^/a, Subn = a, \nNor =Ya2 + r\\ p - r""/ ^d" + t"". \n\n2, r = a{i -j- cos 0). \n\n\n\nSubt = \xe2\x80\x94 rV(a sin 0), Subn = \xe2\x80\x94 a sin 6, / = \\fry2a. \n\n3. r = a^. Subt = Mar, Subn = r/il/a, Tan (p = Ma. \nTan = r ^T+M^\\ Nor = r |/i + i/i^ / = i^a ^ l^i^a\' + i \xe2\x80\xa2 \n\n4. r\'^-a\'J^. Subt =2a 4/0^/=2aV/i/r* + 4a* =2a |/e/|/i + 46^. \n\n\n\n5. r = fl0i/2. Subt = 2rV^^ / = 2rVV\xc2\xab* -\\-\\r^ \n\n6. ;\' = 2(3! cos 0. \n\nSubt = \xe2\x80\x94 2a cot cos 0, Subn = - 2a sin 0, \n\nTan = 2a cot 0, Nor = 2a. \n\n*]. r \xe2\x80\x94 fl0-i. Subt = \xe2\x80\x94 a. Tan = \xe2\x80\x94 0, Subn = \xe2\x80\x94 ry^. \n\n8. r = a sin 0. = 0. \ng. ?\'\'* = a"^ cos 20. \n\n\n\n= It/ 2 + 20, Subt = \xe2\x80\x94 r^l^a} sin 20), Tan = rd^/ i/a* \xe2\x80\x94 r*, \n\nSubn = \xe2\x80\x94 a2 sin 20/r, Nor = a\'^/r, \n\np \xe2\x80\x94 r^/a"^, Perp. to Nor = r sin 20. \n\n10. r = \xc2\xab0/sin 0. Subt = rt0V(sin - cos 0) \n\n11. r = ab/{ae^ + be-^). Subt = - abjiae^ - be\'^). \n\n12. r \xe2\x80\x94 a{\\ \xe2\x80\x94 cos 0). \n\n(p = B/2. Subt - 2a sin^ (0/2) tan (0/2), / = 2a sin^ (0/2). \n\n13. r = ae^ cot a, /* = r sin or. \n\na(l - e^) a\\l - e^) a\'\'{l - (\'\'Y \n\nI \xe2\x80\x94 (f cos ^ 2a/r\xe2\x80\x94i I \xe2\x80\x94 2<f cos + <?\' \n\n\n\n254 \n\n\n\nD IFFEREN TlA L CA LCUL US. \n\n\n\n15. r = a -B- 1/2. \n\xc2\xab(^2 _ i) \n\n\n\n16. r = \n\n\n\np = ia\'\'r/\xc2\xbbi/r^ + ^a\\ \nb i/r \n\n\n\n1 -\\- e cos 0* \n\n\n\n151- \n\n\n\n\ny2a -\\- r \n\nPolar Equation of a Tangent.\xe2\x80\x94 Let PM = r\\ and \nXPM= e\\ be the coordi- \nnates of M, the point of \ntangency of any tangent, as \nNM, and let FN = r, and \nXFN = e, be the coordi- \nnates of any other point of \nNM,2.^N. l.ttPMN=c()\', whence (\xc2\xa7150) tan 0=rV^7^r\'. \nTriangle FMN gives \n\nr^ _ sin MJVF _ sin [{6\' - 6^) + 0] \n/- ~ sin FMN ~ sin \n\n= sin (^\' - ^) cot + cos (/9\' - 6) \n\n= sin (<y\' - 6) dr\'/r\'dd\' + cos {B\' - 6), . \n\nPutting i/r = u and i/r\' = u\\ whence \ndu\'/dS\' = - dr\'lr"dd\\ \n\nwe have, dividing both members of (i) by r\\ \n\nu = u\' cos (6^\' - 6*) - sin (^\' - 6\') ^z^\'/^^\' \n\nfor the tangent. \n\nOtherwise it may be obtained from eq. (i) (\xc2\xa7 148) by \nchanging the reference to a system of polar coordinates. \n\nLet r\' and 0\' be the polar coordinates of the point of \ntangency {x\',y\'), and r and 6 those of all other points of \nthe tangent. Taking the pole at the origin and the refer- \nence line to coincide with the axis of X^ we have, as in \nex. II, \xc2\xa7 115, \n\n\n\n(0 \n\n\n\n(2) \n\n\n\nTANGENTS AND NOEMALS. 2^% \n\nX = r cos d, x\' = r\' cos 6\\ \n\n^ = r sin 6^, y\' \n\nd\xc2\xa3_ _ sin B\'dr\' + r\' cos B\' dB\' \ndx* ~ cos e\'dr\' - r\' sin ^V^\'\' \n\nSubstituting in (i) (\xc2\xa7 148), we may write \n\n. . , . ^, sm 6\' dr\'/de\'-\\-r\' cos 0\\ ^ , ^,. \n\nr sm d\xe2\x80\x94r\' sin 6"= ^, , , , ,^, ^-T--27>(r cos 6/\xe2\x80\x94^ cos t) ), \n\ncos o dr /dd\' \xe2\x80\x94 r sm c/\' \n\nWhence \n\nr^, sin cos 9\' - r\'^, sin 0\' cos 0\' - rr\' sin sin 0\' + r\'^ sin^ 0\' \ndo ao \n\n= r^, sin 0\' cos + r/ cos cos 0\' - r\'^ sin 0\' cos 0\' - r\'^ cos^ 0\', \nor \n\n\xe2\x80\x94\xe2\x80\x94(sin cos 0\' \xe2\x80\x94 sin 0\' cos 0) \xe2\x80\x94 r/(sin sin 0\' + cos cos 0\') = \xe2\x80\x94 r\'\', \nao \n\nor, changing signs of terms, \n\nrdr\' \n\n\xe2\x80\x94 ^ (sin 0\' cos - sin cos 0\') + rr\'(cos 0\' cos + sin 0\' sin 9) = /\xc2\xab, \n\nor sin {B\' -B)dr\' /rUB\' + cos (^\' - ^) = ^\'A- . (i) \n\nPutting i/r = ^/, and i/r\' = u\\ whence, \n\ndu\'/dB\' = - dr\'/{r"dB\'), \n\nwe have, dividing both members of (i) by r\', \n\nu = u\' cos (^\' -B)- sin (^\' - B)du\'ldB\\ \n\nRepresenting the polar coordinates of all points of a nor- \nmal except the point of tangency (x\\ y\') by r and B, we \ndeduce in a similar manner from (2) (\xc2\xa7 148) \n\nu^u\' cos (^ - 6") - 2^\'^ sin (^ - B\')dB\'/du\' \n\nfor the normal. \n\n\n\n256 \n\n\n\nD IFFEKEN TIA L CALCUL US. \n\n\n\nCHAPTER XIII. \n\nASYMPTOTES. \nRECTANGULAR COORDINATES. \n\n152. An Asymptote to a curve is a definite limiting posi- \ntion of a tangent to the curve, under the law that the point \nof tangency recedes from the origin without limit. \n\nUnless otherwise mentioned rectilinear asymptotes only \nwill be considered. \n\nAsymptotes Parallel to the Coordinate Axes. \n153* If in the equation of a curve >/:^->oo as x\'m-^a, then \n(dy/dx)a =^ (\xc2\xa7 71), and x^a is the equation of an asymp- \ntote parallel to Y. If, otherwise, ym^a as ^b-^oo, then \n^ = <z is an asymptote parallel to X, \n\nEXAMPLES. \n\n\n\n\n\\. y^ \xe2\x80\x94 x^/{2a \xe2\x80\x94 x). \nym-^ \xc2\xb1 00 as x\'!i\xc2\xbb-^2a. \nHer4ce x = 2a is an asymptote. \n\n\n\nASYMPTOTE"^ 257 \n\n2. <fl/* =_j/\xe2\x80\x94 1. ;j/B->I as ^^-^\xc2\xb100. \n\nHence ;j/= i is an asymptote to both branches. \n\n_)/:^->Go as \xe2\x80\x94 x\'^-^Q. \nHence F is an asymptote to the left-hand branch. \n\n\n\n\n3. jjr = a*. ym-^o as xb->\xe2\x80\x94 00 \n\nHence, Xis an asymptote. \n\n\n\n4. ^ = \n\n\n\n23^.;*; + Pc \n\n\n\n\n\n\n\xc2\xab7x2 - 1 \xe2\x80\xa2 \n\njI\'B-\xc2\xbboo as xB-^lrt\', and ji\'S^-\xc2\xbbo as jcB-^oo, \nHence, X and ;r = \xc2\xb1 a are asymptotes. \n\nCURVES. ASYMPTOTES. \n\n5. j/2 == {x^ -|- ax^)/{x \xe2\x80\x94a). X = a. \n\n6. xj/ \xe2\x80\x94 ay \xe2\x80\x94 bx \xe2\x96\xa0=\xe2\x96\xa0 o. x \xe2\x96\xa0= a, y = b. \n\n7. \xc2\xabV ~ ^"\'y ~ ^^^- X = \xc2\xb1 a, y = o. \n8 y = x/{iJr^\'). y=^o. \n\n9. X = a log [(rt; + Va^ -f)/y] - V^^^- V = O. \n\n10. y\xe2\x80\x94 a^x/{x \xe2\x80\x94 of. X \xe2\x80\x94 a, 7 = 0. \n\n11. ;/ = log X. X = O. \n\n12. x^ -f- a^^j* = axy. x = b. \n^\'x. y \xe2\x96\xa0= a\'^x !{a^ -\\~ x"^^. y = o. \n14. xy- = 4rt;-(2a \xe2\x80\x94 x\'^). x = o. \n\n\n\n258 \n\n\n\nD IFFEREN TIA L CA L CUL US. \n\n\n\n15. \n16. \n\n17. \n\n18. \n19. \n20. \n21. \n22. \n\n23- \n24. \n\n\n\nx^y -\\- a^y = a^ \n\n\n\n25. y \n\n\n\nx=\xc2\xb1{y + b)\\/a\'\'-yyy. \ny = a^/{a^ \xe2\x80\x94 x"^). \n\ny(a^ -\\- x\'^) = a^(a \xe2\x80\x94 x), \ny = e-^. \n\ny = a-\\- b^/{x - c)K \na?y -(- x^\'^y = a^x. \nx^ \xe2\x80\x94 3^ = 6xy. \ny(a^ - x"") = b{2x + c). \n2 _ a^jx \xe2\x80\x94 d){x \xe2\x80\x94 3<^) \n\n\n\nx"^ \xe2\x80\x94 2a;^r \n\n\n\n^ = 0. \n\njc = \xc2\xb1 \xc2\xab, ^ =0. \n\n>/= \xc2\xb1 \xc2\xab. \n\nJ>\' = 0. \n\nj(/ = o. \n\ny \xe2\x80\xa2= a, X r=zc, \n\nJ\' = o. \n\n:r = o. \n\n_j/ = o, X \xe2\x80\x94 \xc2\xb1a. \n\n^ = 20!, JP = o, ^ = J, a. \n\n\n\nASYMPTOTES OBLIQUE TO THE COORDINATE AXES. \n\n154. Let JJ/be a tangent and SP an asymptote to any \nplane curve, as NM. \n\n\n\n\nThe equation of TM\'x?, (g 148) \n\nand its intercepts are \n\nOR^x\' - y\\dx\'ldy\'), OT = y\' - x\'{dy\'/dx\'). \n\n\n\nASYMPTOTES. 259 \n\nAs the point of tangency M recedes from the origin \nwithout limit, the tangent TM approaches the asymptote \nSP, and the intercepts OR and OT approach OS and OP^ \nrespectively, as limits. \n\nAssuming that in the equation of the curve y :^-^ co as \nx\' ^-> 00 , and placing \n\ntan OSP = K, OP = Y,, and OS=X,, \n\nwe have, omitting the dashes, \n\nlimit p^-j limit p ^-| \n\nlimit \n\n\n\n^ ^""\'^ r dx \n\n. .^^^ ^ L ay _ \n\n\n\ny m-^ CO \n\nany two of which will serve to determine in regard to an \nasymptote. \n\nIf ^ = o or 00 , or if F\xe2\x80\x9e = Xo = 00 , or if either Y^ or \nX^ is imaginary, there is no corresponding real asymptote \nwithin a finite distance from the origin ; otherwise there is \nand its equation is \n\ny = Kx-\\- F\xe2\x80\x9e, or x=y/K-^X,, \n\nor x/X,-^y/Y, = i. \n\nIf either Y^ or X^ is zero, the corresponding asymptote \npasses through the origin and its equation is \n\ny = Kx. \n\nEXAMPLES. \n\nI. y = 2J}X. \n\nX = CO = y, dy/dx = p/y. K \xe2\x96\xa0=\xe2\x96\xa0 O. \n\n\n\n26o \n\n\n\nD IFFEREN TIA L CALCUL US. \n\n\n\nHence, a parabola has no asymptote. \n\n\n\nX\\ \n\n\n\n\xc2\xb1 <x> , jj/ = q: CO . dy/dx = {4a X \xe2\x80\x94 Sx^)/^)\'^. \n\n\n\n2{2ax\'^ \n\n\n\n4ax \xe2\x80\x94 3^ \n\nHence, y = \xe2\x80\x94 x -{- 20/2 is an asymptote. \n2. y^ = 10 \xe2\x80\x94 x^. \n\nX = \xc2\xb1 CO , y = Tco. dy/dx = \xe2\x80\x94 x^/yi, \n\nx. = [.+yA\']^=[.oA\']^=o. \n\nA-= [- ^vy]\xe2\x80\x9e= - [do -y)/y]r= -[\'\xc2\xb0^\' - \xe2\x96\xa0]r= - \'\xe2\x80\xa2 \n\nHence, jj/ = \xe2\x80\x94 ^ is an asymptote. \n\njr = 00 = JJ/. dy/dx = (^ -|- rs^ir)/ \\2px -f rV2. \n\n\n\nJTo = \n\n\n\n\n\n\n/A + \n\n\n\n-1 ==^- \n\n\n\nWhen r\'^ < o, Fo is imaginary. Hence, an ellipse has no asymp- \ntote. \n\nWhen r^ = o, both results are unlimited, showing that a parabola \nhas no asymptote. \n\nWhen r"^ > o, both results are finite, and since Fo has two values. \nan hyperbola has two asymptotes \n\nPutting/ = b\'^/a and r^ = b\'^/d^, we have \n\nJ^o = - \xc2\xab. Fo = \xc2\xb1 b. \n\n\n\nX, \n\n\n\n_ 3V- -] \n\n2^/jr -)- 3-^\'M ^ \n\n\n\n\'za -f- 3x \n\n\n\ndy/dx \xe2\x80\x94 {2ax -\\- 3x2)/3_j/2. \n\n>lx 4- 3) I ~ ~ T \n\n\n\n(2,. A + 3) \n\n\n\nASYMPTOTES. \n\n2ax^ + 3JC^" \n\n\n\n261 \n\n\n\n\n\n\nx\'-ff\' \n\n\n\nHence, v = ^ + \xc2\xab/3 is an asymptote \n\n6. ^3 _ ^ = ;,o _ 4;,. ^jf/^^ = (3^^ - 4)7(3/ - I). \n\nPut X = 0^, giving / - >\' = ^y \xe2\x80\x94 4<y. \n\nHence, ^ = \xc2\xb1 ^{M^^mf^^^ and / = i give;.= \xc2\xb1 \xc2\xab. = ^. \n\n\n\nXo = [--7(|^)]^^- \n\n\n\n3^^ \n\n\n\n[-3/(3^-4A)]\xc2\xb1- =\xc2\xb0- \n\n\n\nir = \n\n\n\n3^- \n\n\n\nV \n\n\n\n- I J\xc2\xb1co L3 - I A\' J \xc2\xb100 \n\n\n\nHence, j>/ = ;c is an asymptote. \n\n\n\n7. ^\' \n\n\n\n3^;,^ +/ = o. ^^A-^ = (\xc2\xab^ - ^^\'VC/ - ^^)- \n\n\n\nPut .^f = ^J, divide by /, and we find y = 3^//(i + ^^), in which \n^ _ _ I gives ^ = CO = \xe2\x80\x94 X. \n\n\n\nXn= X \n\n\n\n\n\n\nr y - axyr\\ _ r - ^^n = r_z.^ \n\n_ r - ^ "1 ^ r _ j^rf^i^n ^ K \n\n- - Li- alx \\~ L / - ^^ Jco \n\n\n\nHence, jj/ = \xe2\x80\x94 ^ \xe2\x80\x94 a is an asymptote. \n\nASYMPTOTES. \n\n_j/ = \xc2\xb1 bxja. \ny \xe2\x80\x94 \xe2\x80\x94 X -{- a/3. \n\n\n\nCURVES. \n\n\n\n9. y = x2(a \xe2\x80\x94 x). \n\n10. y = ax + ^\xe2\x80\xa2^t^\'\'*\' \n\n11. / = xy{x - I). \n\n12. y^ -{- x^ = 3-^\'- \n\n13. y = ;cV(x2 + 3\xc2\xab\'). \n\n14. y = a^ \xe2\x80\x94 x3. \n\n\n\n_j/ =^ \\/bx + a/2^b. \nX \xe2\x96\xa0= 1. \n\ny = \xe2\x80\x94 X -\\- 1. \ny = X. \nV = - X. \n\n\n\n262 DIFFERENTIAL CALCULUS. \n\n15. y^ = a^x \xe2\x80\x94 x^. y \xe2\x96\xa0=. \xe2\x80\x94 x. \n\n16. y\'^ \xe2\x80\x94 6x^ -{- x^. y = X -\\- 2. \n\n17. xy = x^ -\\- X -{-y. X = \xe2\x80\x94 I, \xc2\xb1y = x. \n\n18. Find the perpendicular distance from the focus of an hyperbola \nto an asymptote. Ans. Semi-conjugate axis. \n\nig. Find a tangent to a given curve which forms with the coordi- \nnate axes a maximum or a minimum triangle. \nLet ul2 = area of triangle. Then \n\ndx\'\\l , dy\'\\ I , dv\' y dx\' \n\n\n\n\'TyK ~"^;="r "" dx\' ] df \n\n\n\n-{^\xe2\x96\xa0-^fm^^)zim \n\n\n\ndu \ndx \n\ndy , , ,dy \n\n- ,.) - = O and y \xe2\x80\x94 x \xe2\x80\x94- \n\ndx \'\xe2\x96\xa0 dx \n\n\n\ny du . ,dy , ,dy , \n\ngeneral, - \xe2\x80\x94 = o requires x -\xe2\x80\x947- -\\- y = o, whence y \xe2\x80\x94 x ^, \xe2\x80\x94 2y . \ndx dx dx \n\nTherefore, in general, when the triangle is a maximum or a min- \nimum, the portion of the tangent between the axes is bisected at the \npoint of tangency. \n\nLet the equation of the curve be x^ -{- y^ \xe2\x96\xa0= R^. \n\nThen C = :i^, and -\'\'*\'--\'\'\' \n\n\n\ndx y dx\' y\' \n\nHence, \xe2\x80\x94 -j\xe2\x80\x94 +/ = o gives / = y, \n\nand u/2 = A\'^ is a minimum. \n\n155. The equation of any algebraic curve of the nih. \ndec^ree may be arranged in sets of homogeneous terms and \nv^ritten in the form \n\nxVXy/^) + ^\'^-y.(jA) 4- x--\'f,{y/x) + . . . = o. (i) \n\nThus, having \n\ny - x\'y + 2/ + 4_)^ + ^ = o, \n\n\n\nASYMPTOTES. 263 \n\nthen (/ - x\'y) + 2/ + (^y + ^) = o, \n\nand x\\flx\' - y/x) + x\\2flx\') + x^^y/x + i) - o. \n\nLet y \xe2\x80\x94 mx -\\- c\\>Q the equation of a right line ; combine \nit with (i) by substituting mx + c for j, giving \n\n-f :^\xc2\xab-y>^ + ^A) + ... = o, . (2) \n\nthe n roots of which are the abscissas of the n points com- \nmon to the two lines. \n\nBy causing in and c to vary, the right line may be made \nto have any position in the plane XY. \n\nDeveloping each term of (2) by Taylor\'s formula, we \nhave \nx^fXm) + x^-\\cf:(in) +/,(;;0] \n\n+ ^\'^-f 7/o"W + cf:{jn) +/,(/;0] + etc. = o. (3) \n\nAny set of values of m and c which satisfy the two equa- \ntions \n\nfXni) = o, .... (4) </;w +/,(\xc2\xab) = o, (5) \n\ncause the two terms in (3) containing the highest powers of \nX to disappear, and the resulting equation to have two in- \nfinite roots. That is, two of the points common to the \nright line and curve are thus made to coincide at an infinite \ndistance from the origin, and the corresponding right line \nconsequently is an asymptote. \n\nLet 7n^^ m^^ ... nin be the n roots of (4); the cor- \nresponding values of c from (5) will be ~/i(^i)//o\'(^i), \netc., and \n\ny =z m,x -AM///M> \' \n\n\n\n264 DIFFERENTIAL CALCULUS, \n\ny = m^x -f,{m^)/f,\'{m^), \netc. etc. \n\ny = MnX \xe2\x80\x94 /i(w\xc2\xab)//o\'(^\xc2\xab), \n\nare the equations of the n asymptotes. \n\nHence we have the following rule : \n\nIn the equation of the curve substitute mx -\\- c for y, place \nthe coefficieitts of the two highest powers of x equal to zero^ \nand find corresponding sets of values for m and c. Each set \nwill determine an asymptote real or imaginary. \n\nThus, having x^ \xe2\x80\x94 2xy \xe2\x80\x94 2x^ \xe2\x80\x94 Sy \xe2\x80\x94 o, then \n\nx^ \xe2\x80\x94 2x\'^(mx -{- c) \xe2\x80\x94 2x\'^ \xe2\x80\x94 S{mx -\\- c) = 0, \nand (i \xe2\x80\x94 2m)x^ \xe2\x80\x94 (2^ -f~ 2)^^ \xe2\x80\x94 Zmx \xe2\x80\x94 8^ = o. \n\nI \xe2\x80\x94 2m. = 0, \xe2\x80\x94 2^ \xe2\x80\x94 2 = o, give m =1/2, ^= \xe2\x80\x94 1. \nHence, y = x/2 \xe2\x80\x94 i is an asymptote. \nHaving xy"^ \xe2\x80\x94 x^ \xe2\x80\x94 ay^ \xe2\x80\x94 ax^ = o, then \n\nx{mx -{- cY \xe2\x80\x94 x^ \xe2\x80\x94 a{mx -\\- cY \xe2\x80\x94 ax^ = o, \nand {m"^ \xe2\x80\x94 i)^^ -1- [2mc \xe2\x80\x94 am^ \xe2\x80\x94 a)x\'^ -\\- etc. = o. \nm^ \xe2\x80\x94 1=0, 2mc \xe2\x80\x94 am^ \xe2\x80\x94 ^ = o, give ;;z = \xc2\xb1 i, <: = \xc2\xb1 ^. \nHence, >\xc2\xbb = \xc2\xb1 (^ -|- ^) are asymptotes. \n\nEXAMPLES. \nCURVES. ASYMPTOTES. \n\n1. y\\x \xe2\x80\x94 2d) = x^ \xe2\x80\x94 a^. x = 2a, y = \xc2\xb1 (x -{- a). \n\n2. y = x(x -f- i)V(x \xe2\x80\x94 i)^ X = I, y = X -j- 4. \n\n3. X* \xe2\x80\x94 y* = a\'^xy. y =. X, y ^ \xe2\x80\x94 x. \n\n4. X* \xe2\x80\x94 x\'^y^ -}- a\'^x\'^ -\\- i>* = o. x=\xe2\x80\x94y, x=:y, x = o. \n\n5. x^ ~\\~y^ -\\- 2)d^x + \'ib\'^y A^ a = O. X = \xe2\x80\x94 y. \n\n6. x^ -\\- y^ = sax^y^. \xe2\x96\xa0^ -\\- y = a- \n\n7. xy^ \xe2\x80\x94 ay* -f" \xe2\x80\xa2^!y^ = P- x = a, y = \xe2\x80\x94 x \xe2\x80\x94 a, y = o. \n\n\n\nA S YMP TOTES. 265 \n\nx^ =iO. y ^= \xe2\x80\x94 X, y -\\- \\ =. X, X =\xe2\x96\xa0 o. \n9\xc2\xab (/\'^ \xe2\x80\x94 7>^y + \'2-x\'^)x \xe2\x80\x94 4ay^\' x = 4a, y = x \xe2\x80\x94 ^^a, y = 2{x -f- 8a). \n10 x(y -\\- xfiy \xe2\x80\x94 2x) = a*. x = o, y = 2x, y = \xe2\x80\x94 x. \n\n11. / = x\'\'{x^ - i)/(x2 4-1). y = \xc2\xb1x. \n\n12. y^ \xe2\x80\x94 x\'^ = a^x. y = X. \n\n13. y^ = {x^ - a\'x\'\')l{2x -a). x= a/ 2, y = {x -\\- a ft)/ )/2. \n\n14. x^ \xe2\x80\x94 xy"^ -\\- ay"^ \xe2\x80\x94O. x = a, y \xe2\x80\x94 \xc2\xb1 {x -\\- a/2). \n\n15. y"^ = x\\x^ \xe2\x80\x94 4a^)/{x\'^ \xe2\x80\x94 a"^). y = \xc2\xb1 x, x = \xc2\xb1 a. \n\n16. y = x{x \xe2\x80\x94 2a)/(x \xe2\x80\x94 a). y = x \xe2\x80\x94 a, x ^a. \n\n17. x^ -\\- 2xy \xe2\x80\x94 xy"^ \xe2\x80\x94 2y^ -(- 4y -|- 2xy -\\- y = I. \n\nX -^ 2y = O, X -\\-y = 1, ^ \xe2\x80\x94 \' = \xe2\x80\x94 I. \n\n18. y^ \xe2\x80\x946xy^ -|- iix^y \xe2\x80\x946x^ -\\- x -\\-y = 0, y =z x, y ^ 2x, y = ^x. \n\n19. y^ \xe2\x80\x94 x\'^y -\\- 2y\'^ -{- 4y -\\- x = o. y=o, y = x\xe2\x80\x94i, y=\xe2\x80\x94 x\xe2\x80\x94i. \n\n20. y* \xe2\x80\x94 X* -\\~ 2ax\'^y \xe2\x80\x94 b\'^x := O. _y = x \xe2\x80\x94 a /2, ^ = \xe2\x80\x94 x \xe2\x80\x94 a/2. \n21 X* \xe2\x80\x94 y* \xe2\x80\x94 a^xy \xe2\x80\x94 Py"^ = o, x = ^ y. \n\n22. x\'^y Ar y\'^x \xe2\x80\x94 c^, x = o, y \xe2\x80\x94 o, x = \xe2\x80\x94 y. \n\n23. 2x^ \xe2\x80\x94 x\'^y \xe2\x80\x94 2xy\'^ -\\-y^ -f- 2x^ -\\-xy\xe2\x80\x94 y\'^-\\-x-\\-y-\\-i =0. \n\n_y = x-|-i, _y = \xe2\x80\x94 X, y =[2x. \n\n156. Let the equation of the curve (i) (\xc2\xa7 155) be arranged \naccording to the descending powers of x; thus: \n\nax^ + {^y 4- d)x\'\'-\'^ + . . . = o. . . . (6) \n\nIf a = o andjvbe assumed equal to \xe2\x80\x94d/b^ two of the roots \nof (6) are infinite and the right lineji\' == \xe2\x80\x94d/b is an asymptote. \nHence, when x^ is missing, the coefficient of the next highest \npower placed equal to zero is the equation of an asymptote \nparallel to X. \n\nIf both x^ and x^~^ are missing, the coefficient of .x"~^, \nwhich will be of the second degree with respect to j\', placed \n\n\n\n266 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nequal to zero, determines two asymptotes real or imaginary \nparallel to X. \n\nIn a similar manner asymptotes parallel to Y may be \ndetermined. Thus, having x"\'/ ^x^y\xe2\x80\x94xy^ -\\- oc-\\-y -\\-\\ =o, \nin which x^^ x^^ y\\ y\' are missing, y^ \xe2\x80\x94y is the coefficient \nof x^ and x^ \xe2\x80\x94 x that of y^. \n\nHence, y \xe2\x80\x94 ^ = o, x^ \xe2\x80\x94 x = o, give the asymptotes \ny = o, _y = I, X = o, and x = i. \n\nEXAMPLES. \nCURVES. ASYMPTOTES. \n\n1. y\'x \xe2\x80\x94 ay\xe2\x80\x94 x^\xe2\x80\x94 ax\'*\' \xe2\x80\x94 ^^ = o, ;r = a, y ^= x-\\-at y =\xe2\x80\x94 jc\xe2\x80\x94 <\xc2\xbb, \n\n2. xy^ + -^^y \xe2\x80\x94 a* = o. j^ = o, :f = o. \n\n3. ^y - a^C^t\'^ -\\-y\'\') = 0. :\xc2\xabr = \xc2\xb1 d!, J = \xc2\xb1 <j!. \n\n4. j^y \xe2\x80\x94 a2j2 \xe2\x80\x94 ;i; = o. :v = \xc2\xb1 a, _^ = O. \n\n5. ajj/2 \xe2\x80\x94 ^rj/\'* \xe2\x80\x94 ;t:^ = o. \xe2\x96\xa0 X = a. \n\nPolar Coordinates. \n157. If for any finite value of ^, designated by a, /^ is infi- \nnite, and the corresponding subtangent -P7"=(^V6^/^r)fl=\xc2\xab, \n\n\n\n\ndesignated by -S", is finite, the curve has an asymptote paral- \nlel to the radius vector corresponding to a. \n\n\n\nASYMPTOTES, \n\n\n\n267 \n\n\n\nKnowing a and ^S", the asymptote may be constructed as \nfollows : \n\nThrough P draw PM\\ making the angle XPM\' = a; it \nwill be the direction of the corresponding infinite radius \nvector. Draw PT\' perpendicular to PM\', and lay off \n\npr= S= (r\'dd/dr)e^a, \n\nT\'M\' drawn parallel to PM* is the corresponding asymp- \ntote. \n\nDesignate the angle XPT\' \xe2\x80\xa2= a \xe2\x80\x94 71/2 by ^, and let r \nand d be the polar coordinates of all points of the asymp- \ntote. Then from the triangle T\'PR we have \n\nr = S/sin {oL-O) (i) \n\nHence, knowing a and 6*, the polar equation of the \nasymptote is known. \n\nIn some cases a and S may be readily found. Thus, having \nthe curve r = ad /sin 6^ 6 = tt = a gives r = co ^ and \n\nad\' \\ ^ ^^ \n\n;sin 6\xe2\x80\x946 cos 6^/^=77 \n\nHence, r sin 6^ = <27r is the polar equation of the asymptote. \nPositive values of S are laid \noff in the direction \n\n(3= a\xe2\x80\x94 7r/2, \n\nand negative values in the \ndirection \n\n/?+ i8o\xc2\xb0 = a-}-;r/2. \n\nThus, having r = a/0^ \n6= o = a gives r = 00 , and S = \xe2\x80\x94 a. Hence, P =\xe2\x80\x94 ^/2, \nand direction of S is 71/2. r sin 6 := a \\s the equation of \nthe asymptote. \n\n\n\n\\ dr )0=a \\s \n\n\n\n\n268 \n\n\n\nDIFFEREN TIA L CALCUL US. \nM \n\n\n\n\nFlavin g r = a -\\- a/Oy \n6 =^ o =z a gives r = oo , and \n\nS = {r\'d6/c/r)e^o ^ - a. \nPlence, r = \xc2\xab/sin d is an \nX asymptote. \n\nAs 6*^-^00, r-m-^a, and the \ncircle R = a \\s called an \nasymptotic ci7\'cle. \niSS* In general, let the polar equation of a curve be \n\nr"/\xe2\x80\x9e(&) + r--\'fn-.{fi) + . . . + r/.(&) +/,(^) = O. (l) \nPlacing u = r/r, clearing of fractions, and arranging with \nrespect to u^ we have \n\n\xc2\xaby.(^) + t^-vm + ... + ufn-m +/\xc2\xab(^) = o- (2) \n\nFor any point corresponding to r = oo ?^ must be zero, \nand the roots of fn{P) = o are the values of Q correspond- \ning to r = CO . \n\nDifferentiating (2) with respect to B, making u = o and \n^ == Of, we have \n\n(du/de)^^^fn-x(pi) +/\xc2\xab\'(\xc2\xab) = o. . . . (3) \nTherefore (\xc2\xa7 151), \n\nHence (i) (\xc2\xa7 157), the polar equation of the asymptote is \nr sin {oL \xe2\x80\x94 d) = fn-i{oc) //n\\ay . . (4) \n\nWhen \xc2\xab = I, \n\nrsm(a^e)=Ua)/f^\\a) (5) \n\n\n\nA S YMP TO TES. 269 \n\nTo illustrate, having r cos ^ \xe2\x80\x94 ^ sin ^ = o. \n\n;2 zz. I, f^B) =: cos e, fXO) = -bsin e. \n\ncos 6^ = o gives a = 7t/2, 37r/2, etc. \n\ny^(<ar) = \xe2\x80\x94 <^ sin a, //{^) = ~ sin ex. \n\nHence, r sin {7t/2 \xe2\x80\x94 0) \xe2\x80\x94 \xe2\x80\x94 (d sin r-e)/ \xe2\x80\x94 sin or = ^, \n\nor r cos 6 ^ b ior the corresponding asymptote. \nLet r = a sec ^ + /^ tan ^. \n\nHere i/r = u = cos ^/(^ + ^ sin ^) = o \n\ngives a = ;r/2, (^^^/^^)^^^/^ = - i/{a + <5), \n\nand r cos ^ = ^ -f ^ for the corresponding asymptote. \n\n(4) maybe deduced from (2) (\xc2\xa7 151) by substituting in \nit u\' = o, 6/\' = a, and - du\'/dd\' ^ fn\\oc)/f\xe2\x80\x9e.