9C>2 f^^V'^^^=-^^/ 'L/a^/^M Ztr^*^K /<:< / ^1^//%. /r\ .C;<.2 University of California • Berkeley ,/ >T^.. X/ c t^*^*- ^ TREATISE GEOMETRICAL REPRESENTATION SQUARE ROOTS OF NEGATIVE QUANTITIES. BY THE Rev. JOHN WARREN, A.M. FELLOW AND TUTOR OF JESUS COLLEGE, CAMBRIDGE. CAMBRIDGE: Printed by J. Smith, Printer to the University; SOLD BY T. STEVENSON, AND J. & J. DEIGHTON, CAMBRIDGE ; BALDWIN & CRADOCK, LONDON; PARKER, OXFORD; BELL & BRADFUTE, EDINBURGH; MILLIKEN AND W. F. WAKEMAN, DUBLIN. 1828 The Author hegs leave to acknowledge his ohligatioiis to the Syndics of the Universiti/ Press; ivho have, from the funds at their disposal, con- tributed liberally to the expense of this Work. CONTENTS. Chap. I. Page Definitions. Addition, Subtraction, Proportion, Multiplica- tion, Division. Fractions, and raising of Powers ...... 1 Chap. II. Roots of Quantities, Fractional and Negative Indices 24 Chap. III. Binomial Theorem. Expansion of a" in a series arranged according to the powers of x. Differentiation of a* . . . 76 Chap. IV. Examples illustrative of the Principles established in the preceding Chapters 106 A TREATISE GEOMETRICAL REPRESENTATION SQUARE ROOTS OF NEGATIVE QUANTITIES. CHAP. I. DEFINITIONS, ADDITION, SUBTRACTION, PROPORTION, MULTIPLICATION, DIVISION, FRACTIONS, AND RAIS- ING OF POWERS. (Art. 1.) All straight lines drawn in a given plane, from a given point, are represented in length and direction by Algebraic quantities; and in the fol- lowing Treatise whenever the word quantity is used, it is to be understood as signifying a line, (2.) Def. The given point from which the straight lines are measured is called the origin, (3.) Def. The sum of two quantities is the di- agonal of the parallelogram whose sides are the two quantities. Thus if a represent AB in length and direction, c D and h represent AC in length and direction ; and the parallelogram ABDC be completed,, and the diagonal AD be drawn; a+h represents AD in length and di- rection. (4.) Def. Subtraction is the reverse of Addition, or if the sum of any two quantities a^hhe c; h is called the difference which arises from subtracting a from c. (5.) If three quantities he added together, the sum will he the same, whatever he the order in which the quantities are added together. Let a, b, c be the three quantities. A c Let AB = a, AC^h, AD = c. 3 Let the parallelogram ABEC be completed, and the diagonal AE be drawn AE=a -{-h. Let the parallelogram AEFD be completed, and the diagonal AF be drawn AF=(a + &) 4-c. Again, let the parallelogram ABGD be completed, and the diagonal AG be drawn AG = « + c. Join CF, GF; .*. BG is parallel and equal in length to AD, and EF also parallel and equal in length to AD, BG is parallel and equal in length to EF; /. GF is parallel and equal in length to BEy and therefore to AC; .-. ACFG is a parallelogram, and AF the diagonal; .-. AF=(a + c) + b; .-. {a +b) + c=:{a-{-c) -\- L In like manner (a + b) + c = {b -{- c) -\- a, (6.) The sum of any number of quantities is the same, whatever be the order in which the quantities are added together. First, let there be four quantities, a, b, c, d. By the preceding article, the sum of any three of these will be the same, whatever be the order in which they are added together. 4 Let a -\- b -t c = s, a -\- b + d = t, a + c -{- d := u, b + c 4- rf = v. To prove that s + d = t + c = u + b=:v + a, s = a -\- b + c =: (a-]- b) +c; .\ s + d = {(a '\- b) -{- c} + d == {(a -\- b) -h d] ■{- c, (by Art. 5.) = t ^ c. In the same manner it may be proved that s + d — U'hb=v+a; therefore if four quantities be added together, the sum will be the same, M^hatever be the order in which the quantities are added together. In like manner it may be proved, if there be five quantities, &c.; therefore the sum of any number of quantities is the same, whatever be the order in which the quantities are added together. (7.) Let there be any number of quantities AB, AC, AD; and let BE be parallel and equal in length to AC, and EF parallel and equal in length to AD, and let AP be joined ; AP = AB + AC-f- AD. For join AE, CE, Then ABEC is a parallelogram^ and AE the di- agonal ; B E . . AE ^ AB + AC. In like manner it may be proved that AF=^AE-\'AD; .-. AF= AB+ AC+AIX (8.) Def. If a represent a quantity in any di- rection^ — a will represent a quantity equal in length to the former, but drawn in the opposite direction : thus if AB be represented by a, and BA be produced to C, and AC taken equal in length to AB, AC will be represented by — a, (9.) The difference which arises from subtracting h from a = a + ( ~ b). Let AB == a, AC ^ h. Join BC^ and complete the parallelogram ACBD, AB = AC+ AD, or a = b -\- AD; .\ AD is the ditFerence which arises from subtracting- h from a. Again^ produce CA to E, and make AE equal in length to AC, and join ED; therefore since DB is parallel and equal in length to AC, it is also parallel and equal in length to EA; therefore EDBA is a parallelogram, and AD the diagonal; /. AD ^ AB^- AE = a +{-b); therefore the difference which arises from subtracting b from a — a + {— b). (10.) Def. a -^ (— b) is expressed a — b. (11.) Def. Quantities drawn from the origin in a certain direction, which direction is arbitrarily as- sumed, are called positive quantities ; and those drawn in the opposite direction are called negative quantities. (12.) Def. The first of four quantities is said to have to the second the same ratio which the third has to the fourth ; when the first has in length to the second the same ratio which the third has in length to the fourth, according to Euclid's definition ; and also the angle at which the fourth is inclined to the third, is equal to the angle at which the second is inclined to the first, and is measured in the same di- rection*. /E Thus let AB in length : AC :: AD in length : JE, and also let angle DAE be equal to angle BAC, and be measured in the same direction from AD, * Angles are said to be measured in the same direction, when the arcs, which subtend them, are measured in the same direction round the circle ; and are said to be measured in opposite directions, when the arcs are measured in opposite directions. Thus angle DAE is said to be measured from AD in the same direction that angle BAC is from AB, because arc DE is measured from D in the same direction round the circle that arc BC is from B ; and angle EAD is said to be measured from AE in the opposite direction that angle BAC is from AB, because arc ED is measured from E in the opposite direction that arc BC is from B. 8 that angle BAC is from AB : then AB is said to have to AC the same ratio which AD has to AE. (13.) If d be a fourth proportional to a, b, c; there is no other quantiti/ different from d which is also a fourth proportional to q, h, c. For, (by Euclid, Book v. Props. 