^{a). \n\nEXAMPLES. \nFind the asymptotes of the following curves : \n\n\\, r = a tan 0. r cos 9 = \xc2\xb1 <r, \n\n2. r cos = a cos 2O. r cos 6 = \xe2\x80\x94 <r. \n\n3. {r \xe2\x80\x94 a) sin = ^. r sin = ^. \n\n4. r cos 20 = a. r = tf/[2 sin (45* \xe2\x80\x94 0)], \n\n5. r sin 40 = a sin 3O. \n\n6. r \xe2\x80\x94 a0-iA . r sin = o. \n\n7. /\xe2\x80\xa2\' sin = 0\' cos 20. y sin = o. \n\n8. r = a cosec (0/2\xc2\xbb) r sin = \xc2\xb1 2na. \n\nQ. rsin;z0 = \xc2\xab. nr sin (0 ) = --. \n\n\\ \xc2\xab / cos /$;r \n\n(In which k is any integer.) \n\n\n\n2 JO DIFFERENTIAL CALCULUS. \n\n10. r = \xc2\xab(sec \xc2\xb1 tail 0). r cos G = 2a. \n\n11. r =: 2a sin 6 tan 0. r cos 6 = 2a. \n\n12. r = cz sec niQ -\\- b tan m^. r sin [7r/2w \xe2\x80\x94 0] = (a -[- b)/m. \n\n13. r^ cos = rt^ sin 30. r cos 6 = 0, \n\n14. r = \xc2\xab0V(0^-i). r= -\xc2\xab/[2 sin(i8o77r- 0)]. \n\n15. r = rt;/cos + <5. r = \xc2\xab/sin (90\xc2\xb0 \xe2\x80\x94 6). \n\nFind the asymptotic circles of the following curves : \n\n16. = \xe2\x80\x94 \xe2\x96\xa0 circle r = 2a. \n\n\\2ar \xe2\x80\x94 r\'\'^ \n\n17. r = a0Y(02 _ i). circle r = \xc2\xab. \n\n\n\nDIRECTION OF CURVATURE. \n\n\n\n271 \n\n\n\nCHAPTER XIV. \nDIRECTION OF CURVATURE. SINGULAR POINTS. \n\n\n\nDIRECTION OF CURVATURE. \n\n159. Let j^=/(^) be the equation of any plane curvej \nand let ^ represent the angle \nmade by any tangent to the \ncurve with X. \n\nThen dy / dx^ldoa 6*, \n\nand d\'^y/dx\' = (d tan 6)/dx. \n\nWhen d-y/dx^ is positive^ \ntan increases with x (\xc2\xa7 71), and, as may be seen from any \nfigure, the concave side is above the curve and the direc- \ntion of curvature is upward.^ \n\n\n\n\n\\l \n\n\n( \n\n\nA \n\n\n\n\n\\ ^ \n\n\n\n\n\n\n/^ \n\n\n^0\' \n\n\nv\xc2\xab^ ^ \n\n\n\n\n\n/ \n\n\n/ \n\n\n> \n\n\n\n\n\nWhen dy/dx^ is negative, tan 6^ is a decreasing function \nof X, and t/ie direction of curvature is downward. \n\n* The direction of curvature at a point is along the normal towards \nthe concave side. \n\n\n\n2/2 \n\n\n\nD IFFERENTIA L CAL CUL US. \n\n\n\nIn a corresponding manner it may be shown that positive \nvalues of d\'^x/dy\' indicate concavity towards the right, and \nnegative values concavity towards the left. \n\nTo illustrate, let y ~ 2 -^ {x \xe2\x80\x94 2)^ \n\nThen d-\'y/dx\'\' = 6ix - 2), \n\nand d\'^x/dy\'\' = 2/9(2 \xe2\x80\x94 x)^ \n\nHence, when x < 2, the direction \n\nof curvature is downward and to \n\nthe right. When x > 2, the curvature is upward and to \nthe left. \n\nShow that the direction of curvature oi y = e\'\' is always \nupward and to the left. \n\n160. Polar System. \xe2\x80\x94 Representing by/ the perpendic- \nular distance J^Q from the pole J^ to the tangent at any \n\n\n\n\n\npoint M on a. curve, it is apparent (when the tangent does \nnot pass through /\') that the curve is concave towards \nthe pole when p is an increasing function of r, and that it \ncurves away from the pole when/ is a decreasing function \nof r. \n\nHence, positive values of dp/dr indicate concavity towards \nthe pole and negative values the reverse. \n\n\n\nDIRECTION OF CURVATURE. \n\n\n\n273 \n\n\n\nFrom (i) (\xc2\xa7 150) we have i//^ = 21^ -j- {du/dOy, from \nwhich - dp/p\' = (z^ + d\'u/d6\')du; but du = - dr/r\\ \n\nHence, dp/dr = {p^ /r\'){u + d\'^u/dO\'^)^ which, since / is \nalways positive, changes its sign only with {ii -\\- d\'^u/dO\'^). \n\n\n\nEXAMPLES, \n\n\n\nI. Let r \xe2\x80\x94 aO *. \n\nPlacing \\/r = u^ we have u = ^/^, du/d6 = 6~-/2a, ami \n\nHence, ^^ + d\'u/dd\' = {0^ - 6-^/4) /a. \n\n\n\n\nTherefore, when 6 < \\, dp/dr is negative, and the concav- \nity is away from the poie. \n\nWhen ^ = i, whence r =: a\\/2^ dp/dr \xe2\x96\xa0=^ o. \n\nWhen 6* > I, dp/dr is positive, and the curve is concave \ntowards the pole. \n\n2. Show that r = p/{i \xe2\x80\x94 cos 6^) is always concave towards \nthe pole. \n\n3. Show that the direction of curvature of r = a^ is \nalways towards the pole. \n\n\n\n274 DIFFERENTIAL CALCULUS. \n\nSINGULAR POINTS. \n\nPoints of any curve which, independent of coordinates, \npossess some unusual property are as a class called singular \npoints. \n\n1 6 1. A Point of Inflexion is one at which, as the -variable \nincreases^ a curve changes its concavity from one side of \nI the curve to the other. The cor- \n\n//^ responding tangent intersects the \n\ny\\ curve, and the direction of cur- \n\ny/7 I vature is reversed. \n\n\' |p Let^ = fx be the equation of \n\na curve. It follows from \xc2\xa7 159 \nthat c is the abscissa of a point of inflexion, if as ^increases \nf"x changes its sign in passing through /\'V. \nThe real roots of the equations \n\nf"x = o and /"x = 00 \n\nare therefore critical values, and may be tested by a method \nsimilar to that described in \xc2\xa7 135 in the case of fx = o \nand/\'jc = 00. \n\nHence, the general method for any critical value as c is to \ndetermine whether, as h vanishes from any definite value, \nf\'\\c \xe2\x80\x94 h) and f"{c-\\- h) ultimately have and retain dif- \nferent signs. \n\nWhen/"(x),/\'"(jic:), etc., are continuous for values of x \nadjacent to critical values, those derived from f"{x) = c \nmay be examined, as in the case of fx == o (\xc2\xa7 136), as \nfollows : \n\nHaving f\'\\c) = o, substitute c for x in f\'"{x)^ f^^(x), etc., \nin order, until c result other than o is obtained. If the cor- \nresponding derivative is of an odd order, c is the abscissa of a \npoint of inflexion, otherwise not. \n\n\n\nSINGULAR POINTS. \n\n\n\n275 \n\n\n\nIf a result oo is obtained^ the general method should be em- \nployed. \n\nWhen f\'"x is complex, the general method is usually \npreferable. \n\nIt should be observed that at a point of inflexion the \nslope of a curve is either a maximum or a minimum, and \nthat the determination of such points is the same as finding \nthose at which /"\'(jt:) is a maximum or a minimum. \n\n\n\nEXAMPLES. \n\\, x^ \xe2\x80\x94 ix^y \xe2\x80\x94 2x^ =\xe2\x80\xa2\xe2\x96\xa0 Sy. \n\ndy _ x{x^ -\\- \\2x \xe2\x80\x94 i6) \n\nTx~ i{x-\' + 4)2 \xe2\x80\xa2 \n\n(Py _ - \\{x\'^ - 6x^ - 12^ - 1 -8) \n\ndx\' " (-^^ + 4j\' \n\n\n\no gives \n\n\n\nX=-2, X= 2(2 - 1/3) = 0.54-, ^= :i(2+ Vs) =7.5. \n\nApplying the method \xc2\xa7 138, we have \n\n\n\n/<Py \n\n\n\n(x^ + 4)^ \n\n\n\n-3 [^ - 0-54][-^ - 7-5] = - 0.2. \n\n\n\n(3)....- (-"fc.[\'+*-"i -+\xe2\x80\xa2\xe2\x96\xa0\'\xe2\x96\xa0 \n\n\n\n- [x + 2][^ ~ 0.54] = - .0013. \n\n\n\n,dx^]^=,,, {x\'^4r \n\nHence, (\xe2\x80\x94 2, \xe2\x80\x94 i), (0.54 \xe2\x80\x94 , \xe2\x80\x94 0.05), and (7.5 \xe2\x80\x94 , 2.6 \xe2\x80\x94 ) are points \nof inflexion. \n\n2. ;p \xe2\x80\x94 2 \xe2\x80\x94 s(x \xe2\x80\x94 2)3/5. \ndy _ - 9 \n\n\n\ndx 5(x - 2)^5* \nd-^y 18 \n\n\n\ndx^ 25(x - 2)V\xc2\xbb\' \n\n18 \n\n~~r ^T7^ = o has no real \n\n25(;c \xe2\x80\x94 2)V5 \n\nfinite roots. \n\n18 \n\n25(x \xe2\x80\x94 2)V5 \n\n\n\nM \n\n\n\n00 gives X = 2, \n\n\n\n{dy/dx%<2 is negative, (dy/dx%>2 is positive. \nHence, jt = 2 = ^ is a point of inflexion. \n\n\n\n2"]^ DIFFERENTIAL CALCULUS. \n\n\n\n3- y \xe2\x80\x94 ^^Vi^ \xe2\x80\x94 x)lx. , X \xe2\x80\x94 3^/4 \n\n4. a^y = (x \xe2\x80\x94 by. X ^= b. \n\n5. y{x \xe2\x80\x94 2) = {x \xe2\x80\x94 l){x \xe2\x80\x94 3). X = 2. \n\n6. a\'^y = x^. X =: o ^= y. \n\n7. y\'^(x \xe2\x80\x94 a) = x^ -\\- ax\'K x =\xe2\x96\xa0 \xe2\x80\x94 2a. \n\n8. J = x\'^/a -\\- a\\{x \xe2\x80\x94 a)/dfl^. x = a. \n\ng. y = x"^ log (l \xe2\x80\x94 x). X = O ~ y. \n\n10. x\'^y \xe2\x80\x94 4a^{2a \xe2\x80\x94 y). x = \xc2\xb1 2a/ ^\'i. \n\n11. y = <?1A. X = \xe2\x80\x94 1/2. \n\n12. x^ -\\- y^ ^= a^. X =^ a, x = O. \n\n13. logy = i^x. ^ = 8. \n\n14. jj/ = x^ tan X. X ^= o ^= y. \n\n15. y = a sin (x/b). x = o, bit, etc. \n\n16. jj/ = (\xc2\xab \xe2\x80\x94 jr)V2 + ax, X = a. \n\nI"], y = x"^ \xe2\x80\x94 x^/^. X \xe2\x80\x94 64/225. \n\n1/3 \n\n\\?>. y = e^ ^ = 8. \n\n19. y = x^/{a^ + ^V\' X = o, jr =: \xc2\xb1<3!|/3. \n\n20. J = a tan {x/b). x = o = r. \n\n21. 1/ = x\'^{x -f- <\'7)/[a(x \xe2\x80\x94 a)\\. X = \xe2\x80\x94 a[\\/2 \xe2\x80\x94 i)- \n\n22. y^ = ay^ \xe2\x80\x94 x^. X = a. \n\n23. J = 2 -|- (x \xe2\x80\x94 2)^. :r \xe2\x80\x94 2. \n\n24. y = axy -[- by"^ -\\- cx3. x := o = V- \n\n25. jf(\xc2\xab* \xe2\x80\x94 \'^*) = ^{\xe2\x96\xa0x \xe2\x80\x94 \xc2\xab)* \xe2\x80\x94 \xe2\x80\xa2^^*. X = ^^/s, X = a \n\n26. y = a^xlixP\' -(- a\'^). x = o, x \xe2\x80\x94 \xc2\xb1a\\/2. \n\n27. 7 =: x\'^/a \xe2\x80\x94 x^y/a^. x \xe2\x80\x94 \xc2\xb1 a/i/3. \n\n28. J = X cos (x/a). X = Q = y. \n\n2g. y = X -\\- 2^x^ \xe2\x80\x94 2x3 \xe2\x80\x94 X*. X = 2, X = \xe2\x80\x94 3. \n\n30. / = a\\/x/{2a \xe2\x80\x94 x). X = ^/2. \n\n31. axy = r8 \xe2\x80\x94 a^. x = a. \n\n32. a\'^jK \xe2\x80\x94 -^^/S \xe2\x80\x94 dX"^ -\\- 2^5. X = rt, \n\n33. xy = a^ log {x/a). x = ae^/^. \n34 {ay \xe2\x80\x94 x2)* = (5^:^ X = gb/64. \n\n35. aV ^ ^\'(\xc2\xab\' - ^^)- X = \xc2\xb1 a/s/l. \n\n36. yix\'\' f \xc2\xab\') .-= ^"^(^ - x). \n\n37. ;t V - ^^\'(-^\' - J\'^)- X = o. \n\n\n\nSINGULAR POINTS. \n\n\n\n277 \n\n\n\n38 \n\n\ny = \n\n\n2ai/{2a \n\n\n-x)/x \n\n\n39 \n\n\ny ~ \n\n\nte-^^f^^^ \n\n\n\n\n40 \n\n\na^j \n\n\n= 3^j;2 - \n\n\n- x\\ \n\n\n41 \n\n\ny = \n\n\nxe\'\'. \n\n\n\n\n42 \n\n\n\xc2\xab|/^ \n\n\n= {x - \n\n\na)\\/y. \n\n\n43 \n\n\n;f^ = \n\n\ne-V^. \n\n\n\n\n44 \n\n\na-^y: \n\n\n= x(a2 \xe2\x80\x94 \n\n\nx\'^). \n\n\n\nX = 3\xc2\xab/2, j(/= \xc2\xb1 2a /\\/ 2,. \n\n\n\nX = a\\/ {n \xe2\x80\x94 i)/n. \n\nX = b \n\nX = \xe2\x80\x94 2. \n\na\' = \xe2\x80\x94 2a. \n\nX = 1/2. \n\nX = o = J. \n\n162. Polar Coordinates. \xe2\x80\x94 Having i\'=f{6), it follows \nfrom \xc2\xa7 160 that r = c corresponds to a point of inflexion if \ndp/dr = (u -}- d"^ u / dB\'^^f / r"^ changes its sign in passing \nthrough {dp/d7\')r^g. \n\nHence, the real roots of the equations dp/dr = o and \ndp/dr = CO, or, what is equivalent. // + d\'^u/dO\'^ = o and \nu -|- d\'^ii / dd"\' = 00 , are critical values which may be lested \nby methods similar to those indicated in \xc2\xa7\xc2\xa7 135 and 136. \n\nIt should be observed that/, corresponding to a point of \ninflexion, is a maximum or a minimum. \n\nEXAMPLES \n\nI. r = a6y{e\' - i) = i/u. \n\nWhence du/dO = 2/{a6\'), d\'\'u/d6\' = - 6/(ae\'). \n\nu + d\'\'u/d6\'\' = {6\' - 6\' - 6)/{a6\') = o \ngives 6^= db V3, and changes sign as passes through either. \nHence, B= \xc2\xb1 V ^, r \xe2\x80\x94 3^/2, are points of inflexion. \n\'\xe2\x80\xa2 2. r= (a-\\- a6)/6 = \\/u. \nHence, du/dB \xe2\x80\x94 a/{a -}- adf^ \nd\'u/dd"- = - 2a\'/(a + ad)\\ \n\nd\'u_a \'{d\'+2e\'+e-2 ) _ \n^~^d6\'~ (a + aey \' -"" \n\ngives 6 = 0.\']\xe2\x80\x94, and dp/d^ \nchanges from \xe2\x80\x94 to + as \npasses through 0.7 \xe2\x80\x94 . \n\n3. r = ae-y\\ d=jj2 \n\n\n\n\n278 \n\n\n\nDIFFERENTIAL CALCULUS, \n\n\n\n163. A Multiple Point is a point common to two or more \nbranches of a curve, and is double or triple^ etc., according \nto the number of branches. They are classed into Points \nof Intersection^ Shooting Points^ and Points of Tangency. \n\nA Multiple Point of Inter- \nsection is one through which \nthe branches pass and have \ndifferent tangents. ^ is a double \nand <^ is a triple point of inter- \nsection. A double point of in- \ntersection is also called a node. \nA Shooting Point is a multiple \npoint at which the branches terminate with different \ntangents. \n\n\n\nA Salient Point is a double shooting \npoint. \n\n\n\n\n\n\nA Multiple Point of Tangency* \nis one at which the branches have \na common tangent. \n\n\n\nA Cusp is a double multiple point of tangency at which \nboth branches terminate. \n\nWhen in the vicinity of a cusp the branches are on \nopposite sides of the common tangent (Fig. c\\ it is of the \n\n\n\n\n* Sometimes called points of osculation or double cusps. \n\n\n\nSINGULAR POINTS. 2/9 \n\nfirst sfecies, or a keratoid (horn) cusp; when on the same \n\n\n\n\nside (Fig. ^), it is of the second species, or a ramphoid \n(beak) cusp. \n\nA Conjugate Point is an isolated real point of a curve. \nThus, the origin ^ = o=j^\' is a real point of the curve \nwhose equation \\-> y = \xc2\xb1 x Vx - 2, but y is imaginary for \n\xc2\xb1 X < 2. Hence the origin is a conjugate point. \n\nLet J = ^{^) be the equation of any curve. \n\nFrom \xc2\xa7 124, \n\njr(c -{- /i) = F{c) + F\'{c)h + F"(c)hy2 + etc. \n\nWhen {c, d) is a conjugate point, Fi^c -\\- h) is imaginary \nfor values of h near zero, while F{c) and h are real. Hence, \none or more of the expressions F\\c), F"(c), etc., must be \nimaginary. This important characteristic of a conjugate \npoint is frequently used in testing critical points. Thus, \n(r, d) is a conjugate point provided F{c) is real, and F"{c) \nis imaginary for any entire value of ;/. \n\nIn the example above we find F\'{o) = -^ V \xe2\x80\x94 2. \n\nSince the ordinates of points of a curve adjacent to a \nconjugate point are imaginary, the number of such ordinates \nfor each point is even. It follows that a conjugate point is \na multiple point in the immediate vicinity of which the \nbranches are imaginary. The tangents corresponding to a \nconjugate point may be real or imaginary, coincident or \nseparate. \n\n\n\n28o \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nHaving the equation of any curve with two or more \nbranches, if either variable, as y^ has but one real value, \nd^ corresponding to any real value of the other, as x = c^ \n{c, d) is a critical point. \n\nIf {dy/dx)(c,d} has two or more values, {c, d) is a multiple \npoint of intersection or tangency according as the several \nvalues of {dy/dx)(^c, d) are unequal or equal. \n\nWhen the values oi y for points adjacent to and on both \nsides of {Cy d) are imaginary, {c, d) is a conjugate point. \n\nEXAMPLES. \n\n\n\nI. ^ = 3 \xc2\xb1 (^ \xe2\x80\x94 4) \'^\xe2\x80\xa2^ \xe2\x80\x94 2. \n\n\n\n\n^ < 2, ;; is imaginary, j*; =: 2, j = 3. \n2 < jc < 4, jv has two real values. -^ = 4, ^ = 3. \nX > 4, y has two real values. \nHence, (2, 3) and (4, 3) are critical. \n\ndy/dx = \xc2\xb1 Vx \xe2\x80\x94 2 \xc2\xb1 {x \xe2\x80\x94 4)7(2 Vx \xe2\x80\x94 2). \n{dy/dx)(fi,z) =00. (2, 3) is not a multiple point. \n\nHence, (4, 3) is a double multiple point of intersection. \n\n,. 2.. ;;,= \xc2\xb1 Vx\\x - ~2)/ V3. \n\nx^=^o-=y. _y is imaginary when ^<o, or o<Ji:<2. \nHence, the origin is a conjugate point at which we have \n\n{dyldx),= -^ 2I Sf^l. \n\n\n\nSINGULAR POINTS. \n\n\n\n281 \n\n\n\nX = o=y. X = o \xc2\xb1 h^ y has three \nvalues, (dy/dx)^ = o and \xc2\xb1 y/ 2. \n\n\n\n\nHence, the origin is a triple point of intersection. \n\nY 4. J^\'= 2 -f- .T tan~^(i/x) = 2+x cot~^jc. \n\n^=: o, J)/ = 2. When cot~^Jc<7r/2, \nX and cot"^^ have the same sign, and \xc2\xb1.x \ngive equal positive value for j. \n\n{dy/dx),= [cot-^^-V(i+^\')]o = ^/2 \nand z^/2 and 57r/2, etc., or\xe2\x80\x94 ;r/2 and \n\xe2\x80\x94 37r/2 and \xe2\x80\x94 57r/2, etc., according as \nX ^B-\xc2\xbbo is positive or negative. Hence, \n(o, 2) is a shooting point. \n\nd\'^y/dx^ = \xe2\x80\x94 2/(1 + jr\'^)\' is negative for \xc2\xb1 ^; hence, the \ncurve is concave downward. \n5. j = V(i+^\'/"). \n:xr = o = J. \n\n\n\n\n\n\n\n= o or I \n\naccording as ^^->o is positive or negative. Hence, the \norigin is a salient point. \n\nY 6. ay = cx\'-\\-x\\ \n\n\n\n\ny=\xc2\xb1.x\'\'Vc-\\- x/a, \n\nX =^ o =y, and ji^ has double \nvalues for values of x between \xc2\xb1 c \nand o. \n\n\n\n\\dx^o i2ai^X\']-cJo \n\n\n\n282 DIFFERENTIAL CALCULUS. \n\nHence the origin is a dpuble point of tangency. \n7. y = \'^\xc2\xb1{x \xe2\x80\x94 2fJK \n\n\n\n\ndy/dx = \xc2\xb1 3 \\x \xe2\x80\x94 2/2. \nx<2^ y is imaginary. \nx\'>2^y has double values. \n\nx=2,y=i, and (dy/dx\\^^^)\xe2\x80\x94 \xc2\xb10. \n\nHence, (2, 3) is a cusp. \n\nThe branches are on opposite sides of the common tan- \ngent JF = 3, and the cusp is of the first species. \n\n8. (2x-j)\' = (x-3)\'. \n\n^<3 makes 7 imaginary, ^=3 givesj)^=6, and ^>3 gives \nreal double values to j. \n\n(dy/dx)(^^^) = 2 \xc2\xb1 o. Hence, (3, 6) is a cusp. \n\nSince ji^ = 2^ \xc2\xb1 (a: \xe2\x80\x94 3)^2 ^ the branches are on opposite \nsides of the common tangent j\' = 2x^ and the cusp is of the \nfirst species. \n\nA characteristic of a cusp of ihe first species is a change \nin direction of curvature from one side of the common tan- \ngent to the other ; while at one of the second species the \ndirection of curvature remains upon the same side of the \ncommon tangent. Hence, different signs for d\'^y/dx^ cor- \nresponding to the two real values of y in the immediate \nvicinity of a cusp indicate the first species, and like signs \nthe second species. \n\nThus, in example (7), (^-oh= (\xc2\xb1 \xe2\x80\x94 has val- \n\nues with different signs. \n\nIn some cases it is preferable to consider J^\' as the inde- \npendent variable. \n\n\n\nSINGULAR POINTS. \n\n\n\n283 \n\n\n\n\ng. jj; = 3 -|- (jt: \xe2\x80\x94 2)^/^. y has but one \nvalue for each value oi x. x =^ 2,y ^= ^, \ndy/dx =2/[3(jt: \xe2\x80\x942)^/^] is negative when \n^>2, is 00 when x \xe2\x96\xa0=\xe2\x96\xa0 2^ and is positive \nwhen x<2, (2, 3) is evidently a cusp of \nthe first species as in figure. \n\nSolving with respect to x^ we have \n\nx=2\xc2\xb1[y- 3)V2 , {dx/dy\\,^ 2) = ( \xc2\xb1 3 \'^^F^z/Az, ?)= \xc2\xb1 o. \nj<3, ^ is imaginary. J = 3, ^\'c = 2. j>3 gives double \nvalue for x. Hence, (3, 2) is a cusp. \n\n|\xe2\x80\x94 7^ ] = ( \xc2\xb1 ,/ I has values with different signs, \n\ntherefore the cusp is of the first species. \n\nPoints corresponding to maximum or minimum ordinates \n\nat which dy/dx = 00 are cusps. \n\n5/2 \n\n\n\n\n10. y^=-x\'^-^x \n~=2x\xc2\xb1sxyy2. \n\ndx"" \n\n\n\n2\xc2\xb1i5^vy4. \n\n\n\nj; = o = jj/. .^c < o, ^ is imaginary. \n\nx>Oy y has two real values. \n\n(dy/dx)^ = \xc2\xb1 o. Hence, the origin is a cusp at which \nthe axis X is the common tangent. \n\nFor o<:x:<i both values of y are positive; therefore the \ncusp is of the second species. This is also indicated by the \nfact that both values of [d\'^y / dx\'^\\<:^x<^^/%%^ are positive. \n\nExamine the following curves for multiple points: \n\n11. y^ \xe2\x80\x94 x^/{ia \xe2\x80\x94 x). (o, o) is a cusp of I St species. \n\n12. (y \xe2\x80\x94 x)2 = x^. (o, o) is a cusp. \n\n\n\n284 DIFFERENTIAL CALCULUS. \n\n\n\n13. jj/ = \xc2\xb1 {x\'^^x^ \xe2\x80\x94 4)/4. jc = o =_j/ is a conjugate point, \n\n14. (4/ \xe2\x80\x94 x^Y \xe2\x80\x94 (\xe2\x80\xa2^\xe2\x80\x94 4)^(^ \xe2\x80\x94 3)*. (3, 27/4) is a conjugate point. \n\n15. j/^ = 2x^ -^x^. (o, o) is a double point of inter- \n\nsection. \n\n16. x^t^ \xe2\x80\xa2\\- yV^ \xe2\x96\xa0=\xe2\x96\xa0 (^l\\ (o, \xc2\xb1 a) and {\xc2\xb1 a, <S) are cusps \n\nof ist species. \n\n17. ^5 \xe2\x80\x94 _y4 \xe2\x80\x94 :t:\xc2\xab, (o, o) is a double point of tan- \n\ngency. \n\n18. {y^ --c^f = x\\2.X\'\\-\'iiaf, (\xe2\x80\x94 3a/2, \xc2\xb1d) are cusps of ist \n\nspecies. \n\n19. y^{cfl \xe2\x80\x94 x^) = ^\xc2\xab. (o, o) is a double point of tan- \n\ngency. \n\n\n\n20. J = \xc2\xb1 ^4/1 \xe2\x80\x94 jf*. (o, o) is a double point of inter- \n\nsection. \n\n21. (;c \xe2\x80\x94 a)* = ()\' \xe2\x80\x94 :v)>. (d!, a) is a cusp of ist species. \n\n22. ojj/" = x^, (o, o) is a cusp of ist species. \n12,, y "= a. -\\- x \xe2\x96\xa0\\- bx^ \xc2\xb1 co^l^. (o, a) is a cusp of 2d species. \n\n24. y^ = \xc2\xab\'jc2 \xe2\x80\x94 jf*. (o, o) is a double point of inter- \n\nsection. \n\n25. O\' \xe2\x80\x94 ^ \xe2\x80\x94 ^^*)^ = (^ \xe2\x80\x94 of. (a, ^ + ^^\'^) is a cusp of 2d species. \n\n26. jj/=\xc2\xb1;c[4/\xc2\xab*+-*" / 4^^^"~-^^]\xc2\xab (Oj o) is a double point of inter- \n\nsection. \n\n27. (y \xe2\x80\x94 ^)^ = (^ \xe2\x80\x94 a)6. (\xc2\xab, /J) is a cusp of ist species. \n\n28. (jy/+i)\'^+(-^~i)^(\'^\xe2\x80\x94 2)=o. (i, \xe2\x80\x94 I) is a cusp. \n\n29. ;j/3 -|- jt^ = 2ffx\' (o, o) is a cusp of 1st species. \n\n164., Let u = f{x^y) = o be the equation in a rational \nintegral form of any algebraic curve, then (2) (\xc2\xa7 iii), \n\n^ _ _ 9^ /9f^ , . \n\ndx~ dx\'dy ^\'^ \n\n\n\nSINGULAR POINTS, 28$ \n\nAt a multiple point dy/dx has two or more equal or un- \nequal values. \n\nSince dufdx and \'du/\'dy are rational integral functions, \neach can have but one value for any set of values of x \nand y. \n\nHence, equation (i), two or more values of dy/dx require \n\n"du/dx = o and d^/dy = o (2) \n\nAny set of real roots of these equations as (r, ^), which \nalso ssitisiy /{Xf y) = o, are therefore critical for multiple \npoints. \n\n(dy/dx)(c,d) may be evaluated as in \xc2\xa7 117, otherwise (3) \n{\xc2\xa7111) gives \n\ndx\'\'^^dxdydx\'^dy\\dx) -<\'>\xe2\x80\xa2\xe2\x80\xa2\xe2\x80\xa2 V3) \n\nfrom which the two values of (dy/dx)^c, d) may be found. \n\nIf in (3) d"u/dx\\ d"u/dx dy and d\'u/dy^ vanish for \n(Cf d)j (dy/dx) is indeterminate. Then (7) (\xc2\xa7 in), \n\na^\' "*" ^-dx\'Zy dx "^ ^-dx-df \\dxl "^ df \\dxl " ^\' ^^^ \n\ngives three values for (dy / dx) (c, S). \n\nIt follows that any algebraic curve whose equation in a \nrational integral form contains no term of a degree less \nthan the second, with respect to the variables, has a mul- \ntiple point at the origin. \n\nEXAMPLES. \n\nI. u= y"^ \xe2\x80\x94 x*(i \xe2\x80\x94 x"^) = o. \n\n^u/dy = 2y = o, du/dx = \xe2\x80\x94 2^(1 \xe2\x80\x94 x*) + 2x\' = o, give \nX \xe2\x80\x94y = o. \n\n\n\n286 DIFFERENTIAL CALCULUS. \n\nHence, the origin is critical. {3) gives {dy/dx)^ = \xc2\xb1 i. \ny has double values for + 1 > ^ > \xe2\x80\x94 i, hence the origin is \na double multiple point of intersection. \n\n2. u \xe2\x80\x94 x^ -^ xy \xe2\x80\x94 y = 0, ] \n\ndu/dx = 4x^ -f 2xy = o, r give x = o =y. \ndu/dy = x\' - 3/ = o, J \n\n\'d\'u/\'dx\'\' = i2:r-^ 4- 27, a\'V9-^9j = 2^. a\'^v\'9/ = - 6r, \n\n-d\'u/dx\' = 24:^, a\' V9^\'9j^ == 2, 9\'2^/ajt: a/ = o, \n\ndW^/ = -6. \n\nFrom (4), (dy/dx)^ \xe2\x80\x94 o, and \xc2\xb1 i. Hence, the origin is \na triple point. \n\n3. ti = (4y - 2,xY - {x- 2Y/2 = o, \ndi^/dx \xe2\x80\x94 \xe2\x80\x94 24y-{-24X \xe2\x80\x94 ^x\'^/2 \xe2\x80\x94 6=0, \ndu/dy = S2y \xe2\x80\x94 24X = o, \n\na\'2/t/a^\' = \xe2\x80\x94 3"^ + 24, d^u/dxdy = \xe2\x80\x94 24, a\'V9/=32. \n\nHence, {dy/dx)(^.2, 3/2) = 3/4 \xc2\xb1 o, \n\nsince j^ = [3^ \xc2\xb1 |/(:\\: \xe2\x80\x94 2)72 ]/4. \n\n^ < 2,_y is imaginary, x > 2,y has double values. \n\nSince x > 2 gives one value of y greater and the other \nless than sx/4, the two branches are on opposite sides of \nthe common tangent y = 3X/4. Hence (2, 3/2) is a cusp \nof the first species. \n\n4. z^ =: y"^ \xe2\x80\x94 x^ \xe2\x80\x94 2ax^ \xe2\x80\x94 a^x = o. \ndt^/dx = \xe2\x80\x94 ;^x^ \xe2\x80\x94 4ax \xe2\x80\x94 d^ = o. \ndu/dy = 2y = o._ \n\n\n\ngive X = 2f \n\ny = 3/2- \n\n\n\nSINGULAR POINTS. 28/ \n\nHence, (\xe2\x80\x94 ^, o) is critical. From (i), \n\n\n\n\n\n\no \n\n\n\n<\xc2\xa7"\'^ &i?L,o,=( \n\n\n\n6-+4\'^\\ =\xc2\xb1|/^. \n\n\n\n2dy/dx /(-a,o) \n\nHence, (\xe2\x80\x94 \xc2\xab, o) is a conjugate point. \n\n5. u = y\'^ \xe2\x80\x94 2x\'^y \xe2\x80\x94 x*y + 2x^ = o. \n\'da/\'dx \xe2\x96\xa0= \xe2\x80\x94 4xy \xe2\x80\x94 ^x^y -\\- 8x\'^ = o. \n\'du/\'dy = 2y \xe2\x80\x94 2x^ \xe2\x80\x94 x" \xe2\x80\x94 o. \n\nHence jc = o =^ is a critical point. \nFrom the equation of the curve, \n\ny^i^x\'-Y x*/2) \xc2\xb1 x\' i/- 4 + 4x\' + xi2, \n\nand is imaginary when x is near zero. Hence the origin is \na conjugate point. \n\nExamine the following curves for multiple points : \n\n6. x^ \xe2\x80\x94 2)0\'Xy-\\-y^ = o. (o, o) is a double point of inter- \n\nsection, \n\n7. x*\xe2\x80\x942ay^\xe2\x80\x942>^^y\'^ \xe2\x80\x94 2a\'^x\'^-\\-a*=o. (o, \xe2\x80\x94a) and {\xc2\xb1. a, 6) are double \n\npoints of intersection. \n\n8. X* \xe2\x80\x94 2ax\'^y \xe2\x80\x94 axy\'\'\' -\\- a^y\'^ = o. (o, o) is a cusp of 2d species, \ng. jj/* \xe2\x80\x94 axy^ = \xe2\x80\x94 x*. (o, o) is a cusp of ist species. \n\n10. {x^-\\-y^y = a\'^{x\'^ \xe2\x80\x94 y"^)\' (o, o) is a double point of inter- \n\nsection. \n\n11. ay^ -f- dx^ = x^. (o, o) is a conjugate point. \n\n12. ay"^ \xe2\x80\x94 x^-\\-4ax^ \xe2\x80\x94 c^a\'^x-\\-2a^ =0. (a, o) is a conjugate point, \n\n13. X* \xe2\x80\x94 ax^y -\\- axyi -\\- a-^y^ = o. (o, o) is a conjugate point. \n\n14. y^- = x(x -\\- ay. (\xe2\x80\x94 a, o) is a conjugate point. \n\n\n\n288 DIFFERENTIAL CALCULUS. \n\n15. {y \xe2\x80\x94 2Y = {x \xe2\x80\x94 \\y{x \xe2\x80\x94 3). (i, 2) is a conjugate point. \n\n16. y\\x^ \xe2\x80\x94 a^) \xe2\x80\x94 X*. (o, o) is a conjugate point. \n\n17. x\'^ -}- x^y^ \xe2\x80\x94 tax^y -\\- a\'^y\'^ = O. (0,0) is a double point of tan- \n\ngency. \n\n18. a^y^ \xe2\x80\x94 2abx^y = x^. (o, o) is a double point of tan- \n\ngency. \n\n19. x* \xe2\x80\x94 axy\'^ = ay^. (o, o) is a triple point and a cusp. \n\n165. A Terminating Point {^oinf d\' arret) is one at \nwhich a single branch of a curve terminates. \n\nEXAMPLES. \n\n\\. y =. X log X, \n\n^ = o = jl^. y is real when jc > o, and is imaginary \nwhen X < o. Hence, the origin is a terminating point. \n\n2. y = ^~V^. \n\nAs + Jt: ^>-^ o, y ^^ o, and as \xe2\x80\x94 ^ \xc2\xbb\xc2\xbb-\xc2\xbb o, y ^-^ 00 . Hence, \nthe origin is a terminating point for the right-hand branch. \n\n3. x^ log X -{-y = xy. (o, o) is a terminating point. \n\n\n\nCURVATURE OF CURVES, \n\n\n\n2^9 \n\n\n\nCHAPTER XV. \n\n\n\nCURVATURE OF CURVES. \n\n\n\nPLANE CURVES. \n\n1 66. The Total Curvature of an Arc of Any Curve is \n\nthe angle which measures the change in direction of the motion \nof the generating point while generating the arc. \n\nLet MM\' = ds be the length of any varying arc not in- \ncluding a singular point, of any curve. At M and M\\ \nrespectively, draw the tangents \nMT and M\'T\\ Each tan- \ngent indicates the direction of \nthe motion of the generating \npoint corresponding to its \npoint of tangency. The angle \nTRT\\ denoted by (^^, in- \ncluded between the tangents \nat the ends of the arc, measures the change in direction of \nthe motion of the generating point while generating the \narc Ss, and is its total curvature. \n\nIf the extremities of an arc coincide, forming a closed \ncurve without singular points, the corresponding tangents \ncoincide, but the total curvature is 2 tt and not zero. \n\n167. The Rate of Curvature of a Curve at a Point is \nthe rate of change^ at the point, of its direction regarded as a \nfunction of its length. Thus, in the preceding figure, let ^ \n\n\n\n\n2Q0 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nrepresent the angle which the tangent MT makes with X. \nIt determines the direction, with respect to X, of the \nmotion of the generating point at M^ and, regarding ^ as a \nfunction of the length of any varying arc of the curve, as \n\n\n\nAM \n\n\n\ndip _ limit \n\n\n\n\'\'di = \n\n\n\nSs M-^ o \n\n\n\n67 \n\n\n\nis the ra^e of curvature of \n\n\n\nthe curve i- at J/". (\xc2\xa770.) \n168. Rate of Curvature of a Circle at a Point.\xe2\x80\x94 Let C \n\nbe the centre and r the radius of any circle. Then \n\nds 8s:B-^o\\_dsJ \\_rdipj r \n\n\n\n\nHence, in any circle the rate of curvature is the same at \nall points, and at any point is equal to the reciprocal of its \nradius. \n\n169. Circle and Radius of Curvature \xe2\x80\x94 When the radius \nof a varying circle decreases continuously^ the rate of cur- \nvature of the circle at any point increases continuously. \nHence, a circle may always be assumed having at all points \nthe same rate of curvature as that of any given curve at any \nassumed point. \n\nSuch a circle tangent to the curve at the point assumed, \nand having the same direction of curvature, is called the \n\n\n\nCURVATURE OF CURVES. 29 1 \n\ncircle of curvature of the pointy and its radius and centre are \ncalled, respectively, the radius and centre of curvature. It \nfollows that a radius of curvature is normal to the curve. \n\nAny chord of a circle of curvature which passes through \nthe point of tangency is called a chord of curvature. \n\n170. Curvature of a Curve at a Point. \xe2\x80\x94 Representing \nthe radius of curvature at any point of any curve by p, we \nhave \n\ndtl)/ds = i/p (i) \n\nThat is, the rate of curvature of any curve at any point is \nequal to the reciprocal of the corresponding radius of cur- \nvature, and the rates of curvature at different points are in- \nversely as the corresponding radii of curvature. \n\ndtp/ds, corresponding to any point of a curve, multiplied \nby the unit of length of s is (\xc2\xa7 68) the change that the cor- \nresponding value of ^ would undergo were it to retain its \nrate at the point over the unit of length of s. In other \nwords, dip/ds multiplied by the unit of s is the total curvature \nof a unit of length of the corresponding circle of curvature, \nand it is generally called the curvature of the curve at the \npoint or the curvature of the corresponding circle of curvature. \nIts numerical value is the same as that of the correponding \nrate of curvature, and for reasons similar to those given in \n\xc2\xa7 95 it is generally used instead of the rate. \n\nIt is important not to confound this curvature of a circle, \nwhich measures the so-called curvature of the curve atapoifit, \nwith the total curvature of an arc described in \xc2\xa7 166. \n\nf) and s must be expressed in terms of the same unit of \nlength, and at any point where p = unit of length the rate \nof curvature is unity, and the corresponding curvature of \nthe curve is a radian, which is therefore the unit of cur- \nvature. \n\n\n\n292 \n\n\n\nDIFFERENTIAL CALCULUS, \n\n\n\nEXPRESSIONS FOR RADIUS OF CURVATURE. \n\n171. From (i) (\xc2\xa7170), p=:ds/dip (i) \n\nwhich enables us to determine the rate of curvature at any \npoint from the equation * of the curve in terms of s and ^. \nThus, having s = c tan ^; for a catenary, \n\ni/p=(cos\'^)A = ^/{/ + ^\'). \n\n2. s = ctp p = c. \n\n3. s = asimp p = a cos ^. \n\n172. \'Lety=f(x) be the equation of any plane curve. \nHaving, as before, AM = j, let \n\nT^ />/" = dx. Then dx = cos ipds, \ndy \xe2\x80\x94 sin i^\'ds, tp = tan~^(^/^A:), \n\ndx\' \n\n\n\n\ndtp \n^ dx \n\n\n\ni=\xc2\xbb\'.+(i)\' \n\n\n\nor \n\n\n\nds. ~ ds/dx ~ dxy L\' ^dx) Jdx \n\nP = [i+/wTV/"W (i) \n\n\n\nfrom which the radius, and therefore the rate of curvature, \nmay be found from the equation of the curve in rectangular \n\ncoordinates. \n\nAdopting the positive value of [i +/\'(-^)\'\']\'^\'>Pwill have \nthe same sign as/"(^), which determines the direction of \ncurvature. \n\n\n\n* Called the intrinsic equation of the curve. \n\n\n\nCURVATURE OF CURVES, 293 \n\nIn general, at a point of inflexion (\xc2\xa7 161) \nf"{x) = o, or 00. \n\nHence, at such a point p is generally 00 or o. In general, \nat a multiple-point, /\'(^) has two or more different values;, \nhence, p has two or more values, one for each branch. At \na multiple-point of tangency, f\'(x) has but one value, but \nf"{x) has, in general, a value for each branch; hence, p dif- \nfers for each branch. \n\nComparing (i) with (3) (\xc2\xa7 139), we see that the centre of \ncurvature corresponding to any point of a given curve coin- \ncides with the point {x^y) (\xc2\xa7 139) whose distance from the \ncurve measured along the normal is, in general, neither a \nmaximum nor a minimum, and whose coordinates are (2) \n(\xc2\xa7 139) \n\nIt follows (\xc2\xa7 139) that the circle of curvature correspond- \ning to any point of a curve, in general, intersects the curve \nat their common point of tangency. \n\nThis is not the case, however, at any point where the \ncurvature is a maximum or a minimum, which, in general, \nincludes any point in the vicinity of which a curve is sym- \nmetrical with respect to the corresponding normal. At such \na point the curvature, in general, decreases or increases in \nboth directions; consequently in that vicinity the circle of \ncurvature is interior or exterior to the curve. \n\nTo illustrate, take the ellipse \n\nay + b\'^x\' = a\'b\\ \n\nf(xf = b*xy{ay\\ f"{x) = - ^v(^y). \n\n\n\n294 \n\n\n\nDIFFERENTIAL CALCULUS, \n\n\n\nHence, i/p = abl{a\' -\\- b"" - x" -/)\'/\xc2\xbb \n\n= - a\'b\'l{ay^ b\'xj/^ (3) \n\nAt the vertices (\xc2\xb1^, o) i/p = a/b"^^ a maximum. \n\nAt the vertices (o, \xc2\xb1 b) i/p = /^/V, a minimum. \n\nHence, at the vertices of the transverse axis the circle of \ncurvature is within the curve, at those of the conjugate axis \nit is outside, and at all other points it cuts the ellipse. \n\n\n\n\nAlso (2), x=:x-x{ay-^rb\'x^)/{a\'b\'\') = (a\'-\'b^)x\'/a\'\'; \n\n\n\n(4) \n\n\n\ny-y-y{ay-^b\'x\'\')/{a\'b\') = - {a^- b\')yyb\\ \n\nIn (3) put/=-^\'^ (a^^x\')/{a\\ and {a\'-b\')/a\'=e\\ whence \nb~a i/i^^\'. Then \n\n\n\ni/p=^:aWi-e\'/(ci\'-e\'xy/\'. . . \xe2\x80\xa2 (5) \n\n\n\nEXAMPLES. \n\n\n\nI.\' y=2X-\\-$. \n\n\n\ncf7~\\\'^ \n\n\n\n/W=4, /V) = o, p = \xc2\xab), \n\n\n\n2. x\'\'-[-y = r\\ \n\nJycf = xV(r^ -~x% fix) = rV(r\xc2\xbb-^\xc2\xbb)3/8, \n\n_/ ^^y/s / r\' _ \n\n\n\nCURVATURE OF CURVES. 295 \n\n3. yi = 2pX. \n\n^~y~^2x) / (2/^)3/3- J^ \xe2\x80\xa2 \n\nPx=o = p \xe2\x80\x94 one half the parameter. \n\n4. ity^ + 4-^\' = 64. \n\nFrom (3) and (4) (\xc2\xa7 172) we have at (2, 4/3) \n\np = - 5.86, 7= 3/8, 7= - 9I/3/4. \n\n5. //^\'-^ - ^Va- = - I. \n\n_ {a^x\'-y b\'^x\'\' - a\'^fl\'^ _ (ay -\\~ d^x^f/^ _ {e\'x\' - a\'fl^ \n^^ a^b ~ a\'b\'\' ~ ab \n\nX = \xc2\xb1 a, y = O, and p = b^/a, \n\na=b, p = {2x^ - ay/ya\\ \n\n6. ;jr\xc2\xab = 2px + r\'^xK \n\n_ \\2px + r\'x\'\' + (/ + r^-y)"]^/^ \n/\'^ \nHence (Example 3, \xc2\xa7 149), at any point of a conic, as given, p is \nequal to the cube of the corresponding normal divided by the square \nof half the parameter. \n\n\n\nT. X \xe2\x96\xa0=\xe2\x96\xa0 r vers-i (y/r) \xe2\x80\x94 \\2ry \xe2\x80\x94 y^, \n\nP = - [l + {\'^r/y - l)]3/2/(r/y) = - 2 \'\\f2^. \n\nHence (Example 6, \xc2\xa7 149), at any point of a cycloid p is equal to \ntwice the corresponding normal. \n\ny \xe2\x96\xa0=. Q) gives p = o, and y \xe2\x80\xa2= ir gives p = 4n \n\n8. >r = 4 \xe2\x80\x94 3(x \xe2\x80\x94 2)3/5. \n\n(2, 4) is a point ot inflexion at which p \xe2\x96\xa0= o. Ex. 2, \xc2\xa7 i6r. \n\n\n\n296 DIFFERENTIAL CALCULUS. \n\n9. y = 9^. At(3,i/27) ^= 13.5, 7= - VS \n\n10. ;j/2 = 8;tr. /3 = 2{x + 2)3/2/1/2. \n\n11. ^79 + ;/V4 = I. /3^=o = 9/2. \n\n12. ;fj/ = /\xc2\xab. p = (^\'^ -\\-yy/^/2m, \n\n13. jj/ = a(<fV\xc2\xab + ^-*/\xc2\xab)/2. p = /A, \n\n\n\nJir \n\n\n\n14. Jr2/3 +//3 = a^A p = 3 t^a^ \n\n\xe2\x80\xa2^ = -^ + 3 |/-^/* y=y + 3 Vx\'^y\' \n\n15. 3aV = x\\ p = (a* + xy/^l^a^x, \n\n16. _y = ^3 \xe2\x80\x94 jf\' -|- I. /j^^jyg = 00 (point of inflexion). \n\n17. ^A = sec (^p/a). Q \xe2\x80\xa2=. a sec (jf/a). \n\n18. jj/" = 6^2 ^ ^3. p = - [/ + (4^ + x^Y^y^x^y, \n\n19. jj/ = jf4 _ 4jf3 \xe2\x80\x94 i8jf2. p^^^ = _ 1/36, p;c=3 = \xc2\xab> . \n\n20. y = ae\'^/a, p = (a\xc2\xbb -\\- y\'^)^l\'^/ay. \n\n21. y = fl\'jf. j^ = (o4 -j- i5jf/4)/6)/a\'\xc2\xbb, ^ = (a*^ \xe2\x80\x94 9y^)/2a*, \n\np = (9;/4 -j- aif/z/tay. \n\n22. 3ay = 2X\\ p = \xc2\xb1 (2\xc2\xab + 3Jf)V2 |/jp/\xc2\xab ^3. \n\n23. jf = sec 2y. p = (2;*:\' \xe2\x80\x94 iy/4x. \n\n24. ;/ = loga^. p = (iJ/a\' + X^)3/2/MaX, \n\npa=e = 2 i/2. \n\n25. ;/ = jf\xc2\xbb - jf\xc2\xab + I. p^=o = - 1/2, /Oy=\xe2\x80\x9e^. = 1/2. \n\n26. ^x-\\-^y- \\/a. p = 2{x +yf/y Va. \n\n27. ay"" = x\\ p = i4a + ^xf/\'^xV^Jta. \n\n28. jj/\' = ;\xc2\xbb;V(2\xc2\xab \xe2\x80\x94 ;tr). p = a(8a \xe2\x80\x94 2>x)^/2xyy 2t{\'^a \xe2\x80\x94 ;i;)\'. \n\n29. \xc2\xab^/ = 3x^ -{- cx^y. /3o= CO. \n\n30. J = log sec X. p = sec x. \n\n\n\nCURVATURE OF CURVES. \n\n\n\n297 \n\n\n\n31. In a parabola show that any radius of curvature is twice the \npart of the normal intercepted between the curve and the directrix. \n\nN.P. \n\n32. Applying (5) to a meridian y \nof the earth, we have, since \n\n/ \xe2\x80\xa2=\xe2\x96\xa0 latitude = ^ \xe2\x80\x94 ^/2, \ntan\' / = cot\'^ ^ = i/TX^)\'. \nFor an ellipse we have \n\ntan" / = a\\a^ - x\'\')/b\'\'x\'\' = (a^ ~ ^\')/(i - ^)x\\ \n\n\n\n\nHence, x^ \n\n\n\nI + (i - e") tan^ / ~ sec\' / - / tan\' /\' \n\n\n\n, , 5 2 ^HsecV-^\'sec\'/) a\\i - e") ^ \n\nand c^ \xe2\x80\x94 e^x" \xe2\x80\x94 ^ , ^ ^ ^ = \xe2\x80\x94 ^ \xe2\x80\xa2 -f ; , \n\nsec\' / \xe2\x80\x94 / tan\' / \\ \xe2\x80\x94 e^ sinV \n\nwhich, substituted in (5), gives \n\ni/p = (i - ^\xc2\xbb sin\' /)3/V^(i - ^0- \xe2\x80\xa2 \n\n\n\n(5) \n\n\n\n\n173. In Polar Coordinates \xe2\x80\x94 Differentiating, \nX \xe2\x80\xa2=. r cos B, y ^= r sin ^, \nand substituting in (i) (\xc2\xa7 172), as in Example 11 (\xc2\xa7 115). \n\n\n\nwe have \n\n\n\n-[\'\'+S"]7["+"S)\'-\'S]- (\xe2\x96\xa0) \n\n\n\nRice and Johnson\'s Calculus, p. 355. \n\n\n\n298 \xe2\x80\xa2 DIFFERENTIAL CALCULUS. \n\nPutting r \xe2\x80\x94 i/u, whence dr/dd = \xe2\x80\x94 {i/u\'^){dii/dd)^ and \nd-\'r/de\' = {2/u\'){du/d6y - (ilic\'){d\'\'u/dd\\ and substitut- \ning, we have \n\n^=[--+g]7[-\'(\xc2\xbb+s)]- \xe2\x96\xa0 \xe2\x80\xa2 \xe2\x80\xa2 w \n\nEXAMPLES. \n\n1. r = ae. P = a(l+62)3/2/(2-j-e2)^ (a\'-fr2)3/2/(2a\xc2\xbb+ r2\\ \n\n2. r = a^. p = r 4/1 + log* \xc2\xab. \n\n3. r = asinnB. Pr=o^^ ^^/^\' \n\n4. r \xe2\x80\x942a cos 6 \xe2\x80\x94 iz. p = ^(5 \xe2\x80\x94 4 cos 6)^2/(9 \xe2\x80\x94 6 cos G). \n\n5. ?- = \xc2\xab(i \xe2\x80\x94 ^^)/(i \xe2\x80\x94 e cos 6). \n\np = \xc2\xab(i - ^2)(i _ 2^ cos 6 + ^\'^)VV(i \xe2\x80\x94 ^cos 9)3. \n\n6. /\'\'^ cos 20 = \xc2\xab2. p = \xe2\x80\x94 rYa^. \n\n7. r = a(2 cos \xe2\x80\x94 i). p = ^(5 \xe2\x80\x94 4 cos 0)V2/(g \xe2\x80\x94 6 cos 0). \n\n8. /- COS^ (9/2) =/. p = 2^3/2/ V/. \n\n9. r = fl! sec\'^ (0/2). p = 2a sec^ (0/2). \n\n10. r = a0-*. p = K4\xc2\xab*+^\')^/V2\xc2\xab\'(4\xc2\xab^ \xe2\x80\x94 i^*). \n\n11. r\'Bz=za\'^. p = ri4a^ -\\-r^)y^/2a\\4a^ - r% \n\n12. r = 4a sin^ (0/2). \nFrom which and \xc2\xa7 150, \n\ndr/{raB) = cot (0/2) = cot 0. .*. 0/2 = <p. \n^ = 0-1-0 = 30/2, and dip = 3de/2. \nTherefore, \np = ds/dip = {2/3){ds/de) = (2/3)?- cosec = (8/3 )\xc2\xab sin (0/2). \n\n13. rQ = a. p = r(a^ + r^f/ya^ = a(i -\\- Q^f/yQ\\ \n\n14. r^ = d^ cos 20. p = aVsr. \n\n174. From \xc2\xa7 172 we have \n\ndx = cos tpds, and ^ = sin tpds. \n\nDifferentiating without assuming the independent vari- \nable, and writing ds/p for dip^ we have \n\nd^\'x = cos r/^\'d^s \xe2\x80\x94 sin t/:{dsY/p, \ndy = sin ipd^s + cos tp{dsy/p. \n\n\n\nCURVATURE OF CURVES. 299 \n\nSquaring and adding, we obtain \n\n\n\nHence, p = ds\'\' / V {_d^\' x)\' + {d^yf - (^V)^ . . . (i) \n\nwhich is a general expression for p. \n\nRegarding s as the independent variable d\'^s = o, and (i) \n\n\n\nbecomes p = ds\'/V{d\'xy + (^\xc2\xbb% \n\nfrom which i/p\' = (d\'x/dsy + {dy/ds\'y, ... (2) \n\nwhich is a convenient form when x and j are given as func- \ntions of s. \n\n175. In (i) (\xc2\xa7 172), ^ is the independent variable. Sub- \nstituting (dxdy \xe2\x80\x94dyd\'\'x)/dx\' for d\'^y/dx^^ we have (\xc2\xa7 115, \nExample 11) \n\np = (dx" -f dyy/y{dx dy - dy d\'x), . . (i) \n\nin which neither x nor ^ is independent. This form is con- \nvenient when X and y are given as functions of a third vari- \nable. Thus, having \n\nI. x = a(0 \xe2\x80\x94 sin 0), y = a{i \xe2\x80\x94 cos 0), \n\nwe have, as in Example 12 (\xc2\xa7 115), p = \xe2\x80\x94 4^ sin (<p/2), \n\n!x = c sin 26(1 + COS 29). \njj/ = iT cos 26(1 \xe2\x80\x94 cos 20). \n\n\n\np = 4<r cos 39. \n\n\n\nI y = 6 sxn B. \n\n!x = a{2 cos 9 -{- cos 29). \ny = a{2 sin 9 - sin 29). p = - Sa sin (39/2). \n\n{ x = {a + 6)cosQ - d cos ^-4^9. \n\n5. i , . ^ = , / sin -9. \n\nJ/ = (a + ^) sin 9 \xe2\x80\x94 ^ sin \xe2\x80\x94 ! \xe2\x80\x94 9. \n\n\n\n300 DIFFERENTIAL CALCULUS. \n\n176. p in Terms of r and p.\xe2\x80\x94 From \xc2\xa7 150. \nHence, \n\n\n\ndp_ \ndr \n\n\n\n=[\'\'+-(S)\'-\'\'\xc2\xbb]/[\'\'+sr- \n\n\n\n\nComparing with (i) (\xc2\xa7 173), \nwe have \n\n\n\np = rdr/dp. \n\n\n\n(0 \n\n\n\nLet C be the centre, and \nCM = p, the radius of cur- \nvature corresponding to M. \nLet c = MN represent the \nchord of curvature which \npasses through P, and PM = r. Then \n\nc=2p cos PMC = 2P sin OMP = 2pj>/r = 2pdr/dp. \n\nEXAMPLES. \n\n1. p:=ar, p = r/a, c \xe2\x80\x94 ir. \n\n2. ar =p\\ p = ip^ld" = 2(rVa)i/a, c = 2^V\xc2\xab =2r. \n\n3. r^ = ap. p = a/2, c \xe2\x80\x94 r, \n\n4. a> _|_ 32 _ y3 = a23V/>. p = a2^Vi>^ \n\n5. r^=d^p, p = a^/sr, c = 2^/3. \n\n6. rS = 2fl!/^ <^ = 4^/3- \n\n7. r\'w+i =\xc2\xab\xc2\xbb\xc2\xab/. /3 = ^V(^^ + i)/\xc2\xbb \'^ = 2r/(w + i). \n\n\n\n8. f = r^ \xe2\x80\x94 \xc2\xab2. \n\n\n\nA>=A \n\n\n\n2f/r. \n\n\n\nCURVAI URE OF CURVES, \n\n\n\n301 \n\n\n\n9. r^ = \xc2\xab2 cos 29. \n\nPutting ?*-* = \xc2\xab, we have a^u^ = sec 20. \n\nHence, du/dB = u tan 20, m^ _|_ ^^^2/^92 = a^u^ i// = a*/r\\ and \n<^/^r = z^\'^/a^. Therefore, p = a\'/sr, and c = 2^/3. \n\n10. ;\' = 2a{i \xe2\x80\x94 cos 0). ft = Sa sin (0/2)/3 \xe2\x80\x94 4V\xc2\xabV3, <^ = WS- \n\n11. r = a(i + cos 0). p = 24/2^^/3, ^ = 4,V3- \n\n12. ;- = aB. \n\ndr[dB = a; also (\xc2\xa7 150), ^V\'^\'^ = rj/r^ -/\'//. \nHence, r\\^r^ -p\'^/p = a, or r* - ^V^ = a^\'. \n\nFrom which ^^r/^ = /(r=^ + flt\xc2\xab)/2(2r\'\' - /). \n\nBut /\' = rV(^\' + \xc2\xab\'). Therefore, p = (r\xc2\xbb + a\xc2\xbb)3/2/2(r\' + 2a\xc2\xbb). \nThe chord of curvature through the pole \n\n= 2pdr/dp = 2pHr\'^ + a^)/2r{2r^ -p^) = r(r\'^ + a\xc2\xbb)/(^\xc2\xab + 2a\'). \nHence, /O = ^ when (^r\' + ^2)3/2 = 2r(r2 + a\') or ?- = a/4/3. \n\nCURVES OF DOUBLE CURVATURE. \n\n177.