8 and 13.) a has in length a greater ratio to b than c has in length to any quantity greater than d; and c has in length a greater ratio to any quantity less than d, than a has in length to b; therefore there is no quantity greater or less in length than d, which is a fourth proportional to a, b, c. Neither is there any quantity different in direction from d, which is a fourth proportional to a, b, c. For since rf is a fourth proportional to a, b, c; d is inclined at the same angle to c, which b is to a. Let this angle be A, Since b is inclined to a at the angle A^ it is also inclined to a at the angles A + 36o\ J + 2.36o^ &c. A - 36o', A - 2.360', &c. ; therefore any quantity which is equal in length to d, and inclined to c at any of the angles, A, A + 36o\ A + 2.360', &c. A - 36o\ A - 2.36o^ &c. is by the definition a fourth proportional to a, b, c. But since d is incHned to c at the angle A, it is also inclined to c at the angles 9 A + 36o^ A + 2.360°, &c. A — 3600, A - 2.360^ &C. ; therefore there is no quantity different in direction from d^ which is a fourth proportional to a, h, c ; therefore there is no quantity different from d, which is a fourth proportional to a, b, c. (14.) If four quantifies be proportionals, they are proportionals also when taken inversely or alter- nately. AC '. ;: AD : AE. AB : ;: AE AD, AD : ;: AC : AE. Let AB Then inversely AC and alternately AB For since AB in length : AC :: AD in length : AE, (by Simson's Euclid, Book v. Prop. B.) AC in length : AB :: AE in length : AD, And since angle DAE is equal to angle BAC, and is measured in the same direction from AD, that BAC is from AB; angle BAD is equal to angle CAB, and 10 is measured in the same direction from AE, that CAB is from AC; •. AC : AB :: AE : AD, Also since AB in length : AC :: AD in length : ^^, (by Euclid, Book v. Prop. l6.) AB in length : AD :: ^C in lengtli : AE. And since angle DAE = angle jB^C, angle CAE = angle i^^^D; . . AB : AD :: AC : ^E. (15.) i/' a : b :: c : d^ «?2cZ c : d :: e : f; ^Ae^ a : b :: e : f. For since a in length : ^ :: c in length : d, and c in length : d :: em length : Jl (by Euclid, Book v. Prop. 11.) a in length : h :: e in length : /. And since the angle at which / is inclined to e is equal to the angle at which d is inclined to c, and the angle at which d is inclined to c equal to the angle at which b is inclined to a; the angle at which y" is inclined to e is equal to the angle at which b is inclined to a; ,'. a : h :: e : f. (16.) i/'a : b :: c : d, then a+b : b :: c4-d : d. Let AB - a, AC = h, AD^ c, AE ^ d. 11 Complete the parallelograms ABFC, ADGE, and draw the diagonals AF, AG. G B Then AF = a -^ h, and AG = c -V d. And since AB in length : AC :: AD in length : AE, and angle DAE = angle BAG, parallelogram BACF is similar to parallelogram DAEG; therefore triangle FAC is similar to triangle GAE, therefore angle F^C == GAE., and AF in length : AC :: y^G in length : AE ; .', AF : AC :: ^G : AE, or a + b : b :: c + d : d, (17.) Def. Unity is a positive quantity arbitra- rily assumed^ from a comparison with which the values of other quantities are determined. (18.) Def. If there be three quantities such that^ unity is to the first as the second to the third; the third is called the product which arises from the multiplication of the second by the first. 12 Thus if, I : a :: b : c, c is called the product which arises from the multiplication of b hy a. (19.) If c = the product which arises from the multiplication of h by ^^ c is also equal to the pro- duct which arises from the multiplication of 'd by b. For since 1 : a :: b : c, alternando \ : b :: a . c; therefore c is also the product which arises from the multiplication of a hy b. (20.) a : ac :: b : be. For (by definition of multiplication) \ : c \\ a \ ac, and \ : c :\ b : be; .-. (by Art. lb,) a \ ac v, b \ be. (21.) Cor. Hence alternando a : b :: ac : be. (22.) If three quantities be multiplied together, the product will be the same, whatever be the order in which they are multiplied together. Let a, b, c be the three quantities. Let ab = d, and cd = e, and let ac = f and bf = g. To prove that e = g, I : a :: b : d, and I : a :: c : f; 13 ,\ b : d :: c : f :: be : bf (Art. 21.) :: be : g; also b : d :: be : de (Art. 21.) :: 6 c : e; .-. (by Art. 13.) g = e, or (a . c) . fe = (a . 6) . c. In like manner it may be proved that (a .e) .b = (b.e) .a; therefore the product is the same, whatever be the order^, in which the quantities are multiplied toge- ther. (23.) Cor. Henee the produet of any number of quantities is the same, whatever be the order, in which the quantities are multiplied together, (24.) (a + b) . c = ac + be. For (by Art. 21.) a : b :: ac : be; .'. (by Art. l6.) a + b : b :: ac + be : be; .*. alternando a -{• b . ac -\- be \: b : be. But by definition of multiplication 1 : c :: 5 : 6c ; .*. 1 : c :: a + 6 : ac + be; .'. {a + b) c = ac + be. (25.) (a + b + c) d = ad + bd + cd. For (by preceding* Article) (a + b-hc) d={a + b) d + cd = ad-\-bd-\-cd. 14 (26.) (a + b) . (c + d) =:r ac + ad + be -f bd. For {a + b) , {c -h d) = a . (c + d) -^ b . (c + d) — ac + ad + be + bd. (27.) a X (-b) = ^ ab. For let ab = X, 1 : a :: b : x. But since the angle at which — a? is inclined to — b is equal to the angle at which x is inclined to by b : X :: ~ b : — x ; ,\ I : a :: - b : - X ; .'. a X {— b) — - X ~ — ab. (28.) (- a),(-b) = ab. For by the preceding- article (-«).(- h)= -{^a)b = -{-ab) = ab. (29.) Def. If three quantities be such that the first is to unity as the second is to the third ; the first quantity is called the quotient which arises from the division of the second by the third. Thus if c : 1 :: a : b ; c is called the quotient which arises from the division of a by b. 15 (30.) Def. The quotient which arises from the division of a by 6 is thus expressed a -i- b, or thus b ' (31.) If c = - , then a = be and conversely. Let c = ^ ; ,'. c \ \ \: a '. b ; .*. invertendo 1 : c :: 6 : a; .". a = he. In like manner the converse may be proved. (32.) (3;i.) t-(b)- For let - = c ; then c : 1 :: a h .'. alternando c : a .: 1 : 6 .-. — c : - a :: 1 J alternando — c : 1 :: — « : h ••■T-— -Q■ :^b--a)• LM f = e, b 16 c : 1 :: ci : b :: - « :-b .-. c : ~ a :: 1 : - b; - c : a :: \ : — b; - c : 1 : : a : —b; .-. (34.) — a a - b - b* Let 7 = c, c : 1 :: a : b ::- a : ~ b; -a a (35.) !/'a : b :: c : d, then r = "i b d Let -7 = X, b X : I :: a : h :: c:d; c a ''• d~ """l' (36.) Jfl-l a:b::c:d. For let -7 or -, := .r, (37.) (38.) 17 X : 1 :: a : b, also X : 1 :: c : d. » .*. a : h a b " :: c : ac '' be* d. y Art. 21.) « : b :: ac • (by Art. 35.) 1 = ac Vc' b c b-f c -f- - a a a For let - = 0^, -- = 2/ : :. (by Art. 31.) ^ = a^, and c=^ay\ .'. ^ + c - ax ^ ay ^ (by Art. 24.) « (^ + 3/) ; ^ + c , be a ^ a a (39.) Cor. /fe/2ce ^ + ^^ = ^^±^. a c ac j^ h he , d ad Jf or - = — , and --=: — • a ac c ac h d ^ be -^ ad 'a c "~ ac C (40.) 18 bd a c ac ^ 1 ^ d For let ~ = ^, ~ = ?/ ; a c ^ then b =^ ax and d ~ cy ; :, bd =^ ax X cy ^ ac X xy, (by Art. 23.) bd b d ac *^ a c b d be ' a c ad For let - =: X, - = y ; then b ^ aXy and d ^ cy ; .-. &c = (t?.X')c = (ac) a;^ and ac? = a{cy) = {ac)y; be {ac) X X ^ b d ad'^ {ac)y " y a ' c* (42.) Def. Any quantity a x a x a, &c. to w factors is called the rf" power of a^ and expressed a". (43.) If h he an n^^ power of a, b z.s ^A^ only n^^' power of a. For 1 : a :: a : a^ ; 19 therefore since a^ is a fourth proportional to 1 a, a; it is the only fourth proportional to 1, a, a; therefore there is only one second power of «. In like manner since a^ is a fourth proportional to 1, a, or; it is the only fourth proportional to 1^ a, therefore there is only one-third power of a ; therefore only one-fourth power of a, &c. ; therefore only one n^^ power of a. (44.) a"" X a" = a'"+". For d"^ — a X a X a. . , Ao m factors, and a^ = a X a X a. . . .to n factors; .-. a'" X a" = a x a x a. . . .to m + n factors a (45.) -^ = a"'"", when m is greater than n ; a and -r = , when m is less than n. a° a*""'" For^, when m is greater than n (by Art. 44) a" x «'"^" = a"^ '«-'^ = a"^- a'" .-. (by Art. 31.) — = a'" ". 20 Again when m is less than n, «"* x a'""' =«'*; ^7Z^= (by Art. 37.)^. (46.) (ab)"^ = a'^.b^ For {abf = {ah) .{ah) .{ah) to m factors = a. a ,a . . .io m factors x b . b .b . . .to m factors = a"* . fe^. (*^-) ©" = ?• For (^7) =^XTX7-..-tom factors a X a X tt .... to m factors '~bxbxb.,.,tom factors (48.) (a"^)" = a"^". For («"0'' = «"" X «"" X tt"" .... to w factors = «»"% (by Art. 44.) (49.) If a a/zc? b ^e positive quantities and a greater than h, a"' is greater than b"". Let a ^b + c, (^ = h" + 2 he + c%- 21 ,'. c^ is greater than //. Let d' = ¥ ^-d, a' = a. {b' + d) = (5 + c) . (h' + c/) = If + b^c-\- bd + cd ; .'. a^ is greater than h\ In like manner it may be proved that a^ is greater than b^, &c.; .