* Let MM^ = Ss be the length of any varying arc \nnot including a singular point, of any curve of double \ncurvature. At M and J/\' re- \nspectively draw the tangents \nMT and M\' T\\ and through \nthe origin draw the two right \nlines OE and OB^ parallel to \nthem respectively. Upon each \nlay off a length /, and join the \nextremities jS and jE\' by the \nright line EE\\ forming an \nisosceles triangle in which the \n\nangle ^6>^\', designated by St/;, measures the fofal cnrva.- \nture of the arc Ss. (\xc2\xa7 166.) \n\n* Modification of method in Calcul Differentiel, par J. Bertrand, \npage 614. \n\n\n\n\n302 DIFFERENTIAL CALCULUS. \n\nThe rate of curvature of the curve at M is (\xc2\xa7 167) \n\n\n\nds \n\n\n\n= ,"""\' m (X) \n\n\n\nLet \xc2\xab\', A r* ^^^d a + 6a^ ft -[- ^y5, ;/ + dy, represent the \nangles which the tangents MT and M\' T\' make respec- \ntively with the coordinate axes. \n\nFrom the triangle EOE\' we have \n\n\n\nEE\' /l= 2 sin (<^^/2); hence, \n\n\n\nlimit \n\nd^ B->0 \n\n\n\n[5^/.*] =,\xc2\xab.[.,\xe2\x80\x9e (\xc2\xbb)/\xc2\xab]=.. ,., \n\n\n\nThe coordinates of E and -\xc2\xa3"\' are respectively /cos o\', \n/cos/?, /cosy, and / cos (<ar + c^^\'), /cos (/? + (J^), \n/ cos (y + ^?^)- Substituting in formula \n\nD = V[(:c" - x\'r + (/\' -yy + (z" - z\'Yl \n\nand dividing by /, we have \n\nEE\'/l = i/([cos {a + 6 a) - cos o\']" \n\n+ [cos {(3 + d/?) - cos ftY + [cos (y + (J;/) - cos yY)- \n\nDividing by ds, taking the limit as 6s b-\xc2\xbb o, whence \ndipM-^o, substituting Sip for EE\' /l, (2), we have (i) and \n\xc2\xa7170 for the rate of curvature at M, \n\n\n\nlimit r^"|_ 4// ^ cos <y y / ^cos /? y /^_cosj^y \n\n(3) \n\n\n\n(5 \n\n^^ I \n\n\n\nds p \nFrom \xc2\xa7 88, \n\ncos a = dx/dsy cos /? = dy/dsy cos >^ = dz/ds. \n\n\n\nCURVATURE OF CURVES. 303 \n\nDifferentiating and substituting in (3), we have \n\nin which s is the independent variable. \n\nIn order to obtain a more general expression for i/p, \nplace (\xc2\xa7115) \n\nd\'^x _ dsd\'^x \xe2\x80\x94 dxd\'^s d\'^y _ dsd\'^y \xe2\x80\x94 dyd\'^s \n\n~dl ~ ds\' \' \'dF~ ds\' \' ^^^\'^ \n\ngiving \n\n/{dsd\'x - dxd\'sf + {dsd\'y - dyd\'sY \' \n\nL= / -^-idsd\'z-dzd\'sY \n\nP V \'d? \n\nwhich may be written \n\n/ds\\{d^xy -\\- {d^yY + {d^zy\\ \n\n1 ^ / -2dsd\'s{dxd\'x+dyd\'y+dzd\'\'z)^-{d"\'sY(dx\'\'-Vdy\'\'-^dz\'\' ) \n\np V ^^\' \n\nFrom \xc2\xa7 88, ds" = dx\' 4- dy\' -\\- dz\\ \n\nWhence, dsd\'^s = dxd\'x + dydy-{- dzd^z. \n\nTherefore, \n\n\n\np ~ ^ ds^ * \xc2\xb0 ^5; \n\nwhich is a general expression for the rate of curvature at \nany point of any curve. \n\nIf the curve is of single curvature, its plane may be taken \nas that of XV, z will be zero, and (5) reduces to (i) (\xc2\xa7 174). \n\n\n\n304 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nCHAPTER XVI. \n\n\n\nINVOLUTES AND EVOLUTES. \n\n\n\n178. Each point of any given curve as MM\'M"y has, \nin general, a centre of curvature. \nThe locus of the centre of \ncurvature of any given curve is \ncalled its evolute. Thus, CC\'C \nis the evolute of MM\' M". \n\nThe given curve ^^ J/ \'J/\'\' is \ncalled an involute of its evo- \nlute. \n\n179. Coordinates of the Centre \nof Curvature.\xe2\x80\x94 Let C be the \ncentre, and CM = p the radius of curvature corresponding \nto My whose coordinates are x,y. Then \n\n\n\n\n\nOA = OP \xe2\x80\x94 AF, or x = x \xe2\x80\x94 psmil^\\ \nAC = AB -^ BC, or \'y = y -\\- p cos. tp , \n\n\n\n. . (i) \n\n\n\n^, and (\xc2\xa7172) J = \n\n\n\nINVOLUTES AND E VOLUTES, 305 \n\nwhich correspond with (2), \xc2\xa7 172, since (\xc2\xa7 170) \n\ndx/ds = cos ip, dy/ds = sin ^, giving \n\n\xe2\x80\xa2 . _ ds^dy _ dy _dy dx ^dy V /^V~l /^^-^ \n^^^"^"^ ~d^Js~dtp ~~dx\'d^ ~^L W J/^^\'^\' \n\n180. Differentiating (i), \xc2\xa7 179, with respect to s, we have \n\ndx/ds = dx/ds \xe2\x80\x94 p cos ipdip/ds \xe2\x80\x94 sin ipdp/ds. \n\n\n\n(i) \n\ndy/ds \xe2\x80\x94 dy/ds \xe2\x80\x94 p sin rpdip/ds + cos ^dp/ds. \nBut (\xc2\xa7\xc2\xa7170, 172) \n\n^^y^j = i/p, cos ^ = dx/ds, sin ^ = ^/^^. \n\nTherefore, dx/ds = \xe2\x80\x94 sin ^dp/ds, \n\n\n\ndy/ds = cos il\'dp/ds. \n\nHence, by division, dy/dx = \xe2\x80\x94 oot tp. \n\nRepresent the angle which a tangent to the evolute at \nyXf y) makes with X by ^\'r, then, \ndy/dx = tan ^\\ = \xe2\x80\x94 cot i/j = \xe2\x80\x94 dx/dy, or ^, = ^ + 7r/2. \n\nTherefore, the tangent to the evolute at (>x, 7) is normal \nto the involute at {x,y); or, in other words, the radius of \ncurvature of any curve at any point is tangent to the evo- \nlute at the corresponding centre of curvature. An evolute \nmust therefore be drawn tangent to all radii of curvature of \nthe involute. An evolute is therefore the li\'mi\'f of the locus \nof points of intersection of adjacent normals to the involute, \nas the number of normals corresponding to any definite \nportion of the involute is increased without limit. \n\n\n\n3o6 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nIt follows that a radius of curvature which is unlimited \nin length is an asymptote to the evolute. Hence, in general, \nthe normal at a point of inflexion is an asymptote to the \nevolute. \n\nAlso, when in the vicinity of any point a curve is sym- \nmetrical with respect to the normal at the point, the corre- \nsponding point of the evolute is a. cusp. \n\nl8i. Squaring both members of (2), \xc2\xa7 180, and adding \neach to each, we have \n\n{dx\' + d folds\'" = dp\'/ds\\ Hence, dp = \xc2\xb1 Vdx\'-\\- d/- \nLet s represent the length of a varying portion of the \nevolute, then (\xc2\xa7 87) ds = ^ dx^ -\\- dy^. Hence, dp = ds^ and \n(\xc2\xa7 74) p = -y \xc2\xb1 ^, in which dJ is a constant. \nM" \n\nLet C\'M\'= p\\ and C"if"= \n\xe2\x80\x9e p", be the radii of curvature \ncorresponding to M\' and M\'^ \nrespectively. Measuring the evo- \nlute from C, and denoting the \nlengths of the arcs CC and CC \nby j-j and ^2 respectively, we have \n\n\n\n\np\' \xe2\x80\x94 s^-{- a; \n\n\n\np" = s,-{-a. \ns, = arc C\'C\'\\ \n\n\n\n(i) \n\n\n\nHence, \n\np"_p\' = j, -^^ = arc C\'C", . . . (2) \n\nthat is, in general, ^Ae difference between any two radii of \ncurvature of an involute is equal to the arc of the evolute be- \ntween the correspondifig centres of curvature. \n\nExceptions exist when the arc of the evolute includes a \nsingular point or is discontinuous. \n\n\n\nINVOLUTES AND EVO LUTES. 307 \n\nMeasuring the evolute from C, we have \n\narc CC = p\' \xe2\x80\x94 p, or p\' = arc (TC -4- p; \nalso, arc CC" = p" - p, or p" = arc CC" + p. \nHence, (i), a =^ p. \n\nSimilarly it may be shown that the constant a in equation \n(i) is, in general, equal to the radius of curvature which \npasses through the point of the evolute from which it is \nmeasured. \n\nIt follows that as a right line rolls tangentially upon, or as \na string is unwound from, any curve, each point describes \nan involute to the given curve as an evolute. Hence, \nwhile an involute has but one evolute, each evolute has an \nunlimited number of involutes. Any two involutes corre- \nsponding to the same evolute are separated by a constant \ndistance measured along the normals, and are called parallel \ncurves. \n\n182. In general, at cusps of the first species and at \npoints of inflexion p changes sign. Hence, at such points \nwe have p = o, or 00 , and the evolutes pass through these \npoints or have infinite branches to which the corresponding \nnormals are asymptotes. \n\n\n\n\n308 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\n\nAt a cusp of the 2d species p does not change its sign, \nand the corresponding point of the evolute will, in general, \nbe a point of inflexion or a cusp of the 2d species. \n\n\n\n\n183. Radius of Curvature of an Evolute .\xe2\x80\x94The angle \nM" ^^> between any two tangents \nas those at M\' and J/", is equal \nto that between the correspond- \ning radii of curvature to the in- \nvolute, and since these radii of \ncurvature are tangents to the \nevolute, the angle which they \nmake with each other is equal \nto C"OC\\ included between \nthe corresponding radii of cur- \nvature of the evolute. \n\n\n\n\nINVOLUTES AND EVOLUTES. 309 \n\nLet J", = CC\\ Ss\\ = C\'C" ^ and pi = radius of curvature \nof evolute at C , \n\nThen(\xc2\xa7x7o) p. = ^H^^r|]. \n\nBut Ss^ corresponds to d^ = M\'M" of the involute, \nand vanishes with it. Also (\xc2\xa7181), Ss^ \xe2\x80\x94 f)" \xe2\x80\x94 p\' = Sp. \n\nHence p - ^\'"^\'^ T^l - ^ _ ^ f^^ \n\nin which s is the length of an arc of the involute measured \nup to M\' and tp is the angle which the tangent at M\' \nmakes with a fixed right line. \n\n184. Equation of the Evolute \xe2\x80\x94 Let \n\ny=f{^) <0 \n\nbe the equation of any given plane curve. The coordinates \nof the centre of curvature for any point {x^ y) of the curve \nare (i), \xc2\xa7172, \n\n^-[i+7V)VW//"W;, \n\n(2) \n\n\n\ny =j^+(i+/\xc2\xbb)//"W. \n\nExpressions for /\'(^) and /"(^) in terms of x andjj^, \nobtained by differentiating (i). substituted in (2) give x \nand y in terms of x and y. Combining these equations \nwith (i), eliminating x and y, we have y = -F{x) for the \nevolute oiy =/(x). \n\nEXAMPLES. \n\nI. y^ = 2px. \n\ndy/dx = p/y, d\'y/dx\' = - ///. \n\nSubstituting in (2), we have \n\n\n\nX =^ X \n\n\n\n\n\n\n310 \n\n\n\nDIFFERENTIA L CA LCUL US. \n\n\n\n\nCombining with y" = 2px, and eliminating x and y \nwe have \n\nfor the evolute of the parabola, \n\nCCC" is the evolute of the parabola C OC. \n\ny =z O \nives X =\xe2\x96\xa0 p = OC, \nx= 4p = OP \ngives \n\ny = \xc2\xb1pVS= \xc2\xb1 PC" \n\nfor points common to the pa \nrabola and its evolute. \n\nThe arc CC"^ (3 4/3 - i)/. \nTransferring the origin to C, the axes remaining parallel, \nwe have, denoting the new coordinates by x and j, \n\n^ = / + ^, y \xe2\x80\x94y- \n\nHence, we have ^ = ^x^/{2\']p) for the evolute. The \nbranch CC belongs to OC and CC to OC. \n\nLet r = FM\' represent the focal distance of any point \nas M\\ and let / = FY represent the perpendicular from \nthe focus to the tangent TM\'. \n\nThen FY\' = FM\' X FO, or P = pr/2. \n\nHence, 2ldl = pdr/2, or dr/d/ \xe2\x80\x94 4//^. \n\nFrom \xc2\xa7 176, c = chord of curvature through F^ \n= 2/dr/d/. \nHence, c = SP/p = Zpr/2p = 4r = 4FM\' . \n\nThat is, in a common parabola the chord of curvature \nthrough the focus is equal to four times the focal distance \nof the point of tangency. \n\n\n\nINVOLUTES AND E VOLUTES. \n\n\n\n311 \n\n\n\n2. ay + \'^V = ^\'^\'\xe2\x80\xa2 \n\ndy/dx = - /?\'x/ay, d\'y/dx^ = - b\'/ay. \n\nSubstituting in (2) and reducing, we have \n\n^ = (\xc2\xab^ - b\')x\'id; :. X = (a\'x/[a\' - d\'])yK \n\nJ = - {a^ - b^)yyb^; .\'. y = - {b^/W " ^Y^^- \nCombining with a\'y\'^ + b\'^x\'^ = a^h^, we have \n\n{^xY"> + {b\'yf^\'\' = (a\' - b\'f\'^ \nfor the evolute of the ellipse. \n\n\n\n\nCCC\'C" is the evolute of the ellipse MM\'X; x = o \ngives J = {a\' - b\')/b= OC" . \n\nWhen e" = {a\'\' \xe2\x80\x94 b\'\')/d\' =1/2, we have a \xe2\x80\x94 2b\\ and \nOC" = b, in which case the vertices C" and C are on \nthe curve. They are without (as in the figure) or within \nthe ellipse according as e"^ is greater or less than 1/2. \n\npleasuring the evolute from C, we have p = MC = b\'^a, \nand p\' = M\'C\' = a\'/b. \n\nHence, arc CC \xe2\x80\x94 a\'^/b \xe2\x80\x94 b\'^/a = (a \xe2\x80\x94 b>^)/ab. \nAxis CC" = 2a - 2b\'\' I a = 2{ar- \xe2\x80\x94 b")/a. \nAxis CC" = 2^ -h 2{a\'\'/b - 2b) = 2{a\' - b^b. \n\n\n\n312 \n\n\n\nDIFFERENTIAL CALCULUS, \n\n\n\nHence, the axes of the evolute are inversely as the cor- \nresponding axes of the ellipse. \n\n\n\n3. ^ = r vers\'-^ (jj\'/r) \xe2\x80\x94 S/ 2ry \xe2\x80\x94 y^, \n\ndy/dx = {2rly - i)V2, d\'^y/dx\' = - r/y\\ \n\nSubstituting in (2), reducing and combining with the \ngiven equation, we have \n\nX =^ r vers"^ (\xe2\x80\x94 y/r) + ^ \n\n\n\n2ry \xe2\x80\x94 y \n\nfor the evolute of the cycloid. \n\nProduce BA, making AO\' = BA = 2r. \n\nB \n\n\n\n(a) \n\n\n\no/_^ P \n\n\nr\\ \n\n\n)^ \n\n\np\' \n\nX ^ \n\n\n\n\n/o\' \n\n\n\nOM\'0\'0" is the form of an evolute of a cycloid. The \nbranch 00\' belongs to OB and O\' O" to BO", \n\nTransferring the origin to (?\', taking O\' X\' and O\' A as \nthe new coordinate axes, denoting the new coordinates by \nX and_y, we have for any point of the branch 00\\ as M\\ \n\nx = OA- O\'P\' = 7rr-x,y=:A0\' -{\xe2\x96\xa0 P\' M\' = - 2r-\\-y, \nwhich substituted in (a) give \n\n\n\nX =z 7Tr \xe2\x80\x94 r vers \' [(2^ \xe2\x80\x94 y)/r^ \xe2\x80\x94 y 2ry \xe2\x80\x94 J^ \nBut 7ir \xe2\x80\x94 r vers"^ [(2r \xe2\x80\x94 y)/r~\\ = r vers~^ (jA)* \n\n\n\nHence, x \xe2\x80\x94 r vers ^ O\'/\') \xe2\x80\x94 V 2ry \xe2\x80\x94 y \n\n\n\nINVOLUTES AND EVOLUTES. 313 \n\nis the equation of O\'M\'O referred to the new axes. It is \nof the same form and contains the same constants as the \nequation of the cycloid ; hence, the evolute of a cycloid is \nan equal cycloid. \n\nAt O, p \xe2\x80\x94 o, and at B, p\' = O\'B \xe2\x80\x94 4^. \nHence, arc OM\'O\' ^ 2iYC OB = p\' - p = ^r. \n\nTherefore, arc OBO" =^ Sr, that is, l/ie length of one \nbranch of a cycloid is equal to four times that of the diameter \nof its generating circle. \n\n4. x^ -\\- y^ = B\\ X = y = o. \n\n5. ay - b\'x^ = - a^b\\ {axfl\'\' - (^j\')^/^ --= (a\' + b\'fl^. \n\n6. x"^ = 4ay. x^ = 4{y \xe2\x80\x94 2ay/2\'ja. \n\n7. xV^ +//3 = aV\\ (^+ J)^/\' + {x -}f/^ = 2^2/3, \n\n8. 2xy = a\\ (x + j\')^/^ - (x - \'yf\'\' = 2^2/3^ \n185. Equation of Evolute in Polar Coordinates.\xe2\x80\x94 Let C \n\nbe the centre of curvature corresponding to J/ of a curve \nreferred to /* as a pole, and PX as a fixed right line. Draw \n/Tand PTx perpendicular to MT and MT^ respectively. \n\n\n\n\nT \n\nLet PM=r, PC^r., PT=p, and PTx=p,. \nThen r/ rz p^ -|- r\' - 2rp cos PMC. \n\n\n\n3H DIFFERENTIAL CALCULUS. \n\nBut r cos PMC \xe2\x80\x94 r sin PMT = p. \n\nHence, r^^ = p" -\\- r"" \xe2\x80\x94 2pp. ..... (i) \n\nAlso, \xe2\x96\xa0 p,-\' = TM\' = r"" - p""; (2) \n\nand (i), \xc2\xa7 176, p \xe2\x80\x94 rdr/dp (3) \n\nFrom the equation of the curve find p in terms of /\', \ngiving \n\n/-^W . (4) \n\nBy eliminating r, p, and/, there results an equation be- \ntween r^ and p^ for the evolute. Thus, having r = a^, \nthen / = cr^ in which c = 1/ i^i-{- log\'\xc2\xab. (Ex. 3, \xc2\xa7 150.) \n\ndp = cdr, dr/dp \xe2\x80\x94 i/c. \n\nHence, f> = r/c, p^^ = r"" - c\'r\' = r\\i - c"), \n\nand r^\' = r\'/c\' + r\' - 2r* = r\\i -^ c\')/c\' =p,yc% \n\nor p^^ = c^r^ and /, = cr^^ \n\nfor the evolute, which is a logarithmic spiral similar to the \ngiven curve. \n\n186. Having p in terms of ?/\', equation (i), \xc2\xa7 183, enables \nus to express p, in terms of tp^. Thus, \n\nr = ^a sin\' (^A). \n\nExample 12, \xc2\xa7 173. When tp = 3^/2, \n\np = Sa sin (0/2)/^. \nHence, p = 8^ sin (^/3)/3. \n\n(0, \xc2\xa7183, \np, = ^/p/V/^ = 8\xc2\xab cos (?/V3)/9 = 8^ sin (7r/2 + ^V3)/9- \nLet y^\\ represent the angle which p makes with X, \n\n\n\nINVOLUTES AND E VOLUTES. 315 \n\nThen ^ = ^, - 7t/2, and 11/2 + ^\'/a = (/\'^ + ;r)/3. \nHence, Pj = 8^7 sin [(^^ + 7r)/3]/9. \n\n187. Equation of an Involute \xe2\x80\x94 Combine the equations \n\n~y = ^W, (i) \n\ndy/dx = \xe2\x80\x94 dx/dy^ (2) \n\n{x \xe2\x80\x94 x)^{y\xe2\x80\x94y)dyldx-o, , ... (3) \n\neliminating x and J^^, and there will result in general a differ- \nential equation involving x and ji^. \n\n188. Involute of a Circle.\xe2\x80\x94 Let a = radius of circle, \nand let A be the initial position of generating point. Let \nTB = circum. AT h^ any position of the tangent rolling \nupon circum. AT. Take origin and pole at O. Let \n\n\n\n\n^ = angle AOT^ and let B = angle AOB, which radius \nvector r makes with X. Then tangent TB \xe2\x80\x94 aff^\', and we \nhave, for the rectangular coordinates of any point, as B, by \nprojecting the two lines OT and TB upon X and F re- \nspectively, \n\nX \xe2\x80\x94 OB = a cos ^ + , , \n\ny=OQ=^a sin tp -\' \'" ^ \xe2\x96\xa0 \' \n\n\n\n^^^ sin tpy ) \narp cos ip. ) \n\n\n\n3l6 DIFFERENTIAL CALCULUS. \n\nIn polar coordinates we have, from the right triangle \nTB, \n\n\n\nVr\' -a\' = aip. \nBut 6* = ^ - angle BOT = tp - sec"^ {r/a). \nHence, aip = a6 -\\- a sec\'^ (V^)\xc2\xbb \n\n\n\nand y /\xe2\x80\xa2\' \xe2\x80\x94 a^ =aO -j^ a see"* (r/a) (2) \n\n\n\nORDERS OF CONTACT OF CURVES. \n\n\n\n317 \n\n\n\nCHAPTER XVII. \n\nORDERS OF CONTACT OF CURVES AND OSCULATING \n\nLINES. \n\n189, Let y = f{x) and y = ^{x) be the equations of any \ntwo given lines, as BB and Z>Z>, which have in common a \n\n\n\n\nY \n\n\nD M \n\n\n\n\nD \n\n\n\n\n/ \n\n\nB \n\n\n\n\nB \n\n\n\\ \n\n\n\n\n\n\n\nt \nP \n\n\nP \n\n\np" \n\n\n\n\n\npoint M, whose abscissa is OP = a, and whose ordinate is \n\nPM^b=Ad) = <p(a). \n\nIncrease a by an infinitesimal A = PP" ^ and we have for \nthe corresponding ordinates, P" B and P" D^ designated by \ny andy respectively (\xc2\xa7 124), \n\n/ =Aa + h) =f{d) +/\'{a)h +/"(a)h\'/\\2_+ ... (i) \n/\' = (t>{a + h) = <t>(d) + 4>\'{a)h + ct>"{a)h-\'/\\2 + ... (2) \nSubtracting (2) from (i), member from member, we have \n\n\n\ny -y \n\n\n\nDB=[f\'(a)-4>\'{ay\\h \n\n\n\n+ [/"(\xc2\xab) -0"WF/|l+... (3) \nWhen /\'(\xc2\xab) = 4>\'(a), \n/ - /\' = DB = [/"(\xc2\xab) - 4>"{aW/\\2 \n\n+ [/\'"W - 0"\'(\xc2\xab)F/|3|+ \xe2\x80\xa2 \xe2\x80\xa2 . (4) \n\n\n\n3l8 DIFFERENTIAL CALCULUS. \n\nthe lines are tangent to each other, and are said to have a \ncontact of at least the first order. \nWhen, also,/ "{a) = (p\'\'{a), \n\n/ - /\' ^nB= [/"\' W - 0\'" W]/^y|3 + . . . (5) \n\nthe lines have a contact of at least tke secoiid order. \nV^htn,alsoJ\'\'\\a) = cp"\\d), \n\n/ - /\' = DB = [/-^^(a) - 0iv(^)]^y|4 + . _ (6) \n\nthe lines have a contact of at least f/ie third order ; and, in \ngeneral, the order of contact of any two lines having a point \nin common is denoted by the greatest number^ beginning \nwith the first, of successive derivatives of their ordinates \ncorresponding to the common point, which are, respectively, \nequal each to each. \n\nDenoting the order of contact of any two lines by n, we \nhave also f^ia) \xe2\x80\x94 0"(^), making with f{a) = (p(a), n -\\- 1 \nconditions, and giving \n\nFrom which we see that when {n + i) is odd, the sign \nof y \xe2\x80\x94 y" = DB will change when the sign of h changes, \nthat is, if D is above B when h is negative and vanishing, \nit will be bl;low B immediately after h becomes positive. \nThe lines will then intersect, as shown in Fig. i. When \n{n -\\- i) is even, the sign of j\' \xe2\x80\x94 y" \xe2\x80\x94 DB will not chanee \nwith that of h, and the lines will not intersect. (See Fig. 2 ) \n\nHence, when the order of contact is even, lines intersect at \nthe common pointy a7td when it is odd, they do not. \n\nTo illustrate, take the two equations \n\n/==4^, . . . (i) and (^ - 5)^ -{- (7 + 2)^ = 32. (2) \n\n\n\nOSCULATING LINES. 319 \n\nCombining, we find that the point (i, 2) is common, \n(i) gives/\'(i) = I, /"(i) - - 1/2, and f"\\i) = 3/4. \n(2) gives 0\'(i) = I, 0"(i) = - 1/2, and 0\'"(i) = 3/8. \n\nHence, the circle (2) has a contact of the second order \nwith the parabola (i), and intersects it at the point (i, 2), \n\nDetermine the common points, and order of contact at \neach, in the following pairs of lines : \n\n\n\n\\y = x-\\- I. \n\n\n\n, (i, 2) m common. \nAns. \xe2\x96\xa0{ ^ \n\nContact of ist order. \n\n\n\n2. \\ , Ans. (i, i) 2d order. \n\nly = 3x \xe2\x80\x94 sx-\\ri. \n\n{Ay =\xe2\x96\xa0 x"^ \xe2\x80\x94 4. \no , u , Ans. (o, \xe2\x80\x94 i) 3d order. \n\nr + ^ = 2)^ + 3- \n\nTwo lines having at a common point a contact of the \xc2\xabth \norder with a third line, have a contact with each other of \nat least the ;^th order. \n\n190. Osculating Lines. \xe2\x80\x94 The line of any species of line, \nwhich at a given point of a given line has the highest pos- \nsible .order of contact, is called an oscidatrix or osciilatiiig \nline. Thus, the circle which at the given point has the \nhighest possible order of contact is called an osculating circle. \nThe parabola of highest contact is called an osculating pa- \nrabola. \n\nTo determine the equation of an osculatrix at a given \npoint of a given line, assume the general equation of the \nspecies of line in its reduced form. \n\nThe problem then is ta determine such values for the \narbitrary constants contained therein as will cause the re- \nquired line to have the highest possible order of contact. \n\n\n\n320 DIFFERENTIAL CALCULUS. \n\nSince the osculatrix must pass through the given point, \nsubstitute its coordinates in the general equation, giving \none equation between the required quantities, and diminish- \ning the number of arbitrary constants by unity. \n\nFrom the general equation of the species and the equa- \ntion of the given line determine expressions for the succes- \nsive derivatives of the ordinates to include those whose \norder is denoted by the number, less unity, of the constants \nin the reduced form of the general equation of the species. \n\nSubstitute the coordinates of the given point in each, and \nplace the results corresponding to derivatives of the same \norder equal to each other. The resulting equations with \nthe one before obtained will equal in number the required \nquantities, which in general may be determined. Their \nvalues substituted in the general equation will give the re- \nquired osciilat7\'ix. The order of contact will in general be \ndenoted by the number, less unity, of arbitrary constants \nentering the general equation, but in exceptional cases it \nmay be higher. \n\nEXAMPLES. \n\nI. Find the equation of the osculating right line to the \nparabola y = 9^ at the point (i, 3). \n\n\\x\\ y z=z ax -\\- b substitute the coordinates (i, 3), giving \n\nZ = a-\\-b (i) \n\nFrom,)\' \xe2\x96\xa0= ax -\\- b y^Q find 0\'(^) = a. \nFromy = 9^ we find/\'(jc) = 9/27. \n\nSubstituting the coordinates (i, 3) in each, and placing \nthe results equal to each other, we have a = 9/6 = 3/2, \nwhich in (i) gives b = 3/2. Hence, ^ = 3^/2 + 3/2 is the \n\n\n\nOSCULATING LINES. 