*. a"* is greater than h"\ (50.) If a «wrf b ^e any quantities, and a m length greater than b, a''' is in length greater than b™. Let c be a positive quantity equal in length to a, and d b; c is greater than d; therefore by the preceding Article c'" is greater than d"'; but c'" is in length equal to a'", and c^'" to 6"" ; .". a"' is in length greater than b"^. (51.) If ab = Cj and a he inclined to unity at an angle = A^ and b at an angle = B ; c will be inclined to unity at an angle = A -f B. For let AB = unity, AC = a, AD = b, AE = c, then angle BAC =■ A, angle BAD = B ; and since c = ab, 1 : a :: b : c; 22 (by Art. 12.) angle DAE = iing\e BAC=A; and angle BAD =^ B ; /E therefore angle BAE — A ^ B; therefore c is inclined to unity at an angle =^ A + B. (52.) If h = a™, and a be inclined to unity at an angle = A ; b will he inclined to unity at an angle = mA. For since a^ = a. a, (by Art. 51.) d^ is inclined to unity at an angle ^ A -ir A ^^A; and since ct ^a . a^, (f is inclined to unity at an angle = A ^ 2A = 'dA ', in like manner a"" is inclined to unity at an angle ^ aA\ &c. ; .*. h = aP' is inclined to unity at an angle — mA. (53.) Jf '^ be inclined to unity at an angle = A, and b at an angle = B^ and c = r will he inclined to unity at an angle A - B. 23 Let ylB =:umiy, angle BAC=A, angle BAD = B, AC^a, AD = b, AE^c; since c = b' c : I : a : b ; therefore angle CAD = angle EAB ; therefore angle BAE = angle BAC - angle BAD = A ^ B; therefore c is inclined to unity at an angle = A - B. CHAP. II. ROOTS OF QUANTITIES, FRACTIONAL AND NEGATIVE INDICES. (Art. 54.) Def. Any quantity, which when raised to the 7z^'' power produces a quantity a^ is called the if" root of a, and expressed thus //«. (55.) Def. The irf" power of the n*^ root of «. or {^ aY' is expressed thus rt^^'*. (56.) Def. ,» ^^ expressed thus a"", where m may be either whole or fractional. * This as a general definition of a power of any quantity is defective, for, if the p^^ power of a be required, where |> is a surd, since no fraction, whose numerator and denominator are whole numbers, will accurately represent p, this definition cannot strictly be applied. But, as the fifth definition of the fifth Book of Euclid is a correct and general definition of proportion of magnitudes, and that usually given by Algebraists a de- fective one, so a general definition of a power of any quantity might have been given, bearing the same analogy to the fifth definition of the fifth Book of Euclid, which the definition in Article S5 bears to the common Algebraic definition of proportion. Nevertheless, as the demonstration of the propositions, necessary to establish the properties of powers of quantities, would by this definition have been rendered tedious : the definition in Art. 53, being one in use amongst Algebraists, has been adopted, notwithstanding its imperfection as a general definition. 25 (57.) If h he the n^^ root of a, and d the n^'^ root of c, and c 6e m length equal to a ; <^en shall d fee iw length equal to b. For if not, let one of them, d, be the greater in length. Then since d is greater in length than h ; (by Art. 50.) d'* is in length greater than b"" ; But d^ = c, and b"" = a ; therefore c is in length greater than a, which is contrary to the hypothesis ; therefore c? is in length equal to b. (58.) CoR. Hence all the n^^ roots of any quan- tity a are in length equal to one another. (59.) If ^ he in length equal to b, a" is in length equal to h^ , For, by the preceding Article, >^a is in length equal to ^T); •*. {\/aY is >n length equal to {^b^, that is, a" is in length equal to h", (60.) If ^he in length equal toh,^'^ is in length equal ^o b ". For a'"' = JL , and b~~" = 1.; D m m -i 1 S *" .-. a " : 6*" :: 1 : i, :: b" : «" ; a" b^ and by the preceding Article b" is in length equal to a"' ; .', a " is in length equal to ^ ". (61.) If h = ^a, b has n different values. Per let « be inclined to unity at an angle = A. Then a is also inclined to unity at angles =: A-{-p. 36o^ where p is any whole number, either positive or negative. Now in order that b may be an n^ root of «_, it is necessary that /; should be inclined to unity at an angle B, such that nB = A ^ p. 360' (by Art. 52.) ; _ A -h p. 360' * ' "" w * For p substitute successively o, 1, 2, &c. n — I, w, w + 1, &c. also — 1, ~ 2, &c. The corresponding values of B will be respectively A A + 360" ^ + 2.360^ n' n ' n ' ' 27 &c.. w n , &c. But fl±^i:^^ = d + 36O0, and — ^- — -^- = + 360° ; n n therefore the value of 6, which is inclined to unity at A an angle = — , is also inclined to unity at an angle A ^ . A + n. 360^ = - + 36o^ or — ^^ , n n and all the values of 6 are in length equal to one another (by Art. 58.); therefore the value of A h, which is inclined to unity at an angle = —, coincides with the value of h which is incHned to unity at an angle = , and therefore is equal to it. In like manner the value of h which is inclined ., , , A ■\- 360" . to unity at an angle = , is equal to the value of h which is inclined to unity at an angle A -\- n + \ , 360" 28 , , . A - 360" J -\- n-^ I . 360^ ^ . Also since = - 36o\ n n and ^IZ-i^i^" = ^+^^^-36-0" _ 36o„. w n The value of ^, which is inclined to unity at an , J + n - I .360' . I , ,u I i. angle = , is equal to the value of A — 360^ b, which is inclined to unity at an angle = ; and the value of b, which is inclined to unity at an angle = , is equal to the value of b, which is inclined to unity at an angle ^-2.360^ . = ; &c. ; n therefore b has n different values. (62.) Def. If a be inclined to unity at an angle z=z A, A being positive and less than 360^; a ex- presses a considered as inclined to unity at an angle = A, a at an angle = A + 36V, 1 a = A + 2.36o^ 2 &c. &c. also a =. A - 36o^ 29 a at an angle = A - 2.360°, -2 &C. i&C. and generally a = ^ + /? 36o^ p where /> = o, or any whole number, either positive or negative. Thus ^a expresses that value of ^a, which arises from considering a as inclined to unity at the angle = A, ^ a expresses that value of ^ a which arises from considering a as inclined to unity at the angle — A -^ 36*0^, and generally ^^ a expresses p that value of ^ a which arises from considering a as inclined to unity at the angle = A + p.sGo^, (63.) If /aV = b, b is inclined to unity at an angle = ^ (A + p.36o«). For let ^li = c, then c'"- = h (by Art. 55.) p (By Art. 6l, 62.) c is inclined to unity at an angle _^ A + p. 360^ n let djUL^ = C; 30 then since b = c"S b is inclined to unity at an angle = mC n (64.) If (a)" = bj b is inclined to unity at an angle = - - . ( A -|- p . 36o") . and 1 is inclined to unity at an angle = O, and (aV' is inclined to unity at an angle = ~ (J + p .360°) (by Art. 63.) therefore b is inclined to unity at an angle ryy) := O - - (A -\-p,'d6o'') (by Art. 53.) n ^ ' (65.) CoR. In the two preceding Articles it' a be a ])ositive quantity, A ^ 0\ therefore b is inclined to unity at an an^ie == + —p,3bQr. In this case; if p = 0^ b is inclined to unity at an isngic = O, that h, is a positive quantity ; 31 l( p = I, b is inclined to unity at an angle = ±^36o». n km • m (66.) (a]'"= (ay, a being a positive quantity. For let ^a = h, and /!^h = c, then (by Art, 65.) h and c are positive quantities ; And a = 6", 6 = c^ .-. a = (c^)" = (by Art. 48.) c*", .-. c is a value of ''^a; And c is a positive quantity, But 6"' = (ay (by Art. 55.) ■•■{f'-ilf- 32 m p m q 4- n p (67.) {aV X /aV = /a\ "^ , a being a positive quantity. For (by Art. 66.) («)" = {d^^ =: (v^a)"' .-. (by Art. 44.) («f x («f = (;/«)— = (^) "' (68.) (aV X [a)~ = Ya\ "'^ , a being a positive quantity. First let mg be greater than np, = (-C/")""""' (by Art. 45.) Next let mq be less than njp, 33 1 mq — np (69.) (a^ " X /'a\"' = /a^ "' , a being a posi- tive quantity. For {a\~'' X la\~' = _L x -!^ = — ^ j ^'' ^'' er (?r (?)^Mtf V;r, (by Art. 6;.) — *>'g-"P = (?)"""' (a)- (70.) 7^ = faV"-", where a is a positive quan- (?) ^"^ (bV*' = /cV". (92.) If fi.h — c, and a be inclined to unitj/ at an angle = A, and b at an angle = B, A and B being positive and less than 36o" ; 47 (ay . fb\"' = (^cy, if A + B be less than sGo^, = C c ]"", greater , Vp + q+l/ where m ma^ be either whole or fractional, positive or negative. ^«^er= {?)"