321 \n\nequation of the required line, which is tangent to the \nparabola. \n\nSince the general equation of a right line contains but \ntwo arbitrary constants, it cannot, in general, have a con- \ntact of an order higher than the first with a plane curve. \n\nAn exception exists at a point of inflexion where, in \ngeneral, for both the curve and the right line, we have, de- \nnoting the abscissa of the point by ^, f\'ia) = o. The \ncontact is, therefore, at least of the second order. \n\nAt a point of inflexion the direction of curvature changes, \nand y\' \xe2\x80\x94 y" = DB (Fig. i, \xc2\xa7 189) changes sign with h. \nEquation (7), \xc2\xa7 189, shows that this occurs only when n-\\- \\ \nis odd. Hence, the order of contact is even and the tangent \nintersects the curve. \n\n2. Find the equation of the osculating circle to the parab- \nolay = AfX at the point (i, 2). \n\n(x \xe2\x80\x94 ay ~{- (y \xe2\x80\x94 by = i?" is the general equation of the \ncircle. Substituting the coordinates (i, 2), we have \n\n(,-ay + (2-dy = j^\' (1) \n\nDifferentiating the general equation of the circle, we find \n0\'(a:) = -{x- a)/y - b, and 0"(^) = - R\'/^y - by. \nFrom y = 4X we obtain \n\n/\'{x) = 2/y, and /"W - - 4//. \n\nSubstituting the coordinates (i, 2) in each, and placing \nthe results corresponding to derivatives of the same order \nequal to each other, we have \n\n- (i - a)/{2 - ^) = I and - i?V(2 - by = -4/2\', \nwhich with (i) give \n\n^ = 5j b \xe2\x96\xa0= \xe2\x80\x94 2, and i?\' = 32. \n\nHence, {x \xe2\x80\x94 ^y -\\- {y -\\- 2)^ = 32 is the required equation. \n\n\n\n322 DIFFERENTIAL CALCULUS. \n\n191. Osculating Circle at any point (^\',/) of any plane \ncurve whose equation isj^ =/(-^0. \n\nSubstituting {x\\y\') in {x \xe2\x80\x94 a)\' + {y\xe2\x80\x94 bf = R\\ we have \n\n{x\'-aY-\\-{y-by = R\'^ (.,) \n\nFrom {x \xe2\x80\x94 af -f (y \xe2\x80\x94 by = R"" we obtain \n\nn-) = -J^y and 0"(x) = -[i + (i^]\']/(;,-^); \n\nand from y = f{x) we derive expressions for f\'{x) and \nf"{x). \n\nSubstituting (x\', y) in each, and placing the results cor- \nresponding to the derivatives of the same order equal to \neach other, we have \n\n-{x\'-a)/{y\'-b)=f{x\'), \n\nand -[r+fWYy(y\'-l>)^/"(x\'), \n\nwhich with {a) give, omitting the primes, \n\n\n\nji=[i+f(x)ryf"{x), (,) \n\ng := X - [I +f\'{x)\'y{x)/f"(x), . . . (2) \n\n^=7+[i+/\'wV/"W (3) \n\nComparing ihese with (i) and (2), \xc2\xa7 172, we see that t/ie \noscillatory cu\'cle at any faint of a plane curve is the circle of \n\ncurvature, \n\nEXAMPLES. \n\n1. Find the equation and radius of the osculating circle \nto the curve 4(7 -}- i) = \xe2\x96\xa0^^^ ^.t (o, \xe2\x80\x94 i). \n\nAns. y -f ^^ = 2j^ -[- 3 ; radius = 2. \n\n2. Find the radius of the osculating circle to the parabola \n\ny = QJJC, at (3, i/27). Ans. 16.04. \n\n\n\nOSCULA 7ING LINES. 323 \n\n3. Find the equation and radius of the osculating circle \nto the parabola y\'^ = i6jk: at (1, 4). \n\nAns. {x \xe2\x80\x94 11)^ + (>\xe2\x96\xa0 + i)^ = 125 ; radius = 5 4/5. \n\n192. In general, an osculating circle has a contact of the \nsecond order with any plane curve, but \xc2\xa7 172 shows that at \na point where the curvature of a curve is a maximum or a \nminimum, the circle of curvature, and therefore the oscu- \nlating circle, does not intersect the curve. The order of \ncontact is therefore odd, and of a degree higher than the \nsecond. That the contact in such cases is at least of the \nthird order may be shown as follows: \n\nFrom (i), \xc2\xa7 172, and (i), \xc2\xa7 191, we have \n\n^ = P=[i+7>)TV/"(^); \n\nand in order that p may be a maximum or a minimum, \n\n\n\n^ = ogives 3/ W y W - / WLi \n\n\n+ / i \n\n\nwhence /\'"(x) = 3/"{x)\'/\'{x)/[, + /\'(x)\']. \n\n\nFrom \xc2\xa7 114 we have for a circle \n\n\ndy Id\'yVdv /T , ld}^~ \ndx\' - ^\\dxV dxl L\' "^ \\dx) \xe2\x80\x9e \n\n\n\xe2\x80\xa2 \n\n\n\nHence, when the radius of curvature is a maximum or a \nminimum^ the circle of curvature has a contact with the curve \nof at least the third order. \n\nIt follows that the order of contact of an osculating circle \nat a vertex of a conic is odd, higher than the second, and \nthe circle does not intersect the conic. \n\n\n\n324 \n\n\n\nDIFFERENTIAL CALCULUS, \n\n\n\nCHAPTER XVIII. \n\n\n\nENVELOPES. \n193. In \n\nu =f{x,y, a) = (i) \n\nlet a be an arbitrary constant. By giving all possible values \nto ^, (i) will represent a series of lines, all of the same kind \nor family, unlimited in number, and, in general, intersecting \neach other in order, a is then called a variable parameter. \n\n\n\n\nTo illustrate, let \n\nu = (x \xe2\x80\x94 ay -fy \xe2\x80\x94 9 = 0. \n\nBy giving different values to \xc2\xab, the equation may repre- \nsent a series of circles having the same radius, their centres \non X, and intersecting each other in order in points as \nm^ m\\ etc. \n\nIn general, any value of a in (i) corresponds to a deter- \nminate particular line, and ^ + y^ to another line of the \nsame kind having for its equation \n\nu\' :=^f{x,y,a^h) = o (2) \n\n\n\nENVELOPES. 325 \n\nThis second line, which may be regarded as a second \nstate of the first, will, in general, as h vanishes, ultimately \nintersect the first, in points m^ m\\ etc., the coordinates of \nwhich will satisfy both (i) and (2). If (i) and (2) be com- \nbined and a eliminated, the resulting equation will be that \nof the locus of the points m, m\\ etc., of intersection of all \nof the series of lines represented by (i), each with its second \nstate. If in this resulting equation h be made equal to zero, \nwe will obtain the equation of the limit of the above locus, \nand this /////// is called an envelope of the series of lines. In \nthe case of the circles, the right line MM\' M" is the limit \nof the locus mni\'ni" , and is an envelope of the circles. \n\nIn combining (i) and (2) so as to eliminate a, compli- \ncated expressions frequently arise which may sometimes be \navoided by the following method of Calculus. \n\nSince the coordinates of the points m, m\\ etc., common \nto each of the lines of the series represented by (i), and its \nsecond state in order, satisfy both (i) and (2), they will \nsatisfy the equation \n\nW - u)lh = [fix, y,a + h) - /{x, y, a)]//i = o. (3) \n\nAs /i vanishes, the points m, ;//, etc., approach limiting \npositions M, M\\ etc., and the coordinates of the points \nM, M\\ etc., will satisfy both (i) and the equation \n\n\'du/\'da \xe2\x80\x94 9/(^, y, a) /da = o, ... (4) \n\nwhich (3) approaches as /i vanishes. \n\nIf, therefore, (i) and (4) be combined so as to eliminate \na, the resulting equation will be that of the loci^s of \nthe limiti?ig positions of the points of intersection of the \nseries of lines represented by (i), each with its second \nstate. This locus is the same as the limit of the locus of \n\n\n\n326 DIFFERENTIAL CALCULUS. \n\nthe points of intersection, etc., before obtained, and is, \ntherefore, an envelope of the series. \n\nHence, an envelope of any series of lines determined by \ngiving all possible values to a variable parameter in an equa- \ntion involving two variables only may be defined as the \nlimit of the locus^ or the locus of the limiting positions of \npoints of intersection of the series of lines, each with its \nsecond state, under the law that the difference in position \nbetween each second state and its primitive vanishes. \n\nTo obtain the equation of an envelope of a series of lines \ngiven by an equation with a variable parameter, we have \nthe following rule: \n\nCombine the given equation with its diffei-ential equation \ntaken with respect to the variable parameter^ and eliminate the \nparameter. \n\nFrom (i), regarding a as constant, we obtain \n\ndu/dx = \'du/\'dx -\\- {\'dti/dy){dy/dx) = o \n\nfor the differential equation of each of the lilies of the series. \nFrom (i) and (4), ^= <p{x), which substituted in (i) \ngives the equation of an envelope. Hence, differentiating \n(i) regarding a \xe2\x96\xa0=\xe2\x96\xa0 <p{x), we have \n\ndu _ c)u \'du dy c)u da \n\ndx \'dx dy dx \'da dx * \n\nor^ since du/da = o, \n\ndu/dx = du/dx + {du/dy)(dy/dx) \xe2\x96\xa0= o, \n\nfor the differential equation of an envelope. Each line of \nthe series, and an envelope, have, therefore, the same differ- \nential equation, and dy/dx at any point common to any line \nof the series, and an envelope will be the same for botli. \nAn envelope is therefore tangent to all of the lines of the series. \n\n\n\nENVELOPES. 327 \n\nEXAMPLES. \n\n1. Find the equation of an envelope of the series of circles \ngiven by the equation \n\nu\xe2\x80\x94i^x \xe2\x80\x94 aY +/ \xe2\x80\x94 9 \xe2\x80\x94 o, . . . . (i) \n\nwhen ^ is a variable parameter. \n\ndu/da = \xe2\x80\x94 2{x \xe2\x80\x94 a) = o. , . . . (2) \n\nCombining ^i) and (2) so as to eliminates, we have for \nthe required envelope \n\ny= \xc2\xb13, x= 0/0. \n\nHence, the right lines MM\'M" and MM^M^^ (see figure, \n\xc2\xa7 193) are the envelopes. \n\n2. Find the envelope of the curves given by \n\n_y = ^ tan 6 \xe2\x80\x94 x^ /{\\h cos^ 0), as varies. \n\ndu/d^ =^ jc/cos^ \xe2\x80\x94 x^ sin 6/2/1 cos^ = o. \n\nHence, tan 6 =2/i/x, i -\\- tsui^ 6 = {x\'^-\\- 4h\'^)/x\'^= sec^ ^, \n\nand cos\' = x\'/{x\'-{- 4/1\'), 4/1 cos\' 6 = ^hx^/ix" + 4//\'). \nSubstituting in given equation, we find y =^ h \xe2\x80\x94 x\'^/ij^Ji) for \nthe required envelope. \n\n3. Find the envelope of a right line of a given length c \nwhich moves with its ends on the coordinate axes. \n\nLet B equal the angle which the right line makes with \nX. Then intercepts are, respectively, c cos and c sin By \ngiving \n\nu =^ X sec -\\- y cosec 6 \xe2\x80\x94 c =\xe2\x96\xa0 o . . . (i) \n\nfor the line, in which is the variable parameter. \n\n\'du/\'dd = X sec tan 6 \xe2\x80\x94 y cosec 6 cot 6^ = 0.. (2) \n\nCombining (i) and (2), we find \n\nX ^=^ c cos^ 6^, y =^ c sin^ 0\\ \n\n\n\n328 DIFFERENTIAL CALCULUS. \n\nand eliminating ^, we have for the required envelope \n\nOtherwise, let a and b represent the intercepts, respec- \ntively, giving \n\nx/a+y/b=i . . . (3) and a\' -^ b\'= c\\ . . (4) \nRegarding b as the variable parameter, we may by means \n\n\n\n\nof (4) eliminate a from (3), and proceed as before ; or, since \ntf is a function of by we have \n\nxda/a\' + ydb/b"" = o (s) \n\nada -{- bdb = o (6) \n\nCombining (3), (4), (5), (6), eliminating da/db^ a and by \nwe have \n\n{a^ + b\'\')y = b\'; .\'. b= (^\xc2\xbbV3 , and ^ = {c^x^. s , \n\nSubstituting these expressions in (3), we have for the \n\nenvelope \n\n^2/3 _|_ ^2/3 \xe2\x80\x94 ^2/3 ^ \n\nIn a similar manner, when there are n parameters and \nn ~ I equations of condition between them, we may differ- \n\n\n\nENVELOPES. 329 \n\nentiate the n given equations, regarding n\xe2\x80\x94\\ of the param- \neters as functions of the variable parameter. Then, by \ncombining the n differential equations with the given equa- \ntion of the series, the parameters may be eliminated and \nthe envelope determined. \n\n4. Find the envelope of a series of concentric and co- \naxal ellipses having the same area. \n\nThe given equations are \n\n\xc2\xabV + ^x" = a^b\\ and ab - c\\ \nxy = c^l2 is the envelope. \n\n5. Find the envelope of a right line moving so that its \nperpendicular distance from the origin remains constant. \n\nu = /(jc, y, a) ^^ X cos a -\\- y sin \xc2\xab \xe2\x80\x94 / = o, \n\n\'du/\'da = \xe2\x80\x94 :r sin \xc2\xabf -j-jj; cos a = o, \n\nand x^ +y \xe2\x80\x94 p\'^ is the envelope. \n\n6. Find the envelope of the hypothenuse of a right tri- \nangle moving so that the area of the triangle remains con- \nstant. \n\nLet a = constant area, b = base = variable parameter, \nand c = altitude = 2a/b. Then 2a/b\'\' = tan of angle \nhypothenuse makes with base, and taking the coordinate \naxes to coincide with the legs, we have \n\ny = 2ax/lP\' + 2a/ b for the hypothenuse, \n\nand xy = a/ 2 for the required envelope. \n\n7. Find the envelope of a right line moving so that the \nsum of its intercepts on the coordinate axes is constant. \n\nLet a and b represent the intercepts upon X and Y \nrespectively, and let c ^=- a -\\- b\\ then the equation of the \nline is \n\n\xe2\x80\xa2^A + j/(^ - ^) = I, \n\n\n\n330 DIFFERENTIAL CALChLUS. \n\nand a is the variable parameter. \n\nAns. x^ -\\-y^ \xe2\x80\x94 2xy \xe2\x80\x94 2cx \xe2\x80\x94 2cy -\\- c"^ = o \nor Vx -\\- \\^y = ^ c. \n\n8. Find the envelope of a right line moving so that the \nproduct of its distances from two fixed points is constant. \n\nTake X to coincide with right line joining the two \npoints, and the origin to be at its middle point. Let (^, o) \nand (\xe2\x80\x94 \xc2\xab, o) be the two fixed points, and \n\nX cos a -\\- y <sAXi a ^= p (i) \n\nthe equation of the line. \n\nThe distances from the fixed points to the line are, re- \nspectively, \n\na cos a \xe2\x80\x94 p and \xe2\x80\x94 a cos a \xe2\x80\x94 p. \n\nHence, p^ \xe2\x80\x94 a^ cos^ a = ^r^ = constant. ... (2) \n\nFrom (i) and (2), regarding a as the variable parameter, \nwe find for the envelope \n\n9. Find the envelope of the right lines whose equation is \n\ny-f^2b{x-x\')/{l. -b\'^l . . . . (l) \n\nwheny is the variable parameter, and we have \ny\'"^ = 2px\' and /^ = \xe2\x80\x94 y\' /p. \nEliminating b and x\\ (i) becomes \n\nJK^" -/>+// \xe2\x80\x94 2//^ = o. ... (2) \nHence, \'du/\'dy\' = 2yy\' -\\- p^ \xe2\x80\x94 2px = o, \n\nand / = (2px \xe2\x80\x94 p\')/2y (3) \n\nConjbining (3) and (2), eliminating y, we have \n\nX^ \xc2\xb1V^\'-\\-p/2 \n\n\n\nENVELOPES. 331 \n\nfor the envelope, which is a point at the focus of the \nparabola, and is the Caustic of rays of light reflected from \nthe concave side of a parabola, the incident rays being \nparallel to the axis which coincides with X. \n\n10. Find the envelope of the right lines whose equation is \n\ny-y\'=.^i{x-x\')/{l-b^),. . . . (l) \n\nwhen x^ is the variable parameter, and we have \n\ny^-|.^\'^ = ^2 and d=y/x\\ \n\nEliminating d, (i) becomes \n\n:^-x\'={//x\'-x\'//){/-y)/2. . . (2) \n\nDifferentiating and reducing, we obtain \n\ndu/dx\'= v\' - a\'/y/\' = o, or / = a\'/y/\'. . (3) \n\nSubstituting, in (2), y\'x^\'^/a\'^ for jv\' \xe2\x80\x94 y, we have, after \ncombination with (3) and reduction, \n\nx\' = 2ay\'x/{a\' \' + 2yy\'}. \nSquaring the expressions for y\' and x\', and substituting \nin y\'^ -f~ "^\'^ \xe2\x80\x94 ^^} we have for the envelope \n\nwhich is the equation of the Caustic of rays of light re- \nflected from a circle, the incident rays being parallel to X. \n\n11. Find the envelope of the polar line to the ellipse \n9/ + 4-^^ = 36 as the pole moves along the right line \ny ^^ 2x-\\- \\. \n\nLet x" and y" be the coordinates of the pole, giving \ny" = 2^\' + I. \n\nThen ()yy" -\\- /^xx" = 36 is the equation of the polar line \nSubstituting 2x\'\' -\\- 1 for 7", we have \n\nu = iSyx\'^ + 9V + 4xx\'^ \xe2\x80\x94 36 = o. \n\n\n\n332 DIFFERENTIAL CALCULUS. \n\nin which x*\' is the variable parameter. \n\nCombining the last two equations, we have ox^\' -[- 9J = 2>^i \nor_>\' = 4, .%\xe2\x80\xa2 = \xe2\x80\x94 i8 for the required envelope. \n\n"nes. prrre^\'/rs. Envelopes. \n\n\\2. y = ax -^ bja. a. y^ ^ ^dx. \n\n13 ax \xe2\x80\x94 y = x\'^{i -\\- a^)/2p. a, x^ -{- 2py = p^. \n\n14. x^a +y\'\'/ia - k)=i. a. (^x \xc2\xb1 \\/hf -f / = o. \n\n15. x"^ -\\- y^ =: r^ . r. \n\n16 y^ = ax \xe2\x80\x94 a"^. a. y=\xc2\xb1 x/z. \n\n17. X cos 30+>\' sin 30 \n\n= a(cos 20)^/^ 0. (^\'+/)" =\xc2\xab^^\'\'-y). \n\n^ \xe2\x96\xa0 U^ -f (;\xc2\xab - ^)2 = r\\ \\ n = f{m). + 2da-\'x = a*. \n\nig. ry^ = a\'^x \xe2\x80\x94 a^. a. y^ = 4x^/2^^ \n\nUx-ar-^{y-dy = rK <a. \n\n\n\n;r^ +/ = (^ \xc2\xb1 r)\\ \n21. y = 2ax-\\-a*. a. \\t>y^ -\\^ 2-^x^ \xe2\x80\x94 O. \n\n22 y\'^A^iyX \xe2\x80\x94 a^ \xe2\x80\x94 2pa = O, a. y"^ =p{p + 2x). \n\nX =0. \n\ny=\xc2\xb1 cx/(i - c^). \n\n\n\n256/^ -f- 27X* = o. \n\n\n\n23- \n\n\n/ = 2px. \n\n\nP- \n\n\n24. \n\n\n\\ r \xe2\x80\x94 ca. \n\n\na. \n\n\n25- \n\n\ny"^ =1 ax \xe2\x80\x94 a\'^. \n\n\na. \n\n\n26. \n\n\ny = ax -\\- a*. \n\n\nm. \n\n\n\n\na"" cos d^ sin ^\xc2\xbb \n\n\n6. \n\n\n27- \n\n\nX y a \n\n\n28. \n\n\ny \xe2\x80\x94 mx -\\- i^a^m^ + dK \n\n\nni. \n\n\n29. \n\n\ni {x/af + {ylb)- = I. \n1 a-^b = c. \n\n\na. \n\n\n\nENVELOPES. 333 \n\n194. The envelope of the normals of any given curve is \nits evolute. \n\nThis follows from the definitions of envelopes and evo- \nlutes ; otherwise, the equation of a normal to a curve \n>\'=/(^), at (^\',/), is (\xc2\xa7148) \n\nx-x\'-\\-{y-y)f{x\') = o, . . . (i) \nin which y\' = f{x\'), and x\' may be taken as the variable \nparameter. Differentiating with respect to x\', we have \n\n\n\n-I-/V) +(^-y)/\'V) = o. . . (2) \n\nCombining (i) and (2), we find \n\n\n\nx = x\'-ii +f{x\'Y]f{x\'/f"{x\'), , \n\n(3) \n\n\n\ny=y + b+f(xyyf"(x\'), \n\nfor the limiting position of the intersection of the normal at \n{x\\y\') and its second state ; which is therefore the point of \ntangency of the envelope to the normal at {x\',y\'). Com- \nparing (3) with (i), \xc2\xa7 172, we see that this point is the cor- \nresponding centre of curvature of the given curve. Hence, \nthe envelope of the normals is the evolute of the curve. \n\nCombining (3) and y\' =f(x\'), x\' may be eliminated and \nthe equation of the envelope obtained. \n\nEXAMPLES. \n\nI. Find the envelope of the normals to the parabola \nHere / ^/(^\') = 2^VVV2, \n\nf{x\') = a\'/yx\'\'/\\ f"{x\') = - ayy{2x\'y% \n\nSubstituting in (3) and eliminating y, we have \n\n, (i + a/x\')(ay\'/x\'y\') _ \n\n\xe2\x80\xa2^ - ^ - ayy{2xy/\'\' - 3^ -t- 2^, \ny=y+ ltyv.% --^^\'\'\'Vay% \n\n\n\n334 DIFFERENTIAL CALCULUS, \n\nfrom which, eliminating x\\ we obtain for the required en- \nvelope ay\'^ = 4(x \xe2\x80\x94 2ay/2\']. (See Example i, \xc2\xa7 184.) \n\n2. Find the envelope of the normals to an ellipse by the \nabove method. \n\nOtherwise, the equation of the normal to an ellipse at a \npoint whose eccentric angle is denoted by 6^ is \n\nu^=^ ax sec 6 \xe2\x80\x94 by cosec \xe2\x80\x94 a^ -\\- F =\xe2\x96\xa0 o. \n\nRegarding 6 as the variable parameter, \n\ndu/dG = ax sec B tan d -\\- by cosec 6 cot 6 = 0. \n\nCombining and eliminating 6, we have for the required \n\nenvelope (ax)y^ + {byf/^ = {a\' - b\'f/K \n\n\n\nCURVE TRACING, \n\n\n\n335 \n\n\n\nCHAPTER XIX. \n\n\n\nCURVE TRACING. \n\n\n\nI95* The foregoing principles, with those from Analytic \nGeometry, enable r.s, in general, to trace curves from their \nequations with great accuracy. \n\nRECTANGULAR COORDINATES. \n\nNo fixed rule or directions ap]:)ly in all cases, but, in gen- \neral, it is desirable to determine \xe2\x80\x94 \n\ni". Symmetry with respect to the coordinate \n\n\n\n\n\n\nLimiting coordinates and asymptotes paral- \nlel to them. \nPoints on the coordinate axes. \nTerminating points. \n\n\n\nb/3 tn ^ C \n\ncrts I- u \n\n\n\n1> 13 o l> \n\nllll \n\n\n\n5\xc2\xb0. Direction of curve at points on the coordi- \nnate axes. \n6\xc2\xb0. Asymptotes oblique to coordinate axes. \n7\xc2\xb0. Multiple points. \n\n8\xc2\xb0. Character of cusps. \n9\xc2\xb0. Maximum and minimum ordinates. \nio\xc2\xb0. Direction of curvature and points of \\\\v \nflexion. \n\nEXAMPLES. \n\nEach value of x from \xe2\x80\x94 oo to -|- ^ gives a real value \n\n\n\n33^ \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nfor y. The curve is therefore unlimited in both directions \nwith X, and is limited in both directions of Y. \n\nAs :x: B-> \xc2\xb1 00, J ^H^ \xc2\xb1 o. Hence, X is an asymptote in \nboth directions. \n\n\n\nX =^ o gives ji^ \n\n\n\no, and y=^ o gives a; = o, or \xc2\xb1 oo . \nY \n\n\n\n\n\nHence, (o, o) is the only point at which the curve cuts \nthe coordinate axes. \n\nf{x) = a\\a\' - x\')/{a\' + x\')\\ \n\nf\\6) = I. Hence, the direction of the curve at the ori- \ngin makes an angle of 45\xc2\xb0 with X. \n\nf\'(x) = o gives x-^ \xc2\xb1 a and \xc2\xb1 co . \n\nf\'{x) = 2a\'x{x\' - z^")/{a\' + xy, \n\nf\'\\a) is negative; /. y = a/ 2 is a maximum. \n\n/"( \xe2\x80\x94 a) is positive; .