■• (ir =(«)"'• (in er= i^r-iT-' •••er-er=(?r-(r-(})-^' = (cr . ( 1 Y'^ if ^ 4- J5 be less than 36o°, ^ [by Art. 91. = (cy\ /i\»».p+lj greater J = ( c Y\ if A -^B he less than 36o\ = ( c Yy greater (93.) If T- = c, and a be inclined to unity at an angle = A, and h at an angle = B ; A and B being positive and less than 3 Go" ; 4« (a)" :. A±(_^ = fc\*", if A be greater than B ' : > e.)" w;Aere m mai/ be either whole or fractional, positive or negative. For since 7 = c, b,c = a, .*. (hYJcV' is equal in length to (ay, •' (ay .*• yr^, is equal in length to ley or ley; Now lay is inclined to unity at an angle = mA, Iby , = mB, lay .-. 12i_ = mA-mB = m{A- B); Again *.* c = ^ > c is inclined to unity at an angle = A- B, And if A be greater than B, A-B is positive; and it is also less than 360°, since A, B are each less than 36o°; 49 therefore in this case c represents c considered as inclined to unity at the angle = A — B; .-. /cV" is inclined to unity at an angle = m(A—B) ; (ay .*. when J is ereater than B, ^4- = /^V"- But if A be less than B, A — B is negative ; therefore in this case c represents c considered as —1 inclined to unity at the angle A — B; .*. / c \"' is inclined to unity at an angle = m(A'-B); ' lb itr .-. when A is less than B, AiZ_ = (94.) Cor. Hence if :-> = c. e,)" -^r^ = (c)"", if h be a positive quantity^ \0 / = ( c \"', ifh be not a positive quantity. For putting a = 1, in the preceding Article, A = 0, and if & be a positive quantity, B = 0; .'. A - B :=0; G 50 But if h be not a positive .36o«) + ^^- (^ -> r) 360^ where j/ must either = 0, or some whole number^ either positive, or negative ; .-. ^.i^.36o'^ + 2/. 360" in ^ ^'^^.p.?,W^''l.(,l-r)3^0\ 55 .*. kmx + Iny = kmp + I'fn (q-r)y .'. kx + In^ = kp + I (q-r); m In which equation, that x may be a whole number, it is necessary that y should either = O, or some multiple of m; let y — m%, then kx + Inz — kp + l{q- r), where if k be prime to In, integer values of x and z can always be found, which will satisfy the con- ditions of the equation, but by the hypothesis h is prime to /, .*. whenever k is prime to n, h is also prime to /??, km .'. whenever k is prime to w, c is a value of a'". If k be not prime to w, let e be the greatest common measure of k and n, and let k = ef, n = eg, then efx + legz = efp + I (q -r), 56 the conditions of which equation cannot be satisfied I (q ^ f\ by integer values of x and s, unless —^ — O, or some whole number, either positive, or negative; liq ~r) And -^ — ~ cannot = O, unless o — r = O, € ^ Also since I is prime to k, I is prime to e a part of k, .'. —^ — - cannot = a whole number, unless a ~ r e ' ^ be a multiple of e; ,'. if k be not prime to n, c is not a value of a'", unless, either q ^ r = O, or the greatest common measure of k and n be also a measure of q — r. (100.) CoR. 1. It will appear from a similar investigation, if la\ ~ = b, and ih\ " = c, that c is a value of «'", whenever k is prime to n; and that when k is not prime to n, c is not a value of «'% unless either q — r = O, or the greatest common measure of k and n be also a measure of q — r; _k ^ ■ * also that if (a\ ^ = /;, and /M" = c, or la\~' = h, _ m __ km and //;\ " = c, c is a value of a '" in the same 57 (101.) Cor. 2. Let taV = 6, and let a be in- clined to unity at an angle = A, A being positive and less than 36o^ and let B + 9.360° = s{A +p.3W), where B is positive and less than 360°^ and 9 = or some whole number either positive or negative, and let (hV = c, and let s and t be either whole or fractional, positive or negative; then (aV* = c. First let s and t be both positive whole numbers, then c = a'* (by Art. 48), also a" has no other value different from c (by Art. 43.) Next let s and t be both positive, but one or both fractions. Let s = 7 , e = — , / n then B -^^ q. 360' = J. {A-{'P. 360'); But jB + r . 360° = J {A^p36o') (by Art. 99). .-. 9 = r, or q - r = 0; ,-, the equation kx -f Inz — kp -f l(q^r) (in Art. 99) becomes kx ~\- Inz = kp ; H 58 where if we make z = 0, and x = p, the conditions of the equation are ansvyered ; .-. c = («f = («y'. Next let s be negative and t positive, or s positive and t negative, or both s and t negative. These cases may be demonstrated by means of Art. 100, in the same manner that the preceding case was demonstrated by means of Art. 99 ; there- fore whether s and t be whole or fractional _, positive or negative, c =(«)«'. (102.) If /a\'" = b, where m is either whole or fractional^ positive or negative, and a be inclined to unitT/ at an angle = A, A being positive and less than 360^ and m . (A 4- p.36o") = B + q.36o^ where B is positive and less than 36o°, and q = 0, or some whole number, either positive or negative; then /b\" = a. For let {by = c, 59 then c = /o^" (by Art. 101.) = («)■ = «; (103.) If c be a value of a' and also a value of b" ^ ivhere -r , — are fractions in their lowest terms; then if k be prime to m, b is a value of kn For since c is a value of Z)" , 6 is a value of c"' (by Art. 102.) * ' .'. since k is prime m^ Z> is a value of a'"* (by Art. 99). (104.) Cor, In like manner it may be proved, if c be a value of a ^ and also a value of b" , where -r, — are fractions in their lowest terms ; that, if k In ^ be prime to m, /? is a value of a '*". (105.) The values of the square root of ~ \ are inclined to unity at angles = 90° and 21/ — 1, /- l\i will be represented by — s/ —l- For by the preceding Article / — l\i is inclined to unity at an angle = 90^ (-1)* =270^ .'. (-I)* is inclined to / — 1\^ at an angle = 270° --90° = 180^; .-. since / - 1^ is represented by 4- \/ -l, /-lU will be represented by - >/- 1 (by Art. 8). (107.) The values of l* are 1, +xA^:» - 1^ For(i)i=l. 61 36o^ /l\* is inclined to unity at an angle = = 90°^ O' =^-»"-. (■.)' -'-^-r-' Butv^ = 90', - 1 = 180^ ->y^ = 270%- (108.) (l)-i = -y— 1. For (by Art. 65.) /l")""* is inclined to unity at an angle = — ^sGo^, and (1)* =f36o« = 36o°-j36o^ /. (l)~* coincides with /lU, 62 (109.) Any quantity may he expi^essed in the form ± a ± b ^ ~ 1^ where a, b are positive quantities. Let A be the origin, and ^JB = unity; Prom centre A with radius AB describe circle BCDE, and produce BA to D, and draw EAC perpendicular to DB; Let i5C be the direction in which positive angles are measured, then AC = J'^i, AD^--i, AE = - ^^ ; Let c be the given quantity ; Draw AF = c, and draw FG perpendicular to DA, and FH perpendicular to AC; Since AGFH is a parallelogram, AF = AG'^ AH (by Art. 3.) ov c=. AG + AH; 63 Let a be a positive quantity in length = AG, b .. = AH; Tlien AB : a :: AD : AG, Qv I : a :: " I : AG, .-. AG = a X — I = - a; Anil AB : b :: AC : AH, or lib :: >/ -- 1 : ^/f, . - - .-. ^// = bj~::rj; .'. c ^ - « + b >/- 1 ; In like manner it may be proved in any other case. (110.) CoR. 1. Let c be inclined to unity at an angle = C, C being positive and less than 360*^; Then if C be less than 90^ c is of the form + a + b ^ — 1 ; if C be greater than 90^ and less than 180^ c is of the form — a + b ^ — i; if Cbe greater than 180^ and less than 270°, c is of the form - a - b ^J — 1 ; if C be greater than 270°, c is of the form -f « — ^ >/ — 1 ; 64 (111.) Cor. 2. Hence if c = e fiy, e being a positive quantity, and n positive and less tlian I ; Then if n be less than ^, c is of the form + a + b >y/ — 1 ; if n be greater than ^ and less than ^, c is of the form — a + b ^ - i ; if n be greater than \ and less than f , c is of the form — a -— /; ^ — l ; if w be greater than f , c is of the form + a ~ 6 >/ — i . (112.) Cor. 3. When n is less than |, as n increases, - increases; and conversely; when n is greater than \ and less than J, asnincreases, ^decreases; and conversely; when n is greater than ^ and less than f , asnincreases. -increases; and conversely; when n is greater than f , as n increases, - decreases; and conversely. «5 (113.) To express the values of V in the form ± a + b ^ - 1, where a, b are positive quantities. The values of '' ^'^ (i)'' (ir- {\f' {'S' (1)'' (!)''• of which (l\^ = 1; And since \ is less than \ , 1 i\y is of the form a + h ^ — i ; ,\ since ^ is greater than \ and less than ^, hy is of the form — a + ^ ^ — i ; /. since | is greater than i and less than j, 1^ i\y is of the form —a, - b ^ ^ i ; I 66 .'. since | is greater than | , /l\*' is of the form + a - h /— i ; Let 1^ = X, then x^ =1, OY x^ - \ - o. From the solution of this equation we obtain the following- values of x; -1 JL 4- n/^ ' 2 • rt '^, i- ^3 1 ; 1)^=1, 0/ -1, )" - 2+2 ^/^' 4/ 2 2 / 2 2 1. 67 (114.) If a + b {\Y + c nr =f (1)', and a, b, c, f be positive quantities, Let A be the origin, and AB = a. From B draw 5C parallel and equal in length to the line which represents h (l\*^j and from C draw CD parallel and equal in length to the line which represents c l^Y, and join AD; AD = a^- b (ly -\- c (\y (by Art. 7.) On the other side of AB draw BC making angle ABC = ABC and make BC in length = BC, In like manner make angle BCD' •= angle BCD, and CD in length = CD, Join AD; 68 Since BC is equal in length to BC, and angle ABC equal to ABC, but in the opposite direction, and BC parallel and equal in length to the line which represents h (l)""; BC will be parallel and equal in length to the line which represents h (\y; In like manner CD' is parallel and equal in length to the line which represents c I'^Y ; .'. AD' = a -i- b (l\'^ + c (ly; But since figures ABCD, ABCD' are similar and equals AD is equal in length to AD, and angle BAD equal to BAD, but in the opposite direction^ .-. since AD =f(^Y^ ^il^' = f i^Y > (115.) ^ a + b liy + c l\Y = f (IV, and a, b, c, f he either positive or negative quantities; ^ -^ '' (_\r -^ M_\r = M_\r For let one of the quantities h be negative, let then a - g (V^^^ -^ c ^\y =/(l)^ 69 Now — g = g U\K •*• aA-g (ly-^ -^c (1 Y =f(^y (by Art. 114.) But g (^ly^^ = g {i^y . {ly = -^ ( j j^ or a + b(^iy + c(^iy =/(lj^; In like manner it may be proved if any other of the quantities be negative. (116.) If^ + h /ir + c ny = f (l\p, and a, b, c^ f, ^e exY^er positive or negative quantities. For a + &(l)'^ + c(^iy=f(^iy (by Art. 115.) And (Ij^'^ = n\-^ 70 (117.) If Si + h ^—[ + c (1)" = f (l)^ and a, b, c, f be either positive or negative quantities, a-byZT + e(l)-" = f(l)-. For + ^/^ = i^Y (^^ ^^^* ^^'^'^ But (1)"^ = - x/^^ (by Art. 108.) .-.a ^-6^37 + 0(1)-^ =/(!)--. (118.) if a + by^ =e + fV^, ^^zcZ a, b, e, f, Z>e ez^Aer positive or negative quantities ; a = e and b = f . For since a + b ^ — l = e -\-fsJ - 1, a- h sj'^ = e -/^/^ (by Art. II7.) .-. by addition and subtraction, 2a = 2e, and 2h J^^\ = 2f^.J~^\; .'. a — e, and h =f. 71 (119.) /^ a = ± b ± c^/^^, where b and c are positive quantities, a will he in length = >/b^ + c'^ For let a = e /I ys where e is a positive quantity, then e llY = ±h ± c^y^l^ .-. eliy = ±h + cj^~i (by Art. II7.) .-.by multiplication^ e^ = ^»^ + c^ .-. a is in length = mJB^ + c^ (120.) Lei a = b /1\", where h is a positive quantity y and n positive and less than 1 ; then if n he either not greater than ^, or not less than f , a + 1 is greater in length than 1. For since a = b /IV', and n is either not greater than ^ or not less than f, a is of the form e ±f^—\ (by Art. Ill), where e = 0^ or some positive quan- tity^ and y is a positive quantity ; .-. a + 1 = l+e ±fsr^\. 72 .•. « + 1 is in length equal to ^ (1 + €f +yS* .-. a + 1 is in length greater than 1. (121.) het a = h IXY^ where b is a positive qnantity, and n not less than ^ and not greater than f ; then a — 1 is in length greater than 1 . For a is of the form — e ±f^~^ (by Art. Ill), where e and ^/ either = o, or are positive quantities; .-. a - 1 = - 1 - e ±/V^, .*. a — 1 is in length equal to >/ (1 + e)^ +/^; .' . a - \ is in length greater than 1. (122.) Let a = b i'^Y, ivhere b is a positive , quantity, and n positive and less than 1 ; f Aew if n he either less than i, or greater than J, is in ^ ^ ^ a+1 length less than 1 ; hut if n he greater than ^, and a , I less than f, is in length greater than l. First let n be either less than j, or greater 73 Then a is of the form e ± f^ — \, where e and y* are positive quantities; .-. a— 1 is in length equal to^(e-l)*+y^ and a + 1 ^ i^e^xY ^f\ . o-J. x/(6-ir+/\ ^+1 ""' x/(e+ir +/'' a— 1 ..,,,, .*. — — ~ IS in length less than l. Next let w be g^reater than ^, and less than \^ Then a is of the form - e ±fs/- Ij .-. ^^^ is in length equal to x/(^+^)'+/' . /. is in length greater than 1. (123.) nV + /1\""" IS equal to a positive quantity, when n is positive and less than ^. For if n be positive and less than ^, (IV* is of the form a + b ^Z-l, where a and b are positive quantities, .-. (l)- = a-6y3r (by Art. II7), K 74 (124.) Let Si = h (iV" «wd a 4- 1 = c ilV, where b and c are positive quantities ^ and m and p positive and less than 1 ; Then if m he less than ^, p will be less than m, greater , greater First let m be less than J, c And let AB = unity, AC = a, and let the paral- lelogram ABDC be completed, and the diagonal AD be drawn ; Then AD = a ^ \ = c llV; .-. angle BAD = p.36o'; and angle BAC = m . 36o^ .*. p is less than m. Next let m be greater than i. The same construction being made, AC falls on the other side of AB ; 75 angle BAD = 1 - p . 36o°, 1— m is greater than 1 — p ; .*. p is greater than m. CHAP. III. BINOMIAL THEOREMj EXPANSION OF fl IN A SERIES ARRANGED ACCORDING TO THE POWERS OF X, DIFFERENTIATION OF «*. (125.) If a = 1 + b, it is proved in the bino- mial theorem that a™ = 1 +mb + — py- b- + jp-^-^ b'+&c. which is a converging series when b is in length less than unity; Now if m he not a whole number, a™ has many values; To investigate which of the values of a"" is represented by the series l+mb + -p-^~b^+ 5-^-^ b^ + &c. First let a be a positive quantity, then lay is also a positive quantity; 77 .*. {a\'^ is that value of a'"* which is represented by the series 1 + mo + — ; b^ + &c. 1 .2 Next let a = c (ly, where c is a positive quantity and n positive and less than ^; When b = 0, the series , , m.m—1 ,2 . p 1 + mb H b^ + &c. = 1 ; '. when b = 0, that value of a*'' which is represented by 1 + mb H '- b^ + &c. must = 1 ; But 1 is inclined to unity at an angle = ; /. when b = 0, that value of a"* which is represented 771 . T)7> ~~ 1 by I + mb -{ '- b^ + &c. must be inclined to •^ 1.2 unity at an angle = 0; Now (by Art. 86) fa^ = (cV' . /l\*"«, /. {a\^ is inclined to unity at an angle = mn.sGo^^ And a is of the form e +fs/-^ ^ (by Art. HI), where e and / are positive quantities. 78 .-. b = e- I +f,J^~[; .-. as b decreases, e approaches to 1 and J' to O, f /. - decreases, e ' .'. (by Art. 112.) n decreases. And when 6 = 0, y= o, and e = I, f .-. ^ = O^ .-.72 = O, e .-. mw36o^ = 0, .-. when b = 0, (a\'^ is inclined to unity at an angle =0; /. (ay is that value of a"* which = 1 + w6 + -^^^^^ 6' + &c. m,m—l 1.2 Next let w be not less than ^ and not greater than f ; In this case, since b = a—\, b is always in length greater than 1 (by Art. 121.) .*. the series 1 + mo A b + &c. is not a 1.2 converging series, and therefore it does not represent any value of a^. 79 Next let n be greater than f and less than 1 ; Then a is of the form e — f s/ "^y And Z» = e - 1 -f s/~--^ \ f .'. as b decreases, - decreases, e .*. (by Art. 112.) n increases, f And when 6 = 0, "^ = O, and n = I; e Now(«)™=(a)-(_lJ»=(a)™.(ip = (^r.(i)-'""" .-. («)'" is inclined to unity at an angle = m.w-1 .36o%- But when b = 0, n= I ; .". m . w - l . 36o° = O, when 6 = 0, / a "j*" is inclined to unity at an angle = 0; . in this case /a\'" is that value of a"* which = I + mb -{• — ¥ + &c. 1.2 80 (126..) Let a = c /l)", where c is a positive quantity, and n positive and less than 1, and let a - 1 = bj where b is in length less than unity; Then, if n be less than ^, If n be greater than f , (a)-= n)^-.{i + mb + "^'^"'jH ^ + &c.} First let n be less than ^; Then (by Arc. 125), ay = 1 -\- mh -] b^ + &c (?) 1 .2 »"Mr=ar-(?r=ar-(«)" (a)"'=(l)^"".{l +m6+^?^^^&' + &c.}. Next let w be greater than f, (By Art. 125), («J"' = 1 + mb + "UliIL-I 6' + &c. m.m~l (ay. (1)-" = 1 + w6 + l\ *' + &c. 81 .-. (a)- = (ijp+i- Ji f mb + "IL^L^ b' + &c.