\'. j^ = \xe2\x80\x94 a/ 2 is a minimum. \n\nf"{x) = o gives :r = o and \xc2\xb1 a V^. \n\nf"{x) is negative for values of x from \xe2\x80\x94 00 to \xe2\x80\x94 ^ 1^3, \nand the curve is concave downward. \n\nFor values of x from \xe2\x80\x94 a V^to o f"(x) is positive, and \nthe concave side is above. \n\nAs x varies from o to a V$, /"(\xe2\x80\xa2^) is again negative, and \nthe concavity is downward. \n\nValues of x from \xc2\xab 1^3 to + 00 make f"{pc) positive, and \nthe curve is concave upward. \n\n\n\nCURVE TRACING. \n\n\n\n337 \n\n\n\nIt follows that \n\n( - a 1/3, - \xc2\xab ^3/4), (o. o), and {a 4/3, ^ 1/3/4) \nare points of inflexion. \n\n2. ;t:\' \xe2\x80\x94 2x\'y \xe2\x80\x94 2^\' = 8>\'; .\'. y \xe2\x80\x94 x\'ipc \xe2\x80\x94 2)/2(^\' + 4). \n\n^ = \xe2\x80\x94 00 = J. :v: =: o \xe2\x80\x94 J\'. jc = 00 = J. \n\nThe curve is unlimited in both directions along X and Y, \ny ^=. o gives jc = o and 2. Hence, the curve cuts X at \n\nthe points (o, o) and (2, o). \nf\\x) = ^(;(;^ + 12;^ \xe2\x80\x94 i6)/2(^* + 4)^- \n/\'(o) = o. Hence, at the origin X is a tangent. \n/\'(2) = 1/4. Therefore, at the point (2, o) the curve \n\nmakes tan"^ (1/4) with X. \nf\\x) = o gives .T = o and 1.19 nearly. \n\n\n\n\nExpanding the expression for j\', we have \n\ny \xe2\x80\x94 x/ 2 \xe2\x80\x94 I + (4 \xe2\x80\x94 2x)/{x^ -\\- 4), in which, as x b-^ 00 , \n\ny :m-^ {x/2 \xe2\x80\x94 i). \nHence, _>\' \xe2\x80\x94 x/2 \xe2\x80\x94 i is an asymptote. \n\nf\\x) := - 4{x\' - ex\' - I2X + S)/{x\' + 4)\'. \n\n/"(o) is negative ; hence, j^ = o is a maximum. \n\n/"(1.19) = o is positive; hence, j == ~ o.ii is a \nminimum. \n\nf\'\'(x) = o gives ;t:\xe2\x80\x94 \xe2\x80\x94 2, 4\xe2\x80\x9421/3 = 0.54, and \n4 + 2i^= 7\'5- \n\n\n\n338 \n\n\n\nD IFFEREN TIA L CALCUL US. \n\n\n\nVallies of X from \xe2\x80\x94 oo to \xe2\x80\x94 2 make f"{x) positive, and \nthe corresponding part of the curve is concave upward. \nAs X varies from \xe2\x80\x94 2 to 0.54, f"{x) is negative, and the \nconcavity is downward. When 0.54 < x < 7.5, f\'\'{x) is \npositive, and the concave side is above the curve; but \nX > 7.5 makes f"{x) negative, and the concavity is down- \nward. It follows that (\xe2\x80\x94 2, \xe2\x80\x94 i), (0.54, \xe2\x80\x94 0.05), and \n(7.5, 2.6) are points of inflexion. \n\n2,. y = a^x/{x \xe2\x80\x94 a)\'. \n\nf\\x) = - a\\a + x)/{x - a)\\ \n\nf\\x) = 2a\'{x + 2a)/{x - ay. \n\nThe curve is unlimited in both directions cf X, and in \nthe positive direction of Y. \n\nj; = o gives X ^= o and \xc2\xb1 co . \n\nX is an asymptote in both directions, and since j3\xc2\xbb-> 00 \nas X -w^ a, jc = \xc2\xab is an asymptote. \n\n/\'(o) = I. Hence, at the origin the curve makes \ntan"^ I with X. \n\nf\'{x) = o gives X = \xe2\x80\x94 a, and /"(\xe2\x80\x94 a) is positive. \nHence, y = \xe2\x80\x94 a/4 is a minimum. \n\n\n\n\nTo the left of the point of inflexion (\xe2\x80\x94 2a, \xe2\x80\x94 2a/g) \n\n\n\nCURVE TRACING, \n\n\n\n339 \n\n\n\n\nthe concave side is below, and to the right it is above, the \ncurve. \n\n4. y^ = 2ax^ \xe2\x80\x94 x^. .*._)/ = x^/\'^{2a \xe2\x80\x94 xY^^, \n\nf{x) ={4ax-sx\')/sf. \n\nf\'{x) = -Saygx\'\'/%2a-xy/^ \n\nThe curve is unlimited in both \ndirections along X and F. It cuts \nX at (o, o) and (2a, o). V is \ntangent to both branches at the \norigin, which is a cusp of the first species; and the tangent \nat {2a, o) is perpendicular to A\', y \xe2\x80\x94 \xe2\x80\x94 x -\\- 2^/3 is the \nequation of an asymptote in both directions, y = a ^32/3, \ncorresponding to jv = 4^/3, is a maximum. (2^, o) is a \npoint of inflexion to the left of which the curve is concave \ndownward, and to the right of which it is concave upward. \n\nf(x) ={3- xYis - 5^)/i6. \nV"W = (3 - ^nS^ - 6)/4. \nY \n\n\n\n\nAs X B-^ \xc2\xb1 00 , j; B-> \xc2\xb1 00 . X \xe2\x80\x94 o = y. The curve is \nunlimited in the directions of X and V. {o, o) and (3, o) \nare points on X. /\'(o) \xe2\x80\x94 5.06, and/\'(3) = o. /\'(x) = o \ngives X = 3/5 and 3. fis/s) is negative; hence, y = 1.24 \nis a maximum. /\'(^) changes sign in passing through \n\n\n\n340 DIFFERENTIAL CALCULUS. \n\n/\'(t,) = o (\xc2\xa7 135), and 7 = o is a minimum. (1.2, 0.79) is a \npoint of inflexion to the left of which the curve is concave \ndownward, and to the right of which the concavity is up- \nward. \n\n6. / = x^ + x\\ \n\n\n\nf\'\\x) = \xc2\xb1 {isx\' + 24x-{-S)/4{x+iy/^ \nThe curve is symmetrical with respect to X, \n\nY \n\n\n\n\nValues of x <. \xe2\x80\x94 1 give imaginary expressions, for j. \nX \xe2\x80\x94 \xe2\x80\x94 I gives y = \xc2\xb1 o. x > \xe2\x80\x94 i gives two values for/ \nequal with opposite signs. As x :^-^ 00 , y b-> \xc2\xb1 co . The \ncurve is, therefore, limited in the direction of JC negative \nby the ordinate corresponding to :r = \xe2\x80\x94 i, and is unlimited \nin the other directions along X and V. \n\n(\xe2\x80\x94 I, \xc2\xb1 o) and (o, \xc2\xb1 o) are points on X. \n\n/\'(\xe2\x80\x94 i) = \xc2\xb1 00, and /\'(o) = \xc2\xb1 o. Hence, at (\xe2\x80\x941 = \n\xc2\xb1 o) the tangent is parallel to V, and X is tangent to both \nbranches at the origin, which is a multiple point of tangency. \n(Example 6, page 281.) \n\nf\'{x) \xe2\x80\x94 o gives X \xe2\x80\x94 o or \xe2\x80\x94 4/5. /"(o) is positive for \nthe upper and negative for the lower branch. Hence, the \nzero ordinates at the origin are, respectively, a minimum \nand a maximum. /"(\xe2\x80\x94 4/5) is negative for the upper and \n\n\n\nCURVE TRACING, \n\n\n\n341 \n\n\n\npositive for the lower branch. Hence, the corresponding \nordinates are, respectively, a maximum and a minimum. \n\nf"{x) \xe2\x80\x94 o gives x = (\xe2\x80\x94 12 \xc2\xb1 1/24)715. Points of both \nbranches corresponding to the upper sign are points of in- \nflexion, and the direction of curvature is as indicated in \nthe figure. \n\n\n\n/\'W = \xc2\xb1 \n\n\n\na^ -\\- 2c^x\'^ \xe2\x80\x94 x^ \n\n\n\n\nThe curve is symmetrical with \nrespect to X and Y. It is limited \nin both directions of X by the \nasymptotes Jt: = \xc2\xb1 a, and is un- \nlimited in both directions along \nF. \n\n/\'(o) = \xc2\xb1 I, and /\'(\xc2\xb1 ^) = \xc2\xb1 \xc2\xab). \n\nBoth branches pass through the origin, one inclined at \nan angle of 45\xc2\xb0, and the other at an angle of 135\xc2\xb0, with X. \nf\\x) is an increasing function for the branches above A, \nwhich are, therefore, concave upward, and a decreasing \nfunction for those below X, which are concave downward. \n8. J = i^V^ \n\nf\\x) = i/x\'e^/\\ \nf\'\\x) = (i - 2x)lx\'ey\\ \n\n\n\nAs X ^-> =F 00 , jv :^-> I. K% \xe2\x80\x94 X m^ o, y b-> 00 . As \n-f- X \'Wf-^ o, y M-> o. \n\nThe curve is limited by AT in the direction of F negative, \nand is discontinuous at the origin which is a terminating \n\n\n\n342 \n\n\n\nD IFFERENTIA L CALCUL US. \n\n\n\npoint for the right-hand branch. As \xe2\x80\x94 :r -m-^ o, f\'{x) -b-^ oo ; \nand as + ^ B-> o, /\'(^t) ;^-\xc2\xbb o. Hence, X is a tangent at \nthe origin. F is an asymptote to the left-hand branch, and \nJ = I is an asymptote to both branches. As x varies con- \n\n\n\n\ntinuously, /\'(jk:) does not change sign, and there are no max- \nimum or minimum ordinates. Corresponding to :r = 1/2, \nthere is a point of inflexion, to the left of which the curve \nis concave upward, and to the right of which the curvature \nis downward. \n9, The Logarithmic Curve. \n\nX = ey, .*. y = log X. \n\nfix) = i/x. f\'{x) = - ^|x\\ \n\n\n\nAs jc^-^o, y-^^ \xe2\x80\x94 00. x\xe2\x80\x941, \njj; = o. As Jts^^oo , jj;;^-\xc2\xbboo . The \ncurve is limited in the direction \nof X negative by F, which is an \nasymptote, and is unlimited in \nthe other directions along X and \nF. /(i) = I. Hence, at (i, o) \nthe curve makes tan"^ i with X. \nf\\x) is negative for all points of the curve, and the con- \n\n\n\n\nCURVE TRACING. \n\n\n\n343 \n\n\n\ncave side is below. RS, the sublangent on F, is (\xc2\xa7 149) \nxdy/dx = I = 6>.-^. \n10. y = ax" -|- bx^. \n\n\n\n\nf(x) = \xc2\xb1 (^? + ^bx/2)/ Va + bx. \nf\'{x) = \xc2\xb1 {4ab + 3^\'^)/4(a + ^^Y\'\'- \n\nThe curve is symmetrical with re\xc2\xb0 q \nspect to X. \n\nAs :r^-\xc2\xbboo , j:^-> \xc2\xb1 00 . x = o = J. \n^ = \xe2\x80\x94 ^/<^, y = \xc2\xb1 o. X < \xe2\x80\x94 a/b, \ny is imaginary. \n\nThe curve is limited in the direction of X negative by \nthe ordinate corresponding to x = \xe2\x80\x94 a/b, and is unlimited \nin the ether directions along A" and Y. \n\nf{~ a/b) = \xc2\xb1 CO . f{o) = \xc2\xb1 \\/a. The origin is, there- \nfore, a double multiple point. \n\nf\'{x) \xe2\x80\x94 o gives X = \xe2\x80\x94 2a/;^b, for which j^\' = \xc2\xb1 2a V^^/gb \nare maximum and minimum ordinates. \n\nFor ^ > \xe2\x80\x94 a/b the first value of f (x) is positive, and \nthe second negative. The branch BEOC is, therefore, \nconcave upward, and the other is concave downward. \n\nPOLAR coordinates: examples. \n\n\n\nI, r = rt; sin 2^. \n\ndr/cW = 2a cos 26. \nAs varies from o to 7r/4, r \nX changes from o to a, and as 6 varies \nfrom 7r/4 to 7r/2, ; changes from a \nto o; completing a loop in the first \nangle. As f^ varies from 7r/2 to tt, \nis negative, and changes from o, through \xe2\x80\x94 a^ to o form- \n\n\n\n\n344 \n\n\n\nD IFFEREN TIA L CALC UL US. \n\n\n\ning a loop in the fourth angle. As varies from n to s^r/?, \nr is positive and changes from o, through a^ to o, forming \na loop in the third angle. As ^ varies from 37r/2 to 2;r, r \nis negative and changes from o, through \xe2\x80\x94 a^ to o, forming \na loop in the second angle. \n\nAs 6^ passes through 7r/4, dr/dd changes from + to \xe2\x80\x94 . \nHence, r = (^ is a maximum. As d passes through 37r/4, \ndr/dd changes from \xe2\x80\x94 to -|-> ^^d r = \xe2\x80\x94 \xc2\xab is a minimum. \nAs 6* passes through 57r/4, dr/dd changes from + to \xe2\x80\x94 , \nand r = \xc2\xab is a maximum. As B passes through 77r/4, dr/dd \nchanges from \xe2\x80\x94 to +, and z\' = \xe2\x80\x94 ^ is a minimum. \n\n^ 2. r = a tan 6. \n\ndr/dd = a/cos\' d. \n\nr is always an increasing func- \ntion of d. \n\nValues of d from o to 7r/2 give \nthe branch FM\\ As d varies \nfrom 71/2 to rr^ r is negative and \nincreases from \xe2\x80\x94 00 to o, giving the \nbranch in the fourth angle. Values of d from TT^to Z\'^/2 \ndetermine the branch in the third angle, and the branch in \nthe second angle is due to negative values of r correspond-, \ning to values of d from Z\'^/2 to 2 7r. \n\nThe subtangent = r\'\'dd/dr \xe2\x80\x94 a sin\'\' d. \n\n\xe2\x80\x94 7t/2 or 37r/2 gives r = 00 , and the subtangent = a. \n\nHence (\xc2\xa7 157), ^ cos d = \xc2\xb1 a are asymptotes. \n\n3. The Spiral of Archimedes, r = ad. \n\nEstimating from the pole J^, where r = o = d, r in- \ncreases directly with d. Denoting the value of r after one \nrevolution by ~ \n\n\n\n\nCURVE 7\' RACING, \nr, = PA ~ 27ta, \n\n\n\n345 \n\n\n\n\nwe have a = rj27t, \nand r = r^6/27t. \n\nHence, \nFO = ry4, ^(9\' == rj2, \nPO" = r,3/4, etc. \n\ndr = r^dB/27C ; \n.\'. dQIdr \xe2\x80\x94 27t/r^, \nSubtangent \xe2\x80\x94 r\'^dO/dr \n= rj)\'^/27t, (\xc2\xa7 150.) \n\nHence, subtangents are to \neach other as the squares of \nthe corresponding radii. 6 \xe2\x80\x94 m2n gives subt = m^27ir^. \n\nSubnormal \xe2\x80\x94 dr/dO = rj2n, \n\n4. The Parabolic Spiral, r\'\' = d\'d. \n\nr^ = aVzTt\', .*. a = rj^27t^ and r" = r^B/27t. \n\ndr/d6 = (f/2r = subnormal. \nSubtangent = 2r\' /c^. \n\nThis spiral may be con- \nstructed by first construct- \ning the parabola y^ = x^ and \nthe circle CB with centre at \nP and radius = \xc2\xab^ Then \nlay off from C the arc CB equal to an assumed abscissa of \nthe parabola, and upon the radius PB lay off from P PO \nequal to the corresponding ordinate. O will be a point of \nthe curve, since r = PO =: y = Vx \xe2\x80\x94 Vd^O. \n\n\n\n\n34^ DIFFERENTIAL CALCULUS, \n\n5. The Hyperbolic Spiral, r \xe2\x80\x94 a/6. \n\nr^ = a/27t; .\'. a = 2nr^ and r = 27trJ6, \nB C \n\n\n\nV \n\n\n0\' \n\n\n^ \n\n\nr,/^ \n\n\nM \n\n\n\n\nA \n\n\n?f \n\n\n\\ \n\n\n\n\n\n\nA \n\n\nVj \n\n\nJ \n\n\n\n\n\n\n\n\n\n\n\n0\'" \n\nAt the pole r = o and ^ = 00 . r varies inversely with \ne. Hence, FO\' = 4r, , ^6"\' = 2r, , /\'C)"\' = 4^73, etc. \n^/-/^//^ = \xe2\x80\x94 a/d\'\'-^ subtangent PT = \xe2\x80\x94a. \n\n6 := o gives r =: 00 , and the subtangent PB := \xe2\x80\x94 a. \nHence (\xc2\xa7 157), PC, parallel to PA, is an asymptote. \n\nFrom \xc2\xa7 150, tan PAfT= rdd/dr = \xe2\x80\x94 6^, which leads to \na construction of the curve by points. Thus, with /^ as a \ncentre and radius = a, describe a circle. Draw any radius \n\n\n\n\nvector as PM, and the corresponding subtangent PT, \nLay off PE \xe2\x80\x94 arc HN, and draw ElSf. TM drawn par- \nallel to EIV \\N\\\\\\ determine a point M\' oi the curve; for \ntan PMT = tan PJVE = PE/ci = HNJa ^ 6. \n\n\n\nCURVE TRACING. 347 \n\n6. The Logarithmic Spiral, r \xe2\x80\x94 a^. \n\ndO/dr = Mjr, whence (\xc2\xa7 150) tan/\'J/r = tan =: M^; \nand is constant. \n\nAlso, sin = rdd/ds. \n\nHence, ^ = r sin = r^dO/ds = cr^ \n\nin which c represents the sine of the constant angle made \nby the radius ^^ector with the curve. \n\n\n\n\nIf ^ = o be increased by equal angles, the correspond- \ning values of r will be in geometrical progression. With \nany convenient radius, as FA^ describe a circle, and lay oft \nequal arcs Ab, bm, mc, etc. Draw the right lines PA^ Pb, \nPm^ etc., and let PA be the initial side of ^. PA will then \nrepresent unity. Make ^ = Ab/PA, and determine the \ncorresponding value of ^ = PB. PB/PA will be the ratio \nof the progression, and the distances PM^ PC, etc., from \nP to corresponding points of the curve are readily deter- \nmined. Since r = o requires 6^= \xe2\x80\x94 00, the number of \nspires from A x.q P \\^ unlimited. \n\n\n\nCHAPTER XX. \n\nAPPLICATIONS TO SURFACES. \n\n196. u \xe2\x80\x94 F{x,y, z)= o . (i) or z =/{x,y) (2) \nis the general equation of any surface. \n\nAssuming the co-ordinate axes perpendicular to each \nother, and x and y as the independent variables, we have \n\n, du , . du du J , . \n\ndu =. \xe2\x80\x94- ax -\\ \xe2\x80\x94 \xe2\x80\x94 dy -\\- \xe2\x80\x94- dz ^= o, . . . y-i) \n\ndx \' dy \xe2\x80\xa2" \' dz \' ^^\' \n\nor \n\n\'\'\'=tJ\'\' + %\'^^ -w \n\nfor the general differential equation of the surface. \n\nI97\xc2\xab To find the equation of the tangent plane to any sur- \nface at a give?i point. \n\nLet z = (f>{x, y) be the equation of the surface and \nP(x\\ y* , z\') be the given point. \n\nThen ^. = |;y. + |;</v, (I) \n\nin which -r-, and -r-. are constant for a fixed point (\xc2\xa7 102). \ndx dy\' \n\nThe equations of the tangent line J^Q in the vertical \nplane PMN are, from the figure, \n\nx \xe2\x80\x94 x\' _y ^ y\' _z \xe2\x80\x94 z\' \n\n. . . (2) \n\n\n\ndx dy dz \n\n348 \n\n\n\nAPPLICATIONS TO SURFACES. \n\n\n\n349 \n\n\n\nEquations (i) and (2) are relations existing for any set \nof values of dx^ dy, and dz, corresponding to the point P \nand the corresponding tangent. \n\n\n\n\nEliminating dx, dy, and dz^ we find the locus of all these \ntangents to be \n\n^-^\'=^r{^--\')-\\-%(y-y)\' . . (3) \n\n\n\ndx\' \n\n\n\nIf the equation of the surface be in form \nu = F{x, y, z) = o, \n\nthen du = -\xe2\x80\x94r dx -\\- -^^ dy -{--\xe2\x80\x94. dz = o, \n\ndx dy\' -^ dz\' \n\nin which dx, dy, and dz are the same as in (i). \n\nCombining this equation with (2), the equation of the \ntangent plane becomes \n\n9\xc2\xab / /\\ t 9^^ / /\\ I 9^^ / ,x / X \n\n-,{^-^)+^(>,-^)+-,(.-.\')=o. (4) \n\n\n\n350 DIFFERENTIAL CALCULUS. \n\n(For tangent plane to surface of ^d order only, compare \n\n^ o \xe2\x80\xa2 1 . o 1-j ^ o \\ A- Tr 9^\' 9^ 9^ \n\nC. Smith s bond Geo., <5 1:52.) JNote. if \xe2\x80\x94 -, = t7 = "7~7 = o\xc2\xbb \n^ ^ ^ dx\' dy dz \n\nthe plane is indeterminate. \n\nEXAMPLES. \n\n1. Find the equation of the tangent plane at the point \n\n(2, 3, 1^23) on the surface x^ -\\- y^ -\\- z^ \xe2\x96\xa0=\xe2\x96\xa0 36. \n"dz\' /dx\' = - x\'/z\' = - 2/ V^ \n\na^V^y- -/A\'- -3^^ \n\nSubstituting in (3), we have \n\n^ _ V23 = (^ - 2)( - 2/ 1^23) + (y \xe2\x80\x94 3)(\xe2\x80\x94 3/ 1/^), \n\nor 2Jt: -j- 3^ + 4^230 = 36, for the required platie, \nor \n\n\'du/dx\' = 2x\' = 4, du/dy\' = 2j\' = 6, du/dz\' = 2z\' = 2 V23 \n\nSubstituting in (4), we obtain same result. \n\n2. Find the equation of the tangent plane at {x\\ y\\ z\') \non the ellipsoid x^ /a 4-//^\' + z" /c"" = i. \n\nAns. xx\'/a\' -\\-yy\'/b\'\' + zz\'/^ = i. \n\n3. Find the equation of the tangent plane at {x\' yy\\ 2\') \non the surface whose equation is \n\nmx^ -\\-ny\'^ -\\-pz\' -\\- m\'\'x-\\-l=-o, \nAns. 2mx\'x + 2ny\'y -\\- 2pz\'z + m"{x -\\- x\') + 2/ = o. \n\n4. Find the equation of the tangent plane to the ellipsoid \nwhose equation is \n\n4jp\' + 2/ + ^\' = 10, at (i, \xe2\x80\x94 I, 2). \n\n\xe2\x80\xa2Ans. 2X \xe2\x80\x94y -\\-z=.^. \n\n\n\nAPPLICATIONS TO SURFACES. 35 1 \n\n5. Find the equation of the tangent phme to the ellipsoid \n\n^Vi6 +//9 + 2V4 = I, at (3, I, i/^/36). \n\nAns. 3>:r/i6-h yh + ( ^47/36)^4 = i. \n\n6. Find the equation of the tangent plane at any point of \n\n2222 \nthe surface x^-\\-y^ +2^ = ^^; and show that the sum of the \n\nsquares of the intercepts on the axes made by a tangent \n\nplane is constant. \n\n198. The normal at any point of a surface. \n\nLet equation of surface be ;s = 0(-^>>\')^ and point be \n{x\\ y, 2;\'). The normal passes through (jc\', y\\ z\') and is \nperpendicular to the plane given by (3), \xc2\xa7 197; hence its \nequations are \n\nX \xe2\x80\x94 x^ _ y \xe2\x80\x94y\' __ z \xe2\x80\x94 z\' . . \n\n\xe2\x96\xa0 -dz\'/dx\' ~ dz\'/dy\' ~ ^=T ^\'^ \n\nIf equation of surface be \xc2\xab = -^{x^ y, z) = o, equations of \nnormal are \n\n\n\nX y \xe2\x80\x94y _ z \xe2\x80\x94 \n\n\n\ndu/dx\' du/dy\' \'du/dz\'\' \n\n\n\n(2) \n\n\n\n(EXAMPLES. \n\n\n\nT. Deduce formulas for the direction cosines of the nor- \nmal given by (i); by (2). And find the cosines of the angles \nthe tangent planes corresponding to (i) and (2) make with \nXY, XZ, and YZ. \n\n2. Find the tangent of the angle that the tangent plane \nto x" +y -^ ^"^ = Z^ at (2, 3, ^^23) makes with XY. \n\n199. To find the equations of the tangent line to a given \nsurve at a given point. \n\n\n\n352 DIFFERENTIAL CALCULUS. \n\nLet equations of curve be \n\nF{x,y,z) = o, (a) \n\n(pipe, y,z) = o^ (b) \n\nand let {x\\ y\\ z\') be the point of tangency. \n\nThe required tangent line lies in the tangent plane to \neach of the surfaces {a) and {b) at point {x\\ y\\ z\')\\ hence \nits equations are \n\n\n\n9^/ ^\\ 1 90 , /\\ I 90/ ,x \n\n\n\nIf the curve be given by two of its projecting cylinders as \n\n/(=c, ^)=o, {a\') \n\nt(y,^) = o . (f) \n\nthe equations of the tangent become \n\nf (.-/)+ |;(.-.\')=o, \n\nwhich (\xc2\xa7 148) are the equations of lines in XZ and YZ \nrespectively tangent to the curves {a\') and {b\') at the \nprojections of (x , y\\ z\'). The problem is thus reduced to \nfinding the equations of right lines tangent to two of the \n\n\n\nAPPLICATIONS TO SURFACES. \n\n\n\n353 \n\n\n\nprojections of the given curve at the projections of the given \npoint. \n\nEx. Show that the curve whose equations are \n\n\n\nx^ +y = ^\' aiid z = aC tan \n\n\n\n!>\' \n\n\n\nmakes a constant angle with the axis Z. \n\n200. To find the equation of the normal plane to a curve \nat a given point. \n\nLet the equations of the curve be \n\nF{x, y, z) = o and (p{x, y, z) = o, \n\nand (^\', y, z\') be the given point. \n\nThe normal plane is perpendicular to the tangent to the \ncurve at the given point. \n\nLet equation of normal plane be \n\nx{x- x\') -^^{y -y ) + y{z - z\') = o. \n\nIf this plane be perpendicular to the tangent we have the \nconditions \n\n( .dF . dF ^ dF \nEliminating A, //, and v^ the required equation is \n\n\n\nx \xe2\x80\x94 x, y~y , z \xe2\x80\x94 z \n\ndF dF dF \n\ndx" dy" dz\' \n\n90 90 90 \n\ndx" dy" dz\' \n\n\n\no. \n\n\n\n354 \n\n\n\nD IFFEREN TIA L CALCUL US. \n\n\n\n20I. The numerical value of the expression \n\n\n\ndz/ Vdx\' + d/ = tan QMT \n\nmeasures ^/le slope of the surface z = f {x,y), at the point \nAf, along any section, as 3fB, made bv a vertical plane \nthrough M, and whose trace on VX makes with X \ntan-^ dy/dx = (\xc2\xa7 102). <^ \n\n\n\n\nWriting tan QMT = tan i- = \n\n\n\nwe have, (2), \xc2\xa7 102, \n\n\n\nVi + {dy/dxy \n\n\n\ntan s \n\n\n\ndz/dx -\\- (dz/dy) (dy/dx) \nVi + (dy/dxY \n\n\n\nPlacing \n\ndy/dx = tan (p = m, dz/dx = /, and d^/dy = ^, \nwe have \n\n\n\ntan i" = (/ + m^)/ J 1 -j- 7/^\' (i) \n\n\n\nAPPLICATIONS TO SURFACES. 355 \n\nApplication. \xe2\x80\x94 At tlie point (2, 3, ^^23) on the surface \n\nx" + / + ^^ = 36 . . . . (^) \n\nfind the slope of the curve cut out by the plane \n\ny = 2X \xe2\x80\x94 \\, Z \xe2\x80\x94 0/0. ..... (/5) \n\nFrom (^), \'dz/dx = \xe2\x80\x94 x/z, and dz/dy = \xe2\x80\x94 y/z. \n\nHence, j> = \xe2\x80\x94 2/^23, and ^ \xe2\x80\x94 ~ 3/ ^^23. \n\nFrom (I?), dy/dx \xe2\x80\x94 2 ^^ m. Therefore, \n\ntan J = [\xe2\x80\x94 2/1/23 \xe2\x80\x94 6/ 1/23]/ Vi -J- 4 " \xe2\x80\x94 0.746 -{-. ^ \n\nHence, 0.746 + is the required slope. \n202. At any point, as M, tan s varies with w. To deter- \nmine w, in order that the slope shall be a maximum, we \nplace \n\nd tan s _ q \xe2\x80\x94 mp _ \n~"d^ ~ (i + vef\'\'\'\' ~ ^\' \n\nwhence q \xe2\x80\x94 vip = o, or m = q/p. \n\nWhen tan s is positive, maximum values of tan s and the \nslope are the same ; but when tan s is negative, the slope is \na maximum when tan i" is a minimum. \n\nApplication. \xe2\x80\x94 Find the equation of the vertical plane \nwhich passes through the point (2, 3, I/23) on the surface, \n\xe2\x80\xa2^^ +y -|- s^ = T^d, and which cuts from the surface the \nline with the maximum slope. \n\ny \xe2\x80\x94 mx -\\- b^ z = 0/0, is the general form of the required \nequation. \n\nAt (2, 3, V^) p = dz/dx =-2/ V7s, \n\nand q = dz/dy = \xe2\x80\x94 3/ 1/23. \n\n\n\n35^ DIFFERENTIAL CALCULUS. \n\nHence, m = 3/2. The trace on XY must pass through \n(2, 3). Hence, we have \n\n\n\n3 + /^, or /^ \xe2\x80\x94 o, and y = Z^l\'^i ^ ~ \xc2\xb0/\xc2\xb0 \n\n\n\nis the required plane. The maximum slope is approxi- \nmately .751. \n\nWhen / -f" mq = o, or m = \xe2\x80\x94 p/q^ tan j- = o, and \nthe slope is a minimum, since numerical values only of \ntan s are considered. \n\nIn the above application m = \xe2\x80\x94 p/q = \xe2\x80\x94 2/3. Hence, \n3 = \xe2\x80\x94 4/3 4- /^, giving <^ = 13/3, and ^ = \xe2\x80\x94 2^3 + 13/3, \nz = 0/0, is the plane which cuts out the curve whose \ntangent at M is parallel to XV. \n\nThe intersection of the surface by the horizontal plane \nthrough the given-point is a horizontal line,and J^{x^y,c)=^o, \nz^c^ ars its equations. \n\n\n\nSHORT-TITLE CATALOGUE \n\nOF THE \n\nPUBLICATIONS \n\nOF \n\nJOHN WILEY & SONS, \n\nNew York. \nLondon: CHAPMAN & HALL, Limited. \n\n\n\nARRANGED UNDER SUBJECTS. \n\n\n\nDescriptive circulars sent on application. 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(Austen.) i2mo, i 50 \n\nPoole\'s Calorific Power of Fuels , Svo, 3 00 \n\nPrescott and Winslow\'s Elements of Water Bacteriology, with Special Refer- \nence to Sanitary Water Analysis i2mo, i 2^ \n\n* Reisig\'s Guide to Piece-dyeing 8vo, 25 00 \n\n4 \n\n\n\n12 \n\n\nSO \n\n\nI \n\n\n50 \n\n\n3 \n\n\n00 \n\n\n2 \n\n\n00 \n\n\nI \n\n\n25 \n\n\n2 \n\n\n00 \n\n\n4 \n\n\n00 \n\n\nI \n\n\n50 \n\n\n2 \n\n\n50 \n\n\n3 \n\n\n00 \n\n\n\n\n7S \n\n\n2 \n\n\n50 \n\n\n2 \n\n\nSO \n\n\nI \n\n\noa \n\n\n3 \n\n\n00 \n\n\nI \n\n\n25 \n\n\n2 \n\n\n50 \n\n\nI \n\n\n00 \n\n\n3 \n\n\n00 \n\n\nI \n\n\n00 \n\n\nI \n\n\n00 \n\n\nI \n\n\n50 \n\n\n\n\n60 \n\n\n4 \n\n\n00 \n\n\nI \n\n\n25 \n\n\nI \n\n\n00 \n\n\nI \n\n\n00 \n\n\nI \n\n\n50 \n\n\nI \n\n\n00 \n\n\n2 \n\n\n00 \n\n\nI \n\n\n50 \n\n\n\nRichardsand Woodman\'s Air .Water, and Food from ^ Sanitary Standpoint. 8vo, \n\nRichards\'s Cost of Living as Modified by Sanitary Science i2mo, \n\nCost of Food a Study in Dietaries i2mo, \n\n\xe2\x80\xa2 Richards and Williams\'s The Dietary Computer 8vo, \n\nRicketts and Russell\'s Skeleton Notes upon Inorganic Chemistry. (Part I. \xe2\x80\x94 \n\nNon-metaUic Elements.) 8vo, morocco, \n\nRicketts and Miller\'s Notes on Assaying .8vo, \n\nRideal\'s Sewage and the Bacterial Purification of Sewage Svo, \n\nDisinfection and the Preservation of Food. 8vo, \n\nRuddiman\'s Incompatibilities in Prescriptions .8vo, \n\nSabin\'s Industrial and Artistic Technology of VaJnts and Varnish. (In jyress.) \nSalkowski\'s Physiological and Pathological Chemistry. (Orndorff.;. . . .8vo, \nSchimpf\'s Text-book of Volumetric Analysis i2mo. \n\nEssentials of Volumetric Analysis , i2mo, \n\nSpencer\'s Handbook for Chemists of Beet-sugar Houses i6mo, morocco. \n\nHandbook for Sugar^anufacturers and their Chemists. . i6mo, morocco, \nStockbridge\'s Rocks and Soils 8vo, \n\n* Tillman\'s Elementary Lessons in Heat 8vo, \n\n\xe2\x99\xa6 Descriptive General Chemistry 8vo, \n\nTreadwell\'s Qualitative Analysis. (Hall.) 8vo, \n\nQuantitative Analysis. (Hall.) Svo, \n\nTurneaure and Russell\'s Public Water-supplies Svo, \n\nVan Deventer\'s Physical Chemistry for Beginners. (Boltwood.) i2mo, \n\n* Walke\'s Lectures on Explosives Svo, \n\nWassermann\'s Immune Sera: Haemolysins, Cytotoxins, and Precipitins. (Bol- \n\nduan.) i2mo, \n\nWells\'s Laboratory Guide in Qualitative Chemical Analysis Svo, \n\nShort Course in Inorganic Qualitative Chemical Analysis for Engineering \n\nStudents i2mo, \n\nWhipple\'s Microscopy of Drinking-water Svo, \n\nWiechmann\'s Sugar Analysis Small Svo. \n\nWilson\'s Cyanide Processes. i2mo, \n\nChlorination Process i2mo. \n\nWulling\'s Elementary Course in Inorganic Pharmaceutical and Medical Chem- \nistry. i2mo, 2 .00 \n\nCIVIL ENGINEERING. \n\nBRIDGES AND ROOFS. HYDRAULICS. MATERIALS OF ENGINEERING \nRAILWAY ENGINEERING. \n\nBaker\'s Engineers\' Surveying Instruments i2mo, 3 00 \n\nBixby\'s Graphical Computing Table Paper 19^X24! inches. 25 \n\n** Burr\'s Ancient and Modern Engineering and the Isthmian Canal. (Postage, \n\n27 cents additional.) Svo, net, 3 50 \n\nComstock\'s Field Astronomy for Engineers Svo, 2 50 \n\nDavis\'s Elevation and Stadia Tables Svo, i 00 \n\nElliott\'s Engineering for Land Drainage i2mo, i 50 \n\nPractical Farm Drainage i2mo, z oc \n\nFolwell\'s Sewerage. (Designing and Maintenance.) Svo, 3 00 \n\nFreitag\'s Architectural Engineering. 2d Edition, Rewritten Svo, 3 so \n\nFrench and Ives\'s Stereotomy Svo, 3 50 \n\nGoodhue\'s Municipal Improvements i2mo, t 75 \n\nGoodrich\'s Economic Disposal of Towns\' Refuse Svo, 3 50 \n\nGore\'s Elements of Geodesy Svo, 2 50 \n\nHayford\'s Text-book of Geodetic Astronomy Svo, 3 01: \n\nHering\'s Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 \n\nHowe\'s Retaining Walls for Earth i2mo, i 25 \n\nJohnson\'s Theory and Practice of Surveying Small Svo, 4 00 \n\nStatics by Algebraic and Graphic Methods ,\xe2\x99\xa6...., Svo, a 00 \n\n5 , \n\n\n\n\n\noc \n\n\n\n\n00 \n\n\n\n\n00 \n\n\n\n\n50 \n\n\n\n\n75 \n\n\n\n\n00 \n\n\n\n\n50 \n\n\n\n\n00 \n\n\n\n\n00 \n\n\n\n\n50 \n\n\n\n\n50 \n\n\n\n\n35 \n\n\n\n\n00 \n\n\n\n\n00 \n\n\n\n\n50 \n\n\n\n\n50 \n\n\n\n\n00 \n\n\n\n\n00 \n\n\n\n\n00 \n\n\n\n\n00 \n\n\n\n\nSO \n\n\n\n\n00 \n\n\n\n\n00 \n\n\n\n\n50 \n\n\n\n\n50 \n\n\n\n\n50 \n\n\n\n\nSO \n\n\n\n\nSO \n\n\n\n\nSO \n\n\n\nKiersted\'s Sewage Disposal i2mo, i as \n\nLaplace\'s Philosophical Essay on Probabilities. (Truscott and Emory.) i2mo, 2 00 \n\nMahan\'s Treatise on Civil Engineering. (1873O (Wood.) 8vo, 5 00 \n\n\xe2\x99\xa6 Descriptive Geometry 8vo, i 50 \n\nMerriman\'s Elements of Precise Surveying and Geodesy Svo, 2 50 \n\nElements of Sanitary Engineering Svo, 2 00 \n\nMerriman and Brooks\'s Handbook for Surveyors i6m0( morocco, 2 00 \n\nNugent\'s Plane Surveying. Svo, 3 50 \n\nOgden\'s Sewer Design i2mo, 2 00 \n\nPatton\'s Treatise on Civil Engineering Svo half leather, 7 50 \n\nReed\'s Topographical Drawing and Sketching 4to, 5 00 \n\nRideal\'s Sewage and the Bacterial Purification of Sewage Svo, 3 51 \n\nSiebert and Biggin\'s Modem Stone-cutting and Masonry Svo, i 5* \n\nSmith\'s Manual of Topographical Drawing, (McMillan.) Svo, 2 50 \n\nSondericker\'s Graphic Statics, wun Applications to Trusses. Beams, and \n\nArches Svo, 2 00 \n\n* Traxitwine\'s Civil Engineer\'s Pocket-book i6mo, morocco, 5 00 \n\nWait\'s Engineering and Architectural Jurisprudence Svo, 6 00 \n\nSheep, 6 50 \nLaw of Operations Preliminary to Construction in Engineering and Archi- \ntecture. Svo, 5 00 \n\nSheep, 5 50 \n\nLaw of Contracts Svo, 3 00 \n\nWarren\'s Stereotomy \xe2\x80\x94 Problems in Stone-cutting Svo, 2 50 \n\nWebb\'s Problems in the Use and Adjustment of Engineering Instruments. \n\ni6mo, morocco, i 25 \n\n\xe2\x80\xa2 Wheeler\'s Elementary Course of Civil Engineering Svo, 4 00 \n\nWilson\'s Topographic Surveying , , Svo, 3 50 \n\nBRIDGES AND ROOFS. \n\nBoiler\'s Practical Treatise on the Construction of Iron Highway Bridges. .Svo, 2 00 \n\n* Thames River Bridge 4to, paper, 5 00 \n\nBurr\'s Course on the Stresses in Bridges and Roof Trusses, Arched Ribs, and \n\nSuspension Bridges Svo, 3 50 \n\nDu Bois\'s Mechanics of Engineering. Vol. II Small 4to, 10 00 \n\nFoster\'s Treatise on Wooden Trestle Bridges 4to, 5 00 \n\nFowler\'s Coffer-dam Process for Piers Svo, 2 50 \n\nGreene\'s Roof Trusses Svo, i 25 \n\nBridge Trusses \xc2\xab Svo, 2 50 \n\nArches in Wood, Iron, and Stone Svo, 2 50 \n\nHowe\'s Treatise on Arches Svo, 4 00 \n\nDesign of Simple Roof-trusses in Wood and Steel Svo, 2 00 \n\nJ\xc2\xabhnson, Bryan, and Tumeaure\'s Theory and Practice in the Designing of \n\nModern Framed Structures Small 4to, 10 00 \n\nMerriman and Jacoby\'s Text-book on Roofs and Bridges: \n\nPart I. \xe2\x80\x94 Stresses in Simple Trusses Svo, 2 50 \n\nPart n. \xe2\x80\x94 Graphic Statics Svo, 2 50 \n\nPart III. \xe2\x80\x94 Bridge Design. 4th Edition, Rewritten Svo, 2 50 \n\nPart IV. \xe2\x80\x94 Higher Structures Svo, 2 50 \n\nMorison\'s Memphis Bridge 4to, 10 00 \n\nWaddell\'s De Pontibus, a Pocket-book for Bridge Engineers. . . i6mo, morocco, 3 00 \n\nSpecifications for Steel Bridges i2mo, i 25 \n\nWood\'s Treatise on the Theory of the Construction of Bridges and Roofs. Svo, 2 00 \nWright\'s Designing of Draw-spans: \n\nPart I. \xe2\x80\x94 Plate-girder Draws Svo, 2 50 \n\nPart II. \xe2\x80\x94 Riveted- truss and Pin-connected Long-span Draws Svo, 2 50 \n\nTwo parts in one volume Svo, 3 50 \n\n6 \n\n\n\n^ HYDRAULICS. \n\nBazin*s Experiments upon the Contraction of the Liquid Vein Issuing from an \n\nOrifice. (Trautwine.) 8vo, 2 00 \n\nBovey\'s Treatise on Hydraulics 8vo, 5 00 \n\nChurch\'s Mechanics of Engineering 8vo, 6 00 \n\nDiagrams of Mean Velocity of Water in Open Channels paper, i 50 \n\nCoffin\'s Graphical Solution of Hydraulic Problems i6mo, morocco, 2 50 \n\nFlather\'s Dynamometers, and the Measurement of Power i2mo, 3 00 \n\nFolwell\'s Water-supply Engineering , Svo, 4 00 \n\nFrizell\'s Water-power Svo, 5 00 \n\nFuertes\'s Water and Public Health , i2mo, i 50 \n\nWater-filtration Works i2mo, 2 50 \n\nGanguillet and Kutter\'s General Formula for the Uniform Flow of Water in \n\nRivers and Other Channels. (Hering and Trautwine.) Svo, 4 00 \n\nHazen\'s Filtration of Public Water-supply Svo, 3 00 \n\nHazlehurst\'s Towers and Tanks for Water- works Svo, 2 50 \n\nHerschel\'s 115 Experiments on the Carrying Capacity of Large, Riveted, Metal \n\nConduits Svo, 2 00 \n\nMason\'s Water-supply. (Considered Principally from a Sanitary Stand- \npoint.) 3d Edition, Rewritten Svo, 4 00 \n\nMerriman\'s Treatise on Hydraulics. 9th Edition, Rewritten Svo, 5 00 \n\n* Michie\'s Elements of Analytical Mechanics Svo, 4 00 \n\nSchuyler\'s Reservoirs for Irrigation, Water-power, and Domestic Water- \nsupply : Large Svo, 5 00 \n\n\xe2\x99\xa6* Thomas and Watt\'s Improvement of Riyers. (Post., 44 c. additional), 4to, 6 00 \n\nTurneaure and Russell\'s Public Water-supplies Svo, 5 00 \n\nWegmann\'s Desien and Construction of Dams 4to, 5 00 \n\nWater-supply of the City of New York from 1658 to 1895 4to, 10 00 \n\nWeisbach\'s Hydraulics and Hydraulic Motors. (Du Bois.) Svo, 5 00 \n\nWilson\'s Manual of Irrigation Engineering Small Svo, 4 00 \n\nWolff\'s Windmill as a Prime Mover Svo, 3 00 \n\nWood\'s Turbines Svo, 3 50 \n\nElements of Analytical Mechanics Svo, 3 00 \n\nMATERIALS OF ENGINEERING. \n\nBaker\'s Treatise on Masonry Construction 8vo, 5 00 \n\nRoads and Pavements Svo, s 00 \n\nBlack\'s United States Public Works Oblong 4to, 5 00 \n\nBovey\'s Strength of Materials and Theory of Structures Svo, 7 50 \n\nBurr\'s Elasticity and Resistance of the Materials of Engineering. 6th Edi- \ntion, Rewritten Svo, 7 50 \n\nByrne\'s Highway Construction Svo, 5 00 \n\nInspection of the Materials and Workmanship Employed Jn Construction. \n\ni6mo, 3 00 \n\nChurch\'s Mechanics of Engineering Svo, 6 00 \n\nDu Bois\'s Mechanics of Engineering. Vol. I Small 4to, 7 50 \n\nJohnson\'s Materials of Construction Large Svo, 6 00 \n\nKeep\'s Cast Iron Svo, 2 50 \n\nLanza\'s Applied Mechanics Svo, 7 50 \n\nMartens\'s Handbook on Testing Materials. (Henning.) 2 vols Svo, 7 50 \n\nMerrill\'s Stones for Building and Decoration Svo, 5 00 \n\nMerriman\'s Text-book on the Mechanics of Materials Svo, 4 00 \n\nStrength of Materials i2mo, i 00 \n\nMetcalf\'s Steel. A Manual for Steel-users i2mo, 2 00 \n\nPatton\'s Practical Treatise on Foundations Svo, 5 00 \n\n7 \n\n\n\nI \n\n\n2S \n\n\nI \n\n\noo \n\n\n3 \n\n\n50 \n\n\n2 \n\n\noo \n\n\n2 \n\n\noo \n\n\n8 \n\n\noo \n\n\n2 \n\n\noo \n\n\n3 \n\n\n50 \n\n\n2 \n\n\nso \n\n\n5 \n\n\noo \n\n\n4 \n\n\noo \n\n\n3 \n\n\noo \n\n\nI \n\n\n25 \n\n\n2 \n\n\noo \n\n\n3 \n\n\noo \n\n\n4 \n\n\noo \n\n\n\nRockwell\'s Roads and Pavements in France i2mo, \n\nSmith\'s Materials of Machines i2mo, \n\nSnow\'s Principal Species of Wood 8vo, \n\nSpalding\'s HydrauUc Cement i2mo. \n\nText-book on Roads and Pavements i2mo, \n\nThurston\'s Materials of Engineering. 3 Parts 8vo, \n\nPart I. \xe2\x80\x94 Non-metallic Materials of Engineering and Metallurgy 8vo, \n\nPart II.\xe2\x80\x94 Iron and Steel 8vo, \n\nPart III. \xe2\x80\x94 A Treatise on Brasses, Bronzes, and Other Alloys and their \n\nConstituents 8vo, \n\nThurston\'s Text-book of the Materials of Construction 8vo, \n\nTiUson\'s Street Pavements and Paving Materials 8vo, \n\nWaddell\'s De Pontibus. (A Pocket-book for Bridge Engineers.) . . i6mo, mor.. \n\nSpecifications for Steel Bridges i2mo. \n\nWood\'s Treatise on the Resistance of Materials, and an Appendix on the Pres- \nervation of Timber 8vo, \n\nElements of Analytical Mechanics 8vo, \n\nWood\'s Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. . .Svo, \n\nRAILWAY ENGINEERING. \n\nAndrews\'s Handbook for Street Railway Engineers. 3X5 inches, morocco, i 25 \n\nBerg\'s Buildings and Structures of American Railroads 4to, s 00 \n\nBrooks\'s Handbook of Street Railroad Location i6mo. morocco, i 50 \n\nButts\'s Civil Engineer\'s Field-book i6mo, morocco, 2 50 \n\nCrandall\'s Transition Curve i6mo, morocco, i 50 \n\nRailway and Other EarthworkvTables 8vo, i 50 \n\nDawson\'s "Engineering" and Electric Traction Pocket-book. i6mo, morocco, 5 00 \nDredge\'s History of the Pennsylvania Railroad : (1879) Paper, 5 00 \n\n\xe2\x80\xa2 Drinker\'s Tunneling, Explosive Compounds, and Rock Drills, 4to, half mor., 25 00 \n\nFisher\'s Table of Cubic Yards Cardboard, 25 \n\nGodwin\'s Railroad Engineers\' Field-book and Explorers\' Guide i6mo, mor., 2 50 \n\nHoward\'s Transition Curve Field-book i6mo, morocco, i so \n\nHudson\'s Tables for Calculating the Cubic Contents of Excavations and Em- \nbankments 8vo, I 00 \n\nMolitor and Beard\'s Manual for Resident Engineers i6mo, i 00 \n\nNagle\'s Field Manual for Railroad Engineers i6mo, morocco. 3 00 \n\nPhilbrick\'s Field Manual for Engineers i6mo, morocco, 3 00 \n\nSearles\'s Field Engineering i6mo, morocco, 3 00 \n\nRailroad SpiraL i6mo, morocco, i 50 \n\nTaylor\'s Prismoidal Formulae and Earthwork Svo, 1 50 \n\n\xe2\x80\xa2 Trautwine\'s Method of Calculating the Cubic Contents of Excavations and \n\nEmbankments by the Aid of Diagrams Svo, 2 00 \n\nThe Field Practice of [Laying Out Circular Curves for Railroads. \n\n1 2mo, morocco, 2 50 \n\nCross-section Sheet Paper, 25 \n\nWebb\'s Railroad Construction. 2d Edition, Rewritten i6mo. morocco, s 00 \n\nWellington\'s Economic Theory of the Location of Railways Small Svo, 5 00 \n\nDRAWING. \n\nBarr\'s Kinematics of Machinery Svo, 2 50 \n\n\xe2\x80\xa2 Bartlett\'s Mechanical Drawing Svo, 3 00 \n\n\xe2\x80\xa2 \xe2\x80\xa2\xe2\x80\xa2 \' " Abridged Ed Svo, i 50 \n\nCoolidge\'s Manual of Drawing 8vo, paper, i 00 \n\nCoolidge and Freeman\'s Elements of General Drafting for Mechanical Engi- \nneers. {In press.) \n\nDurley\'s Kinematics of Machines \xe2\x80\x94 . , Svo, 4 oe \n\n8 \n\n\n\nHill\'s Text-book on Shades and Shadows, and Perspective 8vo, 2 oo \n\nJamison\'s Elements of Mechanical Drawing. {In press.) \n\nJones\'s Machine Design: \n\nPart I. \xe2\x80\x94 Kinematics of Machinery 8vo, i 50 \n\nPart II. \xe2\x80\x94 Form, Strength, and Proportions of Parts 8vo, 3 00 \n\nMacCord\'s Elements of Descriptive Geometrj . , 8vo, 3 00 \n\nKinematics; or. Practical Mechanism , , , 8vo, 5 00 \n\nMechanical Drawing < . . 4to, 4 00 \n\nVelocity Diagrams 8vo, i so \n\n\xe2\x80\xa2 Mahan\'s Descriptive Geometry and Stone-cutting 8vo, i 50 \n\nIndustrial Drawing. (Thompson.) 8vo, 3 50 \n\nReed\'s Topographical Drawing and Sketching 4to, 5 00 \n\nReid\'s Course in Mechanical Drawing 8vo, 2 00 \n\nText-book of Mechanical Drawing and Elementary Machine Design . . 8vo, 3 o^ \n\nRobinson\'s Principles of Mechanism 8vo, 3 00 \n\nSmith\'s Manual of Topographical Drawing. (McMillan.) 8vo, 2 50 \n\nWarren\'s Elements of Plane and Solid Free-hand Geometrical Drawing. . i2mo, i 00 \n\nDrafting Instruments and Operations i2mo, i 25 \n\nManual of Elementary Projection Drawing i2mo, i 50 \n\nManual of Elementary Problems in the Linear Perspective of Form and \n\nShadow izmo, i 00 \n\nPlane Problems in Elementary Geometry i2mo, i 25 \n\nPrimary Geometry ^ . i2mo, 73 \n\nElements of Descriptive Geometry, Shadows, and Perspective 8vo, 3 50 \n\nGeneral Problems of Shades and Shadows 8to, 3 00 \n\nElements of Machine Construction and Drawing 8vo, 7 50 \n\nProblems. Theorems, and Examples in Descriptive Geometrv 8vo, 2 50 \n\nWeisbach\'s Kinematics and the Power of Transmission. (Hermann and \n\nKlein.) 8vo, 5 00 \n\nWhelpley\'s Practical Instruction in the Art of Letter Engraving i2mo, 2 00 \n\nWilson\'s Topographic Surveying 8vo, 3 50 \n\nFree-hand Perspective 8vo, 2 50 \n\nFree-hand Lettering 8vo, x 00 \n\nWooif\'s Elementary Course in Descriptive Geometry Large 8vo, 3 00 \n\n"ELECTRICITY AND PHYSICS. \n\nAnthony and Brackett\'s Text-book of Physics. (Magie.) Small 8vo, 3 00 \n\nAnthony\'s Lecture-notes on the Theory of Electrical Measurements i2mo, i 00 \n\nBenjamin\'s History of Electricity 8vo, 3 00 \n\nVoltaic Cell. 8vo, 3 00 \n\nClassen\'s Quantitative Chemical Analysis by Electrolysis. (Boltwood.). .8vo, 3 00 \n\nCrehore and Squier\'s Polarizing Photo-chronograph 8vo, 3 00 \n\nDawson\'s "Encineering" and Electric Traction Pocket-book. .i6mo, morocco, 5 00 \nDolezalek\'s Theory of the Lead Accumulator (Storage Battery). (Von \n\nEnde.) i2mo,"2 50 \n\nDuhem\'s Thermodynamics and Chemistry. (Burgess.) Svo, 4 00 \n\nFlather\'s Dvnamometers, and the Measurement of Power i2mo, 3 00 \n\nGilbert\'s De Magnete. (Mottelay.) 8vo, 2 50 \n\nHanchett\'s Alternating Currents Explained i2mo, i 00 \n\nBering\'s Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 \n\nHolman\'s Precision of Measurements 8vo, 2 00 \n\nTelescopic Mirror-scale Method, Adjustments, and Tests Large Svo, 75 \n\nLandauer\'s Spectrum Analysis. (Tingle.) 