} . (127.) Le^ 1 -f Ax + Bx'' + Cx' be a series such that (1 + Ax 4- Bx'^ + &c.) . (1 + Ay + By' 4- &c.) = 1 + A . (x + y) + B (x + y)' + &c. whatever be the values of x and y ; To Jind the law of the series. By multiplicatioiij {I + Jx + Bx' + &c.) . (1 + ^2/ + jB/ + &c.) = \l -\- Jx -}- Bx' + Cx^ + &c. + ^3/ -rA^xi/ + ABx^y + &c. + Bf + ABxi^^ + &c. + Cf + &c. And by expansion, I + A{x^-y) -^ B{x-^yy + &c. = J5a?' 1 4- r Co:^ *^ 4- &c. 2Bxy[ \ +3Cx^y By\ J WsCx/ L 1 + r -^^^ + r 82 .'. equating- the coefficients of the combinations of the like powers of x and i/, SC^AB; .. C=^= — ; 3 2.3* .-. 1 + ^a: + T— T- + r + &c. is a series such that {l^Ax + ^^+ &c.) . (l + Ay + f^' + &c.) ., ^ AUx + yY ^ = 1 + ^ (a? + 3/) + — ^^ ^^ 4- &c. 1 . 2i whatever be the values of a?, 3/, or A, (128.) Let a = c /1\", ichere c is a positive quantity, and n positive and less than 1, and let a — 1 he in length less than unity, and A = (a - 1 ) ~i{a-lf +i-(a-l)'^ -&c. Then if n he less than \, 1 .2 "*" 1.2.3 /a\^ = 1 + Ax 4- f-^ + T^^^ + &c. If n he greater than f, A'x' A^a-^ BS For it is proved in the Appendix to Woodhouse's Trigonometry, that And the proof given by Woodhouse depends upon the binomial theorem ; /. (by Art. 125.) If n be less than ^, If n be greater than f , aV = I -{- Ax + + -. \-i/ 1.21.2.3 (129.) 7b express (IV in a series of the form a + bx + cx^ + &c. Let ilV ^ a -^hx ^ cx^ ^ &c. Let a; = o^ then /iT = a, .-. 1 = a ; .-. l\y = 1 + 6^ + coc^ + &c. Now (1)^.(1)^ =(.)-; 84 .-. 1 -\- bx -{- cx^ + &c. is a series such that (l -\- bx + cx^ + &c.) . (1 + % + CI/" + &c.) = 1 + b,(x + 2/) + c\x -\- yY + &c. /. (by Art. 127.) c = -- , &c. /. (l)' = 1 + &^ + ^-^ + &c. To find the value of b, Since x may be any value, let x be positive and less than ^; And let ny = i-t^, Vi/ 1 — a Also let l\Y=^k, \ -\- a — m, 1 - a = n, then — = A- ; and a — \-~ -, where since x is positive and less (.y^ + i than \, a is in length less than unity (by Art. 122), /lV-1 2/lV 2 /if and m = 1 + 114 = ^ ^^ = — ; 85 2 Let — ; ^ = g, or Ml) then §• is a positive quantity (by Art. 123.) •••»» = g- (1)"% ••■«=-(lf =M}) '-^i\) ■• :. m is inclined to unity at an angle = - 36o^, n =(l-|).360<'; And 1 — is greater than - , since x is less than ^, (my .•.(byArt.93.)J^ = (_^J = (^)'.(l)-'; (mY (my . iJ^y -- V / _. V / . ••u {«y •{])-'-{»)" But k = (1)^ .'. since x is positive and less than 1^ (kV = (ly (by Art. 86.) 86 (my •••(ir=|if=(?r-(«r^ Since m= g l\\*, and - is positive and less than 4, where 3f=(m-l) - ± {m- if + ^ (m-lf - &c. And since « = g (l)'"''' and l - | is greater than I and less than l. N'y^ , / «\-y = 1 - iVj/ + ^ - &c. (by Art. 128), where N = (n-l) - ^{n-lf + iin-lf - &c. . ^ly^ = (l + iJ/y +^V &c.) . (1 - % + ^' - &c.) = 1 + (M- iV) y + (^-^)'-y' + &c. (by Art. 127.) .: l+bxy + ^-|- + &c. ^j i,,,^,, ., .-. bx = M-iV, ,,n ; v f;i' ... 6 = l.{M-N); 87 Now 7;i = 1 4- a» W = 1 — a, *. m — - 1 = a, n — I = — a ; N = ' 1 9 -ja -ja — jO — <»C. M-N=^ 2 {a + i a' + 1 a^ + &C.} •. 6=?.{a + ia' + |a^ + &C.} a? Let X = \, Then (by Art. 113.) l+V^i/TIT+i 3 + ^3.7-1 (V3 + x/-l) -1 3 + y 3 . y^ ^3 ' 88 '■"-!7s-=(i5)'*K7i)'-M-«' then h = c^y — Ij where c is a positive quantity; / c^x' C^X^ f—- , p ny^ = \ + ex ^ 1.2 I .2.3 r.2 «2 „^;; „3 .'3 ^^ (130.) Cor. l. (l)^ For (1)' = (1)- (131.) Cor. 2. If a be a positive quantity + ifi±£ivzii£^ + &c 1 .2.3 where ^ = 2 {— + | (— ) + .| (— ) + &c.j . For by the Appendix to Woodhouse's Trigono- metry, 89 Vo^ 1.2 1.2.3 ^ Where ^ = 2 (^ + 1 (^jV + 1 (iniV + &c.} .-. («)' = (i.^..f!|\&c.).(ij = (l+^.+f!| + &c.).(l+pc.V-i + (P£^^V&c.) = l + C^+pcy^rDx + ^^"^^^y^~^^'^' + &c.(byArt. 127.) (133.) Let a = b (l\'', where b is « positive quantity, and p positive and less than 1 ; Then if p be less than ^, Then if p be greater than f , 1.2 /a\^ = 1 + Ax +^^^ + &c. where A = 2 {^ + i (^V -f ^ (^)^ + &c.} (a + l 3 Va+l/ '^ \a + 1/ J 14-6 For let a = , and let 1 + e = m, 1 - e = w, M 90 . a—l 2a 2 then e = , m = , n = a + l a+ I' a -\- I Let a + l =jr/iy^ where y is a positive quantity, and q positive and less than 1^ 2b(lY _, «-7^--|-y -1, . h\ 2^ = cos a - sin xA^ (by Art. I17.) .-. by addition /iV^'^ + /l\ 2;r--2.cos 0, By subtraction (ly - (1) ^^ = 2. sin ^-l. "27r cos 6^ = sin (jr - 0)" 2 7~1 ' In like manner it may be proved that e_ sec 9 ny^ = 1 + tan ^ >/ - 1, ,_ .'. sec (1)"^'^ = 1 - tan ^ V^ - 1, 104 .-. by addition sec 6 ((1)^'' + (1) ^''| = 2, by subtraction sec |(l)^ - (l)"^^} = 2 tan ^ \/~^ , '. sec 6 = {T + iW'" \ 2 tan x/ — l = i?-!(ir-(}pi tan e = -j=.^^, ; ^"^ (1)2^ + 1 Cosec = sec (^ - o\ (}f M!)*-(ir^' (If -OP Cotan = tan (^ - e\ cotan = 105 ff-26 (1) '' _29_ (D-d) 7-1 or- 29 (1)-- ^ 1 1 -d) 29 1 2^- 1 y:^- -0] 29 1 2-+1 - 1 1 + 29 (1)2'^+ 1 = V - 1 . 2~$ • (1) 2^ 2 7r _ X (140.) Def. When is said to be equal to any angle, it is to be understood that is equal to the arc which subtends that angle to a radius equal unity; thus if be said to be equal to angle BAF in figure Art. 139, it is to be understood that e = BF, O 106 (141.) To express the relations of the sides and angles of triangles hy means of powers of unity. Let ABC be a triangle,, and let its angles be = A^ B, C, and the opposite sides be in length respectively equal to a, b, c ; Let AB be in the positive direction, and let the parallelogram ACBD be completed, and let BAC be a positive angle; Then BAD = - B, and AD is in length = a ; A -A .-. AC= bny-, AD = a /I) 2.; A _^ .-. c = b (ly +«(!) ^"*' In like manner it may be proved that b = a ny + c il\ 2^^; 107 In which two equations, together with the con- sideration that A, B, C being angles must be each less than tt, are contained all the properties of tri- angles. (142.) Ex. 1. To prove that the three angles of any triangle are together equal to tt. Since h = a /n^^ + c l\\ ^^ , by multiplying both sides of the equation by /1\^'^, A A-\-C h ny^ = a (\\ 2^ + c, A -A But h ny- + a (1)'^" = c, .*. by subtraction - a (\\ '^"^ — a (l\ A+C 2ir A + B+C Now — 1 = (iV"^*, where p either = o, or some whole number either positive or negative; ,1 A^B+C 108 A + B^^C . . . , . Since IS positive, /?+i is positive, .-. p cannot be a negative w^hole number; And since, A^ B, C, are each less than tt, A+B+C . I ,u 3 27r 2 3 • p cannot be a positive whole /. J3 = 0; , A+B+C .-. A + B + C=:7r. number; (143.) Ex. 2. To prove that the sides of tri- angles are to one another as the sines of the opposite angles. First to prove that b : c :: sin B : sin Q =6(1 A B -H] 27r-(B+C) B = i.( :\y-{]) '" +«(!) ^' 109 -6 (!)-"-%« (J)- B C .\a = c(\\ 2^ + & (l)^'^ (by Art. 116.) /. by subtraction o=c{(l)^'-(l)-^}-6{(lf'-(lp-'}, = c . 2 sin B sj —^ - b.2 sin C >/ - 1, O = c . sin JB — 6 sin C, .', b : c :: sin B : sin C It may be proved in like manner that a : 6 :: sin ^ : sin i5^ and a : c :: sin ^ : sin C b^ + c^ — a' (144.) Ex. 3. To prove that cos A = r- c=6(l)- + a(l) '27r 110 • '^(0 B '^^ == c -' b ( A \l/ \ 1/ .-..( B ly- = c-br. A .'. by multiplication 2= c' - be |/1)^ + (1)"^ + b' = c^ - be, 2 cos A ■¥ 6^ b' + e'-a^ .'. cos J[ = 26c In like manner any other properties of triangles may be established. (145.) CoR. Hence all the properties of triangles are contained in the two equations B '2ir c = 5(l)- + «(l) A-Y B+C=7r. For the equation b =^ a (l)^"" -V c (\\ ^^ may be deduced from these two equations in the same manner B^ _c that the equation a = c (lY^'' -{- b {l\ ^^ was deduced from them in Art. 143. Ml (146.) To find the lengths of curves and the position of their tangents. Let NPP' be any curve, A the origin, AB the positive direction, AP, AP' two radii, PT a tangent at P; Let P'P be joined and produced to S ; Let AP be represented in length and direction by p> Let AP' be represented in length and direction by/ Let NP be in length = s, NP' =:S\ chord PP ,.=^, 112 Let angle BJP = e, APT = (I>, .,TPS = S; Then angle BTP =^ 6 + , ' \ , and BSP = + ^ + S; p-p k i±±±I s — s s -s \^} * Now let PP' be diminished sine limite, k Then limit of — — = l (Newton^ Lemma 7.) S o and limit of(l\ ^tt ^/i\27r (Newton, Lemma 6.) dp ^ ^i±f Prom which equation the differential expressions for determining the lengths of curves and the position of their tangents may be deduced. (147.) Ex. 1. Let the curve he a circle, and A its centre. 113 P = r (lY' , where r is a constant quantity, J. c / — de i±? c , — do ± c . do ,_^^- .-. - r-— .V - 1.— = /n 2«^^ 2^ .-. by division — •■■ * - J. ^=( * that is, the tangent u i at right ? ingles to the radius; ■■■ (If = (j)i=. fV- "i, -h^-- y-i, .•. r. Stt (Is ~ J 114 .-. s = r .— - + const. 27r When = 0, let s = o, then const. = 0, 27r (148.) Cor. 1. Let r = l, then s = arc which subtends angle BAP to radius = l, that is, s = 0^ c 2^ ~ ' .'. 27r = C = 12 {73 -» (75)" "'(tj)" -*'■!<*"■"* (149.) Cor. 2. Hence s = rO. (150.) Ex. 2. JLe^ the curve be any spiral^ A its pole. Here p = r l^Y'^ , where r is a variable quantity; 6 .-. p == r [l\^ (by Art. 148.) dp ^ ^i dr ^ ^ ^i do /— - 115 i dr i dd f—- , ,^-±f- ' ' ds ds ^ \i) ' dr do ' ds ds '' = (1)' .-. by multiplication, rdr\^ , /de^ which is the differential equation for the length of a spiral. Also by addition and subtraction. de 2r "7" . 1 = /l\c - /i\ c = 2 sin a/^. dr ... cos = ^ sin = r.- tan d) = f -j- ^^1 equations for determining the po^ sition of the tangent. 116 (151.) Ex. 3. Let the curve be referred to rectangular co-ordinates. Let A be the origin, X be measured in the positive direction^ y in the direction of + ^ — l ; then p — X ■\- y sj — \^ dp _dx dy I ' ' ds ^ ds dx '^ ' dx dy t ,i±i dx dy , _?±1 .•. by multiplication, (S)'-(i)'-'. the differential equation for the length of a curve. Also by addition and subtraction, dx i±l _i±i 2^V-l = {l)' -(1) " =2sin(e + <^)^-l, in COS (0 + 0) = — , s*»n(0 + <^)=^. tan(0 + ) = g; And e^(p = angle BTP (Art. 146.) = the angle at which the tangent cuts the axis ; .*. the differential equations for determining the position of the tangent have been found. (152.) Sin X = X ^— + ^^— ^ ^ 1.2.3 1.2.3.4.5 &c. X X COS X = 1 1 &c. 1.2 1.2.3.4 For sin x = ^-^ 111 . 2V^-1 ^ 1.2 1.2.3 1.2.3.4 -^ 118 1.2.3 1.2.3.4.5 COS X {\)'H\)' ^'-h^rrh-^-^''- (153.) Def. One second is assumed as the unit of time; and any other time f is represented by a line measured in the positive direction,, and bearing the same ratio to unity which t" does to 1" (see Art. 1). (154.) Def. The degree of swiftness or slow- ness with which a body is moving at any time is called its velocity at that time : and the velocity is thus measured, let t be the time, at the end of which the velocity is required, and s the length of the path described by the body in time t (the path being either a straight line or a curve, and the mo- tion of the body in that path being either uniform^ or accelerated, or retarded), let s' be the length of the path described in the time t+h; and \ti V : 1 :: limiting ratio of s' - s : h, when h is diminished sine limite ; then V is a line which in length measures the velocity at the end of time t ; also let the direction in which the body is moving at the 119 end of time t be inclined to unity at an angle = a, a. and let u -^ F , (iV , then w is a line which in length and direction measures the velocity at the end of the time t. (155.) Cor. 1. Hence ^=37. ^^- ^' (^Y ' (156.) Cor. 2. Let NP in figure of Art. 146, be the path of the body; then a = (j>-\-9, ,\ u = j-^,n\ c ; __ ds dp dp "^ dt ' ds " dt ' Au„r. = (i5y. {dr\' . ^/de\ •■•^-(£)"-(s)*+-(S)'-(3i)' 120 . „, _ /dx\' /ds\^ (dy\^ /ds\' Us) Kdt) "^ \ds) Kdt) (157.) Def. If the velocity of a body be in- stantaneously changed either in length or direction, the cause which produces this effect is called an impulsive force or an impulse; but if the velocity be gradually changed, the cause which produces 4,his effect is called a continued force. (158.) Def. A continued force, when its effects are estimated with respect only to the change pro- duced in the velocity, without regard to the magnitude of the body moved, is called an accelerating force ; and is thus measured, let u represent in length and direction the velocity of the body at the end of time f, Uy at the end of time t -^ h, and let / : 1 :: limiting ratio v! — u : A, when h is diminished sine limite ; then / is a line which represents in length and direction the accelerating force at the end of time t, (159.) COK. Hence/=^^ = §. ]2l (160.) IVhen a body revolves in a circle in consequence of the action of a force tending to the centre, to determine the velocity and the periodic time. Let p represent the radius in length and direction at end of time t, Let r be a positive quantity in length = p, Let p be inclined to unity at an angle = 0, then p = r (IV , force =g, And this force tends to the centre, .'. -7-^ is a line measured in the opposite direction to that in which p is measured^ •*■ -T^ is inclined to unity at an angle = 7r + 0, d^p 1L±1 Let ^ = P./i\c ^Pisa positive quantity; now /o = r IXV , Q 122 since r is constant in a circle, de dp _ dt ='{\r^dt df -1, ^ {de\^ . , ^ d^e we have the two following* equations, '!¥ = '' ' (/e •■• rf? = *^°"^t- = ^' .• P = r.C-, a constant quantity; And -77 = dt ■.t = 6\/^ + const. 123 when 9 = 0, let t = o, then const. = o ; .-. t = ev ^; Let = 27r, then periodic time = 2 7r y -p; Let F' be in length equal to the velocity, then generally P = (^)' + 7- (^y (by Art 156.) .'. in a circle F'' — r" ( j~\ = rP, (161.) When a body is acted upon by a centri- petal force, to find the sectorial area described i^ound the centre, and the velocity. d'o '"^^ As in Art. i6o. ^ = P. /n c ^ and p =z r ny ; dp dr i id9 t— d\ d'r 124 dr do df ~ df'{\y '^^dt'dt (jr^- /dSv" d'O i+_9 d'r , J dr de 1 I — (dd d'e Z dW dr de , P'{\y^i¥^^'di'Tf^-' .dOx' d'o f — „ d\ dr de df ^ dt' dt' /de\^ d'e f — ^-Kdi) "-'df'^'''' we have the two following- equations, d'r /de\^ J, 0) w^--{di) =-^| dr de d'e {^ 1^25 multiplying all the terms of (2) by r, we have dr do 2^_ ^^dt'dt "^^^ df ~"^' .'. integrating*, ^777 = ^9 a constant quantity, .'.J'r^dO = ht + const, when fr^'de = O, let t = f, then fr^ do = h (t-f), or sectorial area = - {t — t'). dv Now multiplying every term of (i) by 2.-7- and ' dt do every term of (2) by 2r-y- , we have dr (£r _ dr /dO\^ _ _ ^pdr ^ dt'dt' ^'^dt '\dt) ~ dt\ dr /dO\^ . ^ ,dO d'O ^ ^'Tt\rt) +'"^-^=^ . by addition, dr d'r dr /d£\^ ^dO d'O _ j^dr ^Tt'df^^'^dt\dt) ^^"^ dt'de -~^^di' 126 integrating, (g)^ + r^ (g)' =/ - 2Pdr, (by Art. 156.) V'- = /- 2Pdr. (162.) Cor. 1. From equation (i), dt' \dt) = centrifugal force — centripetal force. (163.) Cor. 2. Since r'^= h, /dtV^ _ r" \de) "~F' • • \de) ^"^ fi' ' ■■■^•-*-{?(Sy-?!- (164.) Cor. 3. Let w = 3 , . du 1 dr then -T^ = 2-7^. ..r. = ..{(f:)+„.}. 127 (165.) Cor. 4. Since V'=f~2Pdr, 2Pdu ^2Pdu ,.„ ^. ^. ,,< du (Pu „ du\ 2Pdu ■■■ differentiating, /*' [2- .^ + 2m^| - M= rf9' (rf-M ) P (166.) Let P oc — _, to find the equation to the curve described. Let P = —^ = mu ; ... _ 4. 2^ _ ^ = O, (by Art. 165.) m Let w — 7^ = a^5 du _^ dx **' dO^ do' d^u _ d^x 128 .-. ^, + .T=0; Let (e^^ , /1\ ^ represent a particular value of x, where e is the base of the hyperboHc logarithms^ .-. (Ar + w >y'^)' + 1=0, .*. k -\- n ^^^ = ± V^"^, .-. A: = o, w = + 1 ; .-. the general value is. Let C = « /l\^ , C = b ny , where a, b are positive quantities and a, /3 positive and less than 27r; then X = a {l\ <^ + ^ /l\ *-' , 129 »-^9 B-» ...x = «(l)- ^ +6(1)- » ; .*. by subtraction, = a.(l). -«.(!)-' +6.(1)' -6(1)-' ={Mi)'-Mir'kf)'-hiP-*-(})'Hir' .•.0 = {«(l)?-*(l)-^}.(l)"-{«.(l)-^-6.(l)-^}; .*. since 9 is variable, we have the two following equations, «(ir- H\ )-^ = o «or -b^ i8 - from either of which we deduce b = a R 130 substituting- these values for h and (l)^> . = «(!) ^ +«(!)- ', = « . 2 COS {a + 0); .-. 2^ _ _ = 2a COS (a + 0)y 1 ^ / ^x .*. -pry — la cos (a + 0), equation to the curve described. (167.) To solve a cubic equation x^- qx +*r = 0^ when -^ IS greater than — . By Cardan's rule, -(-^v/FI)*+(--;-v/^|)* =I-3*(|-t)V-}' I3i 9 where y* is a positive quantity^ and - positive and less than 1; ^'•■■{--.