8vo, 3 00 \n\nLe Chatelier\'s High-temperature Measurements. (Boudouard \xe2\x80\x94 Burgess. )i2mo, 3 00 \n\nLob\'s Electrolysis and Electrosynthesis of Organic Compounds. (Lorenz.) i2mo, i 00 \n\n\xe2\x80\xa2 Lyons\'s Treatise on Electromagnetic Phenomena. Vols. I. and U. Svo, each, 6 00 \n\n\xe2\x80\xa2 Michie. Elements of Wave Motion Relating to Sotmd and Light Svo, 4 00 \n\n\n\n2 \n\n\n50 \n\n\n7 \n\n\noo \n\n\n7 \n\n\nso \n\n\nI \n\n\nSO \n\n\n6 \n\n\noo \n\n\n6 \n\n\nso \n\n\n5 \n\n\noo \n\n\n5 \n\n\nso \n\n\n3 \n\n\noo \n\n\n2 \n\n\n50 \n\n\n\nNiaudet\'s Elementary Treatise on Electric Batteries. (FisHoack. )...... izmo, 2 50 \n\n\xe2\x80\xa2 Rosenberg\'s Electrical Engineering, (Haldane Gee \xe2\x80\x94 Kinzbrunner.). . . .8vo, i 50 \n\nRyan, Korris, and Hoxie\'s Electrical Machinery. VoL 1 8vo, 2 50 \n\nThurston\'s Stationary Steam-engines 8vo, 2 50 \n\n* Tillman\'s Elementary Lessons in Heat 8vo, i 50 \n\nTory and Pitcher\'s Manual of Laboratory Physics Small 8vo, 2 00 \n\nUlke\'s Modern Electrolytic Copper Refining 8vo, 3 00 \n\n\n\nLAW. \n\n* Davis\'s Elements of Law 8vo, \n\n* Treatise on the Military Law of United States 8vo, \n\n* Sheep, \n\nManual for Courts-martial i6mo, morocco, \n\nWait\'s Engineering and Architectural Jurisprudence 8vo, \n\nSheep, \nLaw of Operations Preliminary to Construction in Engineering and Archi- \ntecture 8vo, \n\nSheep, \n\nLaw of Contracts 8vo, \n\nWinthrop\'s Abridgment of Military Law i2mo, \n\nMANUFACTURES. \n\nBernadou\'s Smokeless Powder \xe2\x80\x94 Nitro-cellulose and Theory of the Cellulose \n\nMolecule i2mo, 2 so \n\nBolland\'s Iron Founder i2mo, 2 50 \n\n*\xe2\x80\xa2 The Iron Founder," Supplement i2mo, 2 50 \n\nEncyclopedia of Founding and Dictionary of Foundry Terms Used in the \n\nPractice of Moulding i2mo, 3 00 \n\nEissler\'s Modem High Explosives 8vo, 4 00 \n\nEffront\'s Enzymes and their Applications, (Prescott,) Svo, 3 00 \n\nFitzgerald\'s Boston Machinist iSmo, i 00 \n\nFord\'s Boiler Making for Boiler Makers i8mo, i 00 \n\nHopkins\'s Oil-chemists\' Handbook Svo, 3 00 \n\nKeep\'s Cast Iron Svo, 2 50 \n\nLeach\'s The Inspection and Analysis of Food with Special Reference to State \n\nControl. (In preparation.) \n\nMetcalf\'s SteeL A Manual for Steel-users i2mo, 2 00 \n\nMetcalfe\'s Cost of Manufactures \xe2\x80\x94 And the Administration of Workshops, \n\nPublic and Private Svo, 5 00 \n\nMeyer\'s Modern Locomotive Construction 4to, 10 00 \n\nMorse\'s Calculations used in Cane-sugar Factories. i6mo, morocco, i 50 \n\n* Reisig\'s Guide to Piece-dyeing Svo, 25 00 \n\nSmith\'s Press-working of Metals Svo, 3 00 \n\nSpalding\'s Hydraulic Cement i2mo, 2 00 \n\nSpencer\'s Handbook for Chemists of Beet-sugar Houses i6mo, morocco, 3 00 \n\nHandbooK lor sugar Manutaciurers and their Chemists.. . i6mo, morocco, 2 00 \nThurston\'s Manual of Steam-boilers, their Designs, Construction and Opera- \ntion Svo, s 00 \n\n* Walke\'s Lectures on Explosives Svo, 4 00 \n\nWest\'s American Foundry Practice i2mo, 2 50 \n\nMoulder\'s Text-book , i2mo, 2 50 \n\nWiechmann\'s Sugar Analysis Small 8vo, 2 50 \n\nWolff\'s Windmill as a Prime Mover Svo, 3 00 \n\nWoodbury\'s Fire Protection of Mills Svo, 2 50 \n\nWood\'s Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. .. Svo, 4 00 \n\n10 \n\n\n\nMATHEMATICS. \n\nBaker\'s Elliptic Functions 8vo, \n\n* Bass\'s Elements of Differential Calculus i2mo, \n\nBriggs\'s Elements of Plane Analytic Geometry i2mo, \n\nCompton\'s Manual of Logarithmic Computations i2mo, \n\nDavis\'s Introduction to the Logic of Algebra 8vo, \n\n* Dickson\'s College Algebra Large i2mo, \n\n* Answers to Dickson\'s College Algebra 8vo, paper, \n\n* Introduction to the Theory of Algebraic Equations Large i2mo, \n\nHalsted\'s Elements of Geometry 8vo, \n\nElementary Synthetic Geometry Svo, \n\nRational Geometry i2mo, \n\n* Johnson\'s Three-place Logarithmic Tables : Vest-pocket size paper, \n\nloo copies for \n\n* Mounted on heavy cardboard, 8 X lo inches, \n\n10 copies for \n\nElementary Treatise on the Integral Calculus Small Svo, \n\nCurve Tracing in Cartesian Co-ordinates i2mo. \n\nTreatise on Ordinary and Partial Differential Equations Small Svo, \n\nTheory of Errors and the Method of Least Squares i2mo, \n\n* Theoretical Mechanics i2mo, \n\nLaplace\'s Philosophical Essay on Probabilities. (Truscott and Emory.) i2mo, \n\n* Ludlow and Bass. Elements of Trigonometry and Logarithmic and Other \n\nTables Svo, \n\nTrigonometry and Tables published separately Each, \n\n* Ludlow\'s Logarithmic and Trigonometric Tables Svo, \n\nMaurer\'s Technical Mechanics Svo, \n\nMerriman and Woodward\'s Higher Mathematics Svo, \n\nMerriman\'s Method of Least Squares Svo, \n\nRice and Johnson\'s Elementary Treatise on the Differential Calculus. Sm., Svo, \n\nDifferential and Integral Calculus. 2 vols, in one Small Svo, \n\nSabin\'s Industrial and Artistic Technology of Paints and Varnish. {In press.) \nWood\'s Elements of Co-ordinate Geometry Svo, \n\nTrigonometry: Analytical, Plane, and Spherical i2mo, \n\nMECHANICAL ENGINEERING. \nMATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS. \n\nBaldwin\'s Steam Heating for Buildings i2mo, \n\nBarr\'s Kinematics of Machinery . . , . ; Svo, \n\n* Bartlett\'s Mechanical Drawing Svo, \n\n* " " " Abridged Ed Svo, \n\nBenjamin\'s Wrinkles and Recipes i2mo. \n\nCarpenter\'s Experimental Engineering Svo, \n\nHeating and Ventilating Buildings Svo, \n\nGary\'s Smoke Suppression in Plants using Bituminous CoaL (In prep- \naration.) \n\nClerk\'s Gas and Oil Engine Small Svo, \n\nCoolidge\'s Manual of Drawing Svo, paper, \n\nCooUdge and Freeman\'s Elements of General Drafting for Mechanical En- \ngineers. {In press.) \n\nCromwell\'s Treatise on Toothed Gearing i2mo. \n\nTreatise on Belts and Pulleys i2mo, \n\nDurley\'s Kinematics of Machines Svo, \n\nFlather\'s Dynamometers and the Measurement of Power i2mo. \n\nRope Driving , , i2mo, \n\n11 \n\n\n\nI \n\n\n50 \n\n\n4 \n\n\n00 \n\n\nI \n\n\n00 \n\n\nI \n\n\n50 \n\n\nI \n\n\n50 \n\n\nX \n\n\n50 \n\n\n\n\n25 \n\n\nI \n\n\n25 \n\n\nI \n\n\n75 \n\n\nI \n\n\n50 \n\n\nI \n\n\n75 \n\n\n\n\n15 \n\n\n5 \n\n\n00 \n\n\n\n\n25 \n\n\n2 \n\n\n00 \n\n\nI \n\n\n50 \n\n\nI \n\n\n00 \n\n\n3 \n\n\n50 \n\n\nI \n\n\n50 \n\n\n3 \n\n\n00 \n\n\n2 \n\n\n00 \n\n\n3 \n\n\n00 \n\n\n2 \n\n\n00 \n\n\nI \n\n\n00 \n\n\n4 \n\n\n00 \n\n\n5 \n\n\n00 \n\n\n2 \n\n\n00 \n\n\n3 \n\n\n00 \n\n\n2 \n\n\n50 \n\n\n2 \n\n\n00 \n\n\nI \n\n\n00 \n\n\n\n2 \n\n\nso \n\n\n2 \n\n\nso \n\n\n3 \n\n\n00 \n\n\nI \n\n\n50 \n\n\n2 \n\n\n00 \n\n\n6 \n\n\n00 \n\n\n4 \n\n\n00 \n\n\n4 \n\n\n00 \n\n\nI \n\n\n00 \n\n\nI \n\n\nSO \n\n\nI \n\n\nSO \n\n\n4 \n\n\n00 \n\n\n3 \n\n\n00 \n\n\n2 \n\n\n00 \n\n\n\nGill\'s Gas and Fuel Analysis for Engineers i2mo, i 25 \n\nHall\'s Car Lubrication i2mo, i 00 \n\nBering\'s Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 \n\nButton\'s The Gas Engine 8vo, s ou \n\nJones\'s Machine Design: \n\nPart I. \xe2\x80\x94 Kinematics of Machinery 8vo, i 50 \n\nPart II. \xe2\x80\x94 Form, Strength, and Proportions of Parts 8vo, 3 00 \n\nKent\'s Mechanical Engineer\'s Pocket-book i6mo, morocco, 5 00 \n\nKerr\'s Power and Power Transmission Svo, 2 00 \n\nMacCord\'s Kinematics; or, Practical Mechanism 8vo, 5 00 \n\nMechanical Drawing 4to, 4 00 \n\nVelocity Diagrams 8vo, i 50 \n\nMahan\'s Industrial Drawing. (Thompson.) 8vo, 3 50 \n\nPoole\'s Calorific Power of Fuels ,^vo, 3 00 \n\nReid\'s Course in Mechanical Drawing 8vo, 2 00 \n\nText-book of Mechanical Drawing and Elementary Machine Design. ,8vo, 3 00 \n\nRichards\'s Compressed Air i2mo, i 50 \n\nRobinson\'s Principles of Mechanism 8vo, 3 00 \n\nSmith\'s Press-working of Metals ,8vo, 3 00 \n\nThurston\'s Treatise on Friction and Lost Work in Machinery and Mill \n\nWork 8vo, 3 00 \n\nAnimal as a Machine and Prime Motor, and the Laws of Energetics. 1 2mo, i 00 \n\nWarren\'s Elements of Machine Construction and Drawing 870, 7 50 \n\nWeisbach\'s Kinematics and the Power of Transmission. Herrmann \xe2\x80\x94 \n\nKlein.) Svo, 5 00 \n\nMachinery of Transmission and Governors. (Herrmann \xe2\x80\x94 Klein.). . Svo, 500 \n\nHydraulics and Hydraulic Motors. (Du Bois.) Svo, 5 00 \n\nWolff\'s Windmill as a Prime Mover Svo, 3 00 \n\nWood\'s Turbines Svo, 2 50 \n\nMATERIALS OF ENGINEERING. \n\nBovey\'s Strength of Materials and Theory of Structures Svo, 7 50 \n\nBurr\'s Elasticity and Resistance of the Materials of Engineering. 6th Edition, \n\nReset Svo, 7 50 \n\nChurch\'s Mechanics of Engineering Svo, 6 00 \n\nJohnson\'s Materials of Construction Large Svo, 6 00 \n\nKeep\'s Cast Iron Svo, 2 50 \n\nLanza\'s Applied Mechanics Svo, 7 50 \n\nMartens\'s Handbook on Testing Materials. (Henning.) Svo, 7 50 \n\nMerriman\'s Text-book on the Mechanics of Materials Svo, 4 00 \n\nStrength of Mater>als i2mo, i 00 \n\nMetcalf\'s SteeL A Manual for Steel-users i2mo, 2 00 \n\nSmith\'s Materials of Machines i2mo i 00 \n\nThurston\'s Materials of Engineering 3 vols., Svo, 8 00 \n\nPart II.\xe2\x80\x94 Iron and Steel Svo, 3 50 \n\nPart III. \xe2\x80\x94 A Treatise on Brasses, Bronzes, and Other Alloys and their \n\nConstituents Svo 2 50 \n\nText-book of the Materials of Construction Svo, 5 00 \n\nWood\'s Treatise on the Resistance of Materials and an Appendix on the \n\nPreservation of Timber Svo, 2 00 \n\nElements of Analytical Mechanics Svo, 3 00 \n\nWood\'s Rustless Coatings: Corrosion and Electrolysis of Iron and Steel, . .Svo, 4 00 \n\nSTEAM-ENGINES AND BOILERS. \n\nCarnot\'s Reflections on the Motive Power of Heat. (Thtirston.) i2mo, l 50 \n\nDawson\'s "Engineering" and Electric Traction Pocket-book. .i6mo, mor., 5 00 \n\nFord\'s Boiler Making for Boiler Makers iSmo, i 00 \n\n12 \n\n\n\nGoss\'s LocomotiTe Sparks 8vo, 2 00 \n\nHemenway\'s Indicator Practice and Steam-engine Economy i2mo, a 00 \n\nButton\'s Mechanical Engineering of Power Plants 8vo, 5 00 \n\nHeat and Heat-engines 8vo, 5 00 \n\nKent\'s Steam-boiler Economy Svo, 4 00 \n\nKneass\'s Practice and Theory of the Injector Svo i 50 \n\nMacCord\'s Slide-valves Svo, 2 00 \n\nMeyer\'s Modem Locomotive Construction 4to\xc2\xbb 10 00 \n\nPeabody\'s Manual of the Steam-engine Indicator i2mo, i 50 \n\nTables of the Properties of Saturated Steam and Other Vapors Svo, i 00 \n\nThermod3mamics of the Steam-engine and Other Heat-engines Svo, 5 oo \n\nValve-gears for Steam-engines Svo, 2 50 \n\nPeabody and Miller\'s Steam-boilers Svo, 4 00 \n\nPray\'s Twenty Years with the Indicator Large Svo, 2 50 \n\nPupln\'s Thermodynamics of Reversible Cycles in Gases and Saturated Vapors. \n\n(Osterberg.) i2mo, i 25 \n\nReagan\'s Locomotives : Simple, Compound, and Electric i2mo, 2 50 \n\nRontgen\'s Principles of Thermodsmamics. (Du Bois.) Svo, 5 00 \n\nSinclair\'s Locomotive Engine Running and Management i2mo, 2 00 \n\nSmart\'s Handbook of Engineering Laboratory Practice i2mo, 2 50 \n\nSnow\'s Steam-boiler Practice Svo, 3 00 \n\nSpangler\'s Valve-gears Svo, 2 5\xc2\xa9 \n\nNotes on Thermodynamics i i2mo, i 00 \n\nSpangler, Greene, and Marshall\'s Elements of Steam-engineering Svo, 3 00 \n\nThurston\'s Handy Tables Svo, i 50 \n\nManual of the Steam-engine 2 vols. . Svo, 10 00 \n\nPart I. \xe2\x80\x94 History, Structuce, and Theory Svo, 6 00 \n\nPart n. \xe2\x80\x94 Design, Construction, and Operation Svo, 6 00 \n\nHandbook of Engine and Boiler Trials, and the Use of the Indicator and \n\nthe Prony Brake Svo 5 oe \n\nStationary Steam-engines Svo, 2 50 \n\nSteam-boiler Explosions in Theory and in Practice i2mo i 50 \n\nManual of Steam-boilers, Their Designs, Construction, and Operation .Svo, 5 00 \n\nWeisbach\'s Heat, Steam, and Steam-engines. (Du Bois.) Svo, 5 o\xc2\xab \n\nWhitham\'s Steam-engine Design Svo, 5 00 \n\nWilson\'s Treatise on Steam-boilers. (Flather.) i6mo, 2 50 \n\nWood\'s Thermodynamics Heat Motors, and Refrigerating Machines. . . .Svo, 4 00 \n\n\n\nMECHANICS AND MACHINERY. \n\n\n\nBarr\'s Kinematics of Machinery Svo, \n\nBovey\'s Strength of Materials and Theory of Structures Svo, \n\nChase\'s The Art of Pattern-making i2mo, \n\nChordal. \xe2\x80\x94 Extracts from Letters i2mo, \n\nChurch\'s Mechanics of Engineering Svo, \n\nNotes and Examples in Mechanics Svo, \n\nCompton\'s First Lessons in Metal- working lamo, \n\nCompton and De Groodt\'s The Speed Lathe i2mo, \n\nCromwell\'s Treatise on Toothed Gearing i2mo, \n\nTreatise on Belts and Pulleys i2mo, \n\nDana\'s Text-book of Elementary Mechanics for the Use of Colleges and \n\nSchools i2mo. \n\nDingey\'s Machinery Pattern Making i2mo. \n\nDredge\'s Record of the Transportation Exhibits Building of the World\'s \n\nColumbian Exposition of iSgs 4to, half morocco, 5 00 \n\n13 \n\n\n\n2 \n\n\n50 \n\n\n7 \n\n\n50 \n\n\n2 \n\n\n50 \n\n\n2 \n\n\n00 \n\n\n6 \n\n\n00 \n\n\n2 \n\n\n00 \n\n\nI \n\n\n50 \n\n\nI \n\n\n50 \n\n\nI \n\n\n50 \n\n\nI \n\n\n50 \n\n\n1 \n\n\n50 \n\n\n2 \n\n\n00 \n\n\n\nDu Bois\'s Elementary Principles of Mechanics : \n\nVol. I. \xe2\x80\x94 Kinematics 8vo, \n\nVol. II.\xe2\x80\x94 Statics .\' 8vo, \n\nVol. III.\xe2\x80\x94 Kinetics 8vo, \n\nMechanics of Engineering. Vol. I Small 4to, \n\nVol. II Small 4to, \n\nDurley\'s Kinematics of Machines 8vo, \n\nFitzgerald\'s Boston Machinist i6mo, \n\nFlather\'s Dynamometers, and the Measurement of Power i2mo. \n\nRope Driving i2mo, \n\nGoss\'s Locomotive Sparks 8vo \n\nHall\'s Car Lubrication i2mo. \n\nHolly\'s Art of Saw Filing i8mo , \n\n* Johnson\'s Theoretical Mechanics i2mo. \n\nStatics by Graphic and Algebraic Methods 8vo, \n\nJones\'s Machine Design: \n\nPart I. \xe2\x80\x94 Kinematics of Machinery 8vo, \n\nPart n. \xe2\x80\x94 Form, Strength, and Proportions of Parts 8vo, \n\nKetr\'s Power and Power Transmission 8vo, \n\nLanza\'s Applied Mechanics 8vo, \n\nMacCord\'s Kinematics; or. Practical Mechanism 8vo, \n\nVelocity Diagrams 8vo, \n\nMaurer\'s Technical Mechanics 8vo, \n\nMerriman\'s Text-book on the Mechanics of Materials 8vo, \n\n\xe2\x80\xa2 Michie\'s Elements of Anal3^ical Mechanics 8vo, \n\nReagan\'s Locomotives: Simple, Compound, and Electric i2mo, \n\nReid\'s Course in Mechanical Drawing 8vo, \n\nText-book of Mechanical Drawing and Elementary Machine Design. .8vo, \n\nRichards\'s Compressed Air i2mo, \n\nRobinson\'s Principles of Mechanism 8vo, \n\nRyan, Norris, and Hoxie\'s Electrical Machinery. Vol. 1 8vo, \n\nSinclair\'s Locomotive-engine Running and Management i2mo, \n\nSmith\'s Press-working of Metals 8vo, \n\nMaterials of Machines i2mo, \n\nSpangler, Greene, and Marshall\'s Elements of Steam-engineering 8vo, \n\nThurston\'s Treatise on Friction and Lost Work in Machinery and Mill \nWork 8vo, \n\nAnimal as a Machine and Prime Motor, and the Laws of Energetics. i2mo, \n\nWarren\'s Elements of Machine Construction and Drawing 8vo, \n\nWeisbach\'s Kinematics and the Power of Transmission. (Herrmann \xe2\x80\x94 \nKlein.) 8vo, \n\nMachinery of Transmission and Governors. (Herrmann \xe2\x80\x94 Klein. ).8vo. \nWood\'s Elements of Analytical Mechanics 8vo, \n\nPrinciples of Elementary Mechanics i2mo. \n\nTurbines 8vo, \n\nThe World\'s Columbian Exposition of 1893 4to, \n\nMETALLURGY. \n\nEgleston\'s Metallurgy of Silver, Gold, and Mercury: \n\nVol. I.\xe2\x80\x94 Silver 8vo, 7 So \n\nVol. II.\xe2\x80\x94 Gold and Mercury 8vo, 7 50 \n\n\xe2\x99\xa6\xe2\x99\xa6 Iles\'s Lead-smelting. (Postage 9 cents additional.) i2mo, 2 50 \n\nKeep\'s Cast Iron 8vo, 2 50 \n\nKunhardt\'s Practice of Ore Dressing in Europe 8vo, i 50 \n\nLe Chatelier\'s High-temperature Measurements. (Boudouard \xe2\x80\x94 Burgess.) . i2mo, 3 00 \n\nMetcalf\'s SteeL A Manual for Steel-users i2mo, 2 00 \n\nSmith\'s Materials of Machines i2mo, i 00 \n\n14 \n\n\n\n3 \n\n\n50 \n\n\n4 \n\n\n00 \n\n\n3 \n\n\n50 \n\n\n7 \n\n\n50 \n\n\n10 \n\n\n00 \n\n\n4 \n\n\n00 \n\n\nI \n\n\n00 \n\n\n3 \n\n\n00 \n\n\n2 \n\n\n00 \n\n\n2 \n\n\n00 \n\n\nI \n\n\n00 \n\n\n\n\n7S \n\n\n3 \n\n\nCO \n\n\n2 \n\n\n00 \n\n\nI \n\n\nSO \n\n\n3 \n\n\n00 \n\n\n2 \n\n\n00 \n\n\n7 \n\n\n50 \n\n\n5 \n\n\n00 \n\n\nX \n\n\nSO \n\n\n4 \n\n\n00 \n\n\n4 \n\n\n00 \n\n\n4 \n\n\n00 \n\n\n2 \n\n\nSO \n\n\n2 \n\n\n00 \n\n\n3 \n\n\n00 \n\n\nI \n\n\nSO \n\n\n3 \n\n\n00 \n\n\n2 \n\n\nSO \n\n\n2 \n\n\n00 \n\n\n3 \n\n\n00 \n\n\n1 \n\n\n00 \n\n\n3 \n\n\n00 \n\n\n3 \n\n\n00 \n\n\nI \n\n\n00 \n\n\n7 \n\n\nSO \n\n\n5 \n\n\n00 \n\n\n5 \n\n\n00 \n\n\n3 \n\n\n00 \n\n\nI \n\n\n25 \n\n\n2 \n\n\nSO \n\n\nI \n\n\n00 \n\n\n\n2 \n\n\n5o \n\n\n3 \n\n\noo \n\n\n2 \n\n\noo \n\n\n4 \n\n\noo \n\n\nI \n\n\noo \n\n\nI \n\n\n25 \n\n\n3 \n\n\n50 \n\n\n[2 \n\n\n50 \n\n\nI \n\n\n00 \n\n\n4 \n\n\n00 \n\n\nI \n\n\n50 \n\n\nI \n\n\n00 \n\n\n2 \n\n\n00 \n\n\nI \n\n\n25 \n\n\n2 \n\n\n50 \n\n\n2 \n\n\n00 \n\n\n4 \n\n\n00 \n\n\n\nThurston\'s Materials of Engineering, In Three Parts 8vo, 8 00 \n\nPart II. \xe2\x80\x94 Iron and Steel Svo, 3 5o \n\nPart III. \xe2\x80\x94 A Treatise on Brasses. Bronzes, and Other Alloys and their \n\nConstituents Svo, 2 50 \n\nUlke\'s Modem Electrolytic Copper Refining Svo, 3 00 \n\nMINERALOGY. \n\nBarringer\'s Description of Minerals of Commercial Value. Oblong, morocco, \n\nBoyd\'s Resources of Southwest Virginia Svo, \n\nMap of Southwest Virginia Pocket-book form. \n\nBrush\'s Manual of Determinative Mineralogy. (Penfield.) Svo, \n\nChester\'s Catalogue of Minerals Svo, paper. \n\nCloth, \n\nDictionary of the Names of Minerals \xe2\x80\x9e Svo, \n\nDana\'s System of Mineralogy Large Svo, half leather. \n\nFirst Appendix to Dana\'s New "System of Mineralogy.". .. .Large Svo, \n\nText-book of Mineralogy Svo, \n\nMinerals and How to Study Them i2mo. \n\nCatalogue of American Localities of Minerals Large Svo, \n\nManual of Mineralogy and Petrography i2mo, \n\nEakle\'s Mineral Tables Svo, \n\nEgleston\'s Catalogue of Minerals and Synonyms Svo, \n\nHussak\'s The Determination of Rock-forming Minerals. (Smith.) Small Svo, \nMerrill\'s Non-metallic Minerals: Their Occurrence and Uses Svo, \n\n* Penfield\'s Notes on Determinative Mineralogy and Record of Mineral Tests. \n\nSvo, paper, o 50 \nRosenbusch\'s Microscopical Physiography of the Rock-making Minerals. \n\n(Iddmgs.) Svo, 5 00 \n\n\xe2\x80\xa2 Tillman\'s Text-book of Important Minerals and Docks Svo, 2 00 \n\nWiUiams\'s Manual of Lithology Svo, 3 00 \n\nMINING. \n\nBeard\'s Ventilation of Mines i2mo, 2 50 \n\nBoyd\'s Resources of Southwest Virginia Svo, 3 00 \n\nMap of Southwest Virginia Pocket-book form, 2 00 \n\n\xe2\x99\xa6 Drinker\'s Tunneling, Explosive Compounds, and Rock Drills. \n\n4to, half morocco, 2S 00 \n\nEissler\'s Modem High Explosives Svo, \n\nFowler\'s Sewage Works Analyses i2mo, \n\nGoodyear \'s Coal-mines of the Western Coast of the United States i2mo, \n\nIhlseng\'s Manual of Mining Svo, \n\n** Iles\'s Lead-smelting. (Postage 9c. additional) i2mo, \n\nKunhardt\'s Practice of Ore Dressing in Europe Svo, \n\nO\'Driscoll\'s Notes on the Treatment of Gold Ores Svo, \n\n* Walke\'s Lectures on Explosives Svo, \n\nWilson\'s Cyanide Processes i2mo, \n\nChlorination Process i2mo. \n\nHydraulic and Placer Mining i2mo. \n\nTreatise on Practical and Theoretical Mine Ventilation i2mo \n\nSANITARY SCIENCE. \n\nCopeland\'s Manual of Bacteriology. (In preparation.) \n\nFolwell\'s Sewerage. (Designing, Construction and Maintenance.) Svo, 3 00 \n\nWater-supply Engineering Svo, 4 00 \n\nFuertes\'s Water and Public Health i2mo, i 50 \n\nWater-filtration Works i2mo, 2 50 \n\n15 \n\n\n\n4 \n\n\n00 \n\n\n2 \n\n\n00 \n\n\n2 \n\n\n50 \n\n\n4 \n\n\n00 \n\n\n2 \n\n\n50 \n\n\nI \n\n\n50 \n\n\n2 \n\n\n00 \n\n\n4 \n\n\n00 \n\n\nI \n\n\n50 \n\n\nI \n\n\n50 \n\n\n2 \n\n\n00 \n\n\nz \n\n\n25 \n\n\n\nGerhard\'s Guide to Sanitary House-inspection i6mo, i \n\nGoodrich\'s Economical Disposal of Town\'s Refuse Demy 8vo, 3 \n\nHazen\'s Filtration of Public Water-suppUes 8vo, 3 \n\nKiersted\'s Sewage Disposal i2mo, i \n\nLeach\'s The Inspection and Analysis of Food with Special Reference to State \n\nControl. (In preparation.) \nMason\'s Water-supply. (Considered Principally from a Sanitary Stand- \npoint.) 3d Edition, Rewritten 8vo, 4 \n\nExamination of Water. (Chemical and BacteriologicaL) i2mo, i \n\nMerriman\'s Elements of Sanitary Engineering , .- , 8vo, 2 \n\nNichols\'s Water-supply. (Considered Mainly from a Chemical and Sanitary \n\nStandpoint.) (1883.) Svo. 2 \n\nOgden\'s Sewer Design i2mo, 2 \n\nPrescott and Winslow\'s Elements of Water Bacteriology, with Special Reference \n\nto Sanitary Water Analysis i2mo5 i \n\n* Price\'s Handbook on Sanitation i2mo, i \n\nRichards\'s Cost of Food. A Study in Dietaries i2mo, i \n\nCost of Living as Modified by Sanitary Science i2mo, i \n\nRichards and Woodman\'s Air, Water, and Food from a Sanitary Stand- \npoint 8vo, 2 \n\n\xe2\x99\xa6 Richards and Williams\'s The Dietary Computer 8vo, 1 \n\nRideal\'s Sewage and Bacterial Purification of Sewage 8vo, 3 \n\nTurneaure and Russell\'s Public Water-supplies 8vo, s \n\nWhipple\'s Microscopy of Drinking-water 8vo, 3 \n\nWoodhull\'s Notes and Military Hygiene i6mo, i \n\n\n\nMISCELLANEOUS. \n\nBarker\'s Deep-sea Soundings 8vo, \n\nEmmons\'s Geological Guide-book of the Rocky Mountain Excursion of the \n\nInternational Congress of Geologists Large 8vo \n\nFerrel\'s Popular Treatise on the Winds 8vo \n\nHaines\'s American Railway Management I2mo^ \n\nMott\'s Composition.\'Digestibility . and Nutritive Value of Food. Mounted chart. \n\nFallacy of the Present Theory of Sound i6mo \n\nRicketts\'s History of Rensselaer Polytechnic Institute, 1824-1894. Small 8vo, \n\nRotherham\'s Emphasized New Testament Large 8vo, \n\nSteel\'s Treatise on the Diseases of the Dog 8vo, \n\nTotten\'s Important Question in Metrology 8vo \n\nThe World\'s Columbian Exposition ot 1893 4to, \n\nWorcester and Atkinson. Small Hospitals, Establishment and Maintenance, \nand Suggestions for Hospital Architecture, with Plans for a Small \nHospital i2mo. \n\n\n\nHEBREW AND CHALDEE TEXT-BOOKS. \n\nGreen\'s Grammar of the Hebrew Language 8vo, 3 \n\nElementary Hebrew Grammar i2mo, i \n\nHebrew Chrestomathy 8vo, 2 \n\nGesenius\'s Hebrew and Chaldee Lexicon to the Old Testament Scriptures. \n\n(Tregelles.) Small 4to, half morocco, 5 \n\nLetteris\'a Hebrew Bible 8vo, 2 \n\n16 \n\n\n\nk% \n\n\n\nLIBRARY OF CONGRESS \n\n\n\n\n003 507 107 6 \n\n\n\n1 \n\n\n\n\n\n\n\n\n\n'