-(*-i)'^^'!*--^(:p- ••• ^=/(})^ +/(!)"' =/-2 cos e; By multiplication^ \4 2/ 4/ ^ * cos 6: Also - - + (J^ ) ,^n^\ = f^ /i\T r £5 3 6) = /'.2cos(30); /. COS (3^) = - 132 r _ _ __r r sj^ '(I) V f27 Let — t-^ = cos 05 where = COS = COS (36o«+0), (2.36o"-f0), .'. COS (3 0) = COS = COS (360^ + 0) = COS (2 . 360^ '+); .-. the values of are |, 120° + ^, 240° + |; the values of x are 2\/|.C0s(f), 2\/|.COs(l20" + |), 2 V |.COS (^240° + |), which may be found from trigonometrical tables. 133 (168.) To find the integral of — . dx ^^^"f>e^•of•(-+^-^--^)^ = a:.(m + w>y -1), 1 dx X dy ^ /dx f — — = my^nysJ-\\ Now X is in length = /eV^^, .*, my is the hyperbolic logarithm of the length of x\ And ny is the angle, at which x is inclined to unity, .-. if this angle be called angle of x, — = hyp. log. of length of a? + \/ — 1 . angle of x. X (169.) To find the integral of -^ s* 134 2a 1 1 /2adx _ n dx p dx a" + x^ ~J a-^xj~^\ J a-x x/^-^ — ^ n dx ^ — l 1 p- dx s/ — l aJ ~l J a^x ,J — X ^ ~\*J a-x />/— 1 = /-— r ■] hyp. log. of length of a + .r \/--l + ^ — \ . angle of a + .r ^ — 1 r /-— : ■] hyp, log. of length of a — x ^— 1 + ^^. angle of . flj-^V^-l |- , / 7 1 <, « + ^ x/ -^ ) + ^/ — 1 . anii'le or ^ , > ^ ^ a-x^-\} = -^ fhyp. log. of length of "^"^-^-N^-li j + anij^le or ^ , — ^ . 185 (170.) Cor. Hence, if a and x he positive quantities, and v be an angle whose tangent is - ; ' 2adx /2aax _ 2V. For in this case "^"^"^ /-4 ^^ ^^ *^"St^ = ^' a-'X ^ — 1 a-V^ sj — 1 ^ .-. hyp. log. of length of — y== = 0; .a + a:>y^ , .{a-k-xj -if And angle of ^77^^ = a"S»e of ^^^^j^i , = angle of ri + - \/-l j > = 2. angle of \\ +-x/^j, = 2t?; /ladx d'-\-x'' dx (171.) To find the integral of _^ /^._^^. • Let ± Va?'+a' =S/y then 0?' + a' = y, 136 dx ^ y' di/ + dx _^ x-\-y dx " y ' /dx _ ^ ndx-\-dy y ~'^ ^'^y ' /dx _ pdx+dy — hyp. log. of length of {x-^y) + sT—^ .angle of {x-^y) — hyp. log. of length of (^± x/PTo*) + ^y—l. angle of {x± ^x^-\-d), (172.) CoR. In like manner the integral of . will be found to be ± \/ x"^ + lax hyp. log. of length of {x^-a± ^x'-\-2ax) + A^ - 1 . angle of {x-\-a± ^x'-\-2ax) ; And by similar investigations the integrals of other logarithmic and circular functions may be ob- tained. 137 (173.) When a hody is moving about a centre of force as in Art, Ib'l; a radius vector^ and the velocity of the hody at the extremity of that radius vector being known both in length and direction; to find the value of the constant quantity h.

-0) + F.sin (^-0).yZT dr _ -^ do ~ dt '^^df -1, .-. rsm(f-9) = r.^; But h = r\~, dt .-. h = r^.sin (J —^\ 140 dp V , ( !, AT 1 I de Now -5 = 7— rr--n> r InF dt -■ p'~inr- il) ' dt' p'-dt n\l 'V^{\) ]•[{) dt ._3.^.|^,4i|, const. When e = o, p=l, 1 e+2 •. const. = / 2(l+e).Z 6+2-2-26 + const.^ 2(l+e).Z 2 (l+e)J/ 141 r 1 Let a = 1-e^ rx,, 1 1 + e COS 6 r"^ a i\—e^) ' ^^"^*'^" ^^ ^"^ve described. Next to find the time of describing' any part of this curve, = 2(l-e^).«{^ + «(})"^' + 2(i)-^}, 2(l-e') a i-e^ e /> e' 142 •••(1) '^e(!) '-^^-'—Ti^-^ •■ [[) e - ep^ •••(f) -^-^ ^? --e = ^^ dp V , f i\ N"^d7 = 7^^^^•^+(Jr^ rf< n 1 rfp~ /: v-.-..(. e (!) 1 ^ F^-l 'dp + 1 (f)^^e _ M I _ />* 1 143 Now pihdp = 2/05 (^ + 2ae)^ — /(/o + 2ae)^.-T ^J {p-\-2aef ^ ^P ^ ^^ ■- J {p' + 2aepy- = + (p''+2aepy + ae f , , o ^ ri = + (p^+2aepy 4- ae.hyp. log. of length of (p + ae± ^J p" -\-2aep) 4- ae s/ — I • angle of (p + ae ± ^ p^ -{• 2 ae p) ; 4- ae.hyp. log. of length of (|0 + ae ± ^ p^ + 2aep) 4- aesj - 1 .angle of (/o + ae ± ^ p^ -\-2aep) r + const.; 144 ep^ Vi/ ^ l+e(l) ^ .-. When i = 0, = 0, /o=:l-e.a, and ±(,'^ + 2.e,)i = (i:i^^ + ae.hyp. log. of length of (a + a >/l — e^) + ae.V^.angleof (a + «x/l--^') | + const. ^ ae. hyp. log. of length of^a -^-a^l-e') + ae. J~^\ . angle oi{a^a V 1 - e^ | + const.. •. « = ^^VT^ ^^ J\~-e'.{p^ae)^{p'-V'2aepf + ae . hyp, log. of length of ^_^^j^_^. 145 an equation which gives t in terms of p, where that value of + ,y^~+2aep must be used, which satisfies the conditions of the equation + (p4-2ae)2 ^ „., ^ _i 1 " ^^ , i-(l-.e^)^ = /n c+ . (175.) To ^nd the time t iVi terms of the angle 0. 4- ^p' + 2ap = - />. ^i^ey r. /iV + er .*. \/l -e^ .{p-\~ae) + {p^^ -\-2aep)^ f - 1 ^ (l-e^) Jr /iV + ae\ ~ r . llV — er ' f(, ,> '-'•'-■{■ + ''(!)-l | (l_e*)r(.^' «^« 1+ecose 3 146 = «e.(l-e')i 1 + e cos 9 ^,i cos — cos - sin ^ - 1 ^ae^l-ey. 1 + , cos ^ ae^(l -g^)^.sin ^y^ , ~ 1 + e cos ^ ' Also p + ae ± ^ p^ + '2aep = «5e + ?^ (Vl-e^ + l).(iy' +e x/l-^ = ae + a. >/l —( (7l-e^+l).(l)^"-he 1 + e cos e j-f e^cose + (l-^^+ >s/l-e'). (1)^ +e^l-< = « = a . -e'-^+e'cos^+C 1 + e cos ^ 0+sin^/ y^ 6 + 6^/1^ l-e^ + s/l- -e').(cos 1) 1 + e cos ?).COS0 + (l- -e^ + s/T -e^)sin6 'n/- e+e^l- -e^+(l+x/l- -1 1 + e cos 147 = a.(l + 7l-e=). :^^L^.^^^ , p + ae± sf p^-\-2aep __ e + cos + >s/l— e^ . sin Q sj^-i . a + Us/T^' " 1+ecos^ ' .-. t — — — — , — 7= . J ae. hyp. log. of length of e + cos^+ ^l--e\sin 6.^-1 1 +e cos Q , , .e + cos^ + ^/lT^.sin^.V^'^ 4- ae,jj — \ anffle or ^ -^ ^^^ ^ ^ ^ 1 + e cos ^ 1 + e cos ^ ) = ^ . I —p== . hyp. log. of length of e + cos ^ + V^ 1 - e"^ . sin ^ V^ — 1 1 + e cos d + angle of e + cos + V 1 - e\ sin ^ . >^ ~1 - 1 + e cos 1 + e cos ) ' now V ^ ^ T ' 148 ^ la a a^ •'• rJT^e" Jin' ... t ^^ " 7^' {r/^'^^^* ^^^' ^^ *^"^^^ ^^ e + cos + ^1 ~e\ sin 6 J -\ 1 + e cos , n e+ cos 0+ x/l-^^-sin ^.x/- 1 + anii^le of ^ — ^ ® 1+ecos^ e(l - e^)^ . sin 1 + e cos 6 ]• an equation which gives the value of t in terms of e, (176.) To determine what this value of t becomes according as different values are given to n. First let n be less than 1^ Then e is negative. Let e= — e, 149 then t - -^j= \ , . hyp. log of length of -e +cos^ + x/l — g'^^.sin ^V- 1 1 -- e' cos d 4- angle of -^^^ ^ ^ ° 1 — e cos Now length of eVl-e^^sine l "*" 1-e'cos^ P - e^ + cos a + x/ 1 - e'" . sin ^ \/^^ 1 - e' cos Q {(-e^+cos^)^ + (l -Q.sin^^}^ _ ^ 1 — e' cos /. hyp. log. of length of - e^ + cos ^ + J\^^. sin d J^\ _ ^ I — e cos 1 ^ - e' + cos0 4- \/T^^e^.sin ^^/"^ and anffle or ; -, s — ■ o I — e cos - e + cos = anfifle. whose cosine = ; , ° ' 1 - e cos and whose sine = y/l-e'\s'm \ — e' cos S 150 Let this angle = u, Then t = —7= . i w + e' sin ^^ [- . Next let 71 = 1, Then e = 0, /. i = — 1==-| i hyp. log. of length of (cosa + sin e V^--^) + angle of (cos + sin ^ — l) [• = —7=^. J —7=. hyp. loff. of lensrth of /iV + angle of (iV V ai s/r .e. Next let n be greater than 1 and less than \/2, Then e is positive and less than 1 ; r ^ u 1 u • e + cos Let u be an ano^le whose cosine = ~, ^ 1 + e cos e 151 Then it may be proved nearly as in the first case that t == —piz ^ i u-e sin u h , s/ni I 3 Next let n = ^2, Then e = I; Here we must find t by the method of deter- mining the value of fractions^ whose numerators and denominators are evanescent; By the preceding case, when e is less than l, —7= . -J u — e sin u h t = l^ u—e sm u s/m' (l-e)t 2.4' sin w +i- jsin^w + ^--r-g sin^w + &c. — e sin u m' (l-# ,| (1 - e) sin M + g . sin^ ^ "^ 7~ ' ^^^^ ^ "^ ^^* Now sin u = ^^^^-^'"Z = N/^^-N^Te-^inj^ 1 + e.cos 1 -f e cos 152 >/»*■ 1 1+ecose "*" « (1 +e cos 0/ Let e — 1, rru . ^^ ( /TT sin , 2t sin' 1 Then < = —7— . i ^2 . — : + yr . > tt^ [■ - ^!x/^ [ sine ^ sin' '] x/m *\l+cose"^ 3 -(J ^cos^)'/ = ;— .< tan -- + 4 . tan^ - > . Next let n be greater than ^2, Then e is greater than 1, In this case a* = (y^J = (^(^31)) = ^^ = i VHl . Let = a, e—l Then «* = a'*-^-l; 153 . log. of length of 1 + e cos e + cos^ + x/e^—l . ^~1 . sin ^. ^ — 1 1 + e cos ^ -I- angle of 1+ecos^ J = — P= . I hyp. log. of length of e + cos — s/e^— 1 . sin ^ 1 + e cos ^ / , . e + cos^ — Ve^^-l .sin^ + x/ — 1 • anffle or , . 3 ' ^ o 1 + e cos 1 + e cos J ' e + cos^- x/e^ - 1 . sin . 1 + e cos 9 Now ' "^"° ", /^ ^ — -^-i^llL^ is a positive quantity, .*. hyp. loiir. of length of -— ~ — ^^^ — '- — ■. — ■^^ ^ ^ l+ecos0 = hyp. log. e + cos — ^e^-i .sin S 1 + e cos ^ ~~ ' and ancle ot ^ — r— = ; ^ l+ecos0 154 .■..=-fi,{, = • \ *^yp- J^g- e 4- cos - jj e^—\ . sin Q 1 + e cos 1 + e cos J ' .■'rr/y^ '^^' *■ .*^*-