sec.X= COS. 1) = cos. na sin."w, tan.no e e 9 e e 1 1 multiplying the first by vī, and adding the recosec.X=sin.x sults, we obtain dz + V - Idy=(z+V-ly)dx v -1, THE ANALYSIS OF ANGULAR SECTIONS. dzt-ly) Idx. z+V --ly The first member of this equation is the differ ential of the hyperbolic logarithm of ztv -1.4, The formulæ which exhibit the relations between and :,: two arcs and their sum and difference, may be easily Uz + ✓ -- ly) = V-1.x, transformed into others which determine the relations between the sines, cosines, &c. of successive ...cos.x + V1.sin.x = 6V—, multiples of the same arc. If in the formulæ established in plane trigonometry (18) et seq. we sub no constant is added, since both sides become equal when x = 0. stitute nu for w and w for a' we obtain the following results: If the sign of x be changed, this becomes sin.(n+1) = sin.na cos.c. + sin.co cos.no, COS.X i sin.x = –V—T, cos.(n + 1) = cos. næ cos. I sin.nu sin.c, which being added to and subtracted from the sin.(n + 1). + sin.(n. 2sin. na COS.609 former, gives sin.(n + 1). — sin.(n 2sin.u cos.no, cos.x=}(exv 1 +e-XV =) cos.(n + 1)2 + cos. .(n = 2cos.No Cos. , [1]. cos.(n+1)— cos.(n − 1). 2sin.nu sin., v=ī sin.x=}(x+xvFT-XV1) sin.(n + 1)" sin.(n-1) sin.'na sin'a, From these formulæ we can immediately deduce cos.(n+1) cos.(n-1). exponential values for tan.x. Dividing the second sin.(n + 16 tan.na + tan.co by the first, we find Xv -X-1 2x -I [3] cos.n-1) 1+tan.na tan.co XV - --XV 2x v te The whole theory of angular sections, and all the +1 various relations between the trigonometrical func From the formulæ [1] another celebrated formutions of an angle and its multiples, may be derived la may be immediately derived, called from its disfrom two remarkable formulæ established by Euler, coverer, Moivre's formula. Since by which the sine and cosine of an angle are expressed as exponential functions of the angle itself. cos.x+v-1sin.x=0 These formulæ we propose, in the first instance, to establish, and from them to derive all the theorems (cos.x+visin.x) =EMXV—1 which form the subject of this part. They will But also, thus hold the same relation to the analysis of angu mx-1 lar sections as the formula for the sine of the sum cos.mxtv-lsin.mx=e of two angles does to plane trigonometry, and as the formulæ established in (25.) do to spherical trigo :(cos.c+v--- Isin.2)m=cos.mx+v—isin.mc [4). nemetry. Were it not that the principles upon By the principles which have just been establishwhich these exponential formulæ are established ed, we are enabled to resolve the formula are not sufficiently elementary, the original formula z2m-2azm + 1 for the sine of the sum of two angles itself might into its simple factors when a is not greater than be deduced from them, and these celebrated theo unity. rems might thus be made the foundation of the whole superstructure of trigonometry. Let a = cos. X, and the formula becomes The exponential formulæ for the sine and cosine z2m — 2cos.x .zm + l. which we are now to establish are By solving the equation 22m 2cos. xz'm +1=0, sin.x = zm = cos. x + I sin.x, .::2=(cos.x +V-I sia.x), XVT +v1 sin. Let of these two formulæ for z', each is susceptible y = sin.x, Z = cos.it, of m different values found by substituting success•:• dy = cos.xdx, dz= -sin.3dx, ively for x, :: dy = zdx, dz = -- ydz; 2, 2x + x, 47 + x,.... 2m - 1)T + 2, X z = COS. m m -(cos. and therefore the proposed formula resolved into (z-co. + +v=1 sin. ) -v=1 sin. :) ) m PA,=2cos.x, PA,=2cos. 2x, PAz=2cos. 3x, &c. Let a be assumed, so that PA, =p, which is always possible, since p is not > 2. Hence 2zcos.X + 1 = 0, ...2+ = 2cos. X = PA,, and by what has been already proved, it follows that za + = 2cos. 2x = PA,. 27 +3 (cos. m (z—(cos. 2(m—13*++V – Isin. m m m m m 21+x SECTION II. Isin.m m of the Development of Sines and Cosines, of Multi ple Arcs in Powers of the Sines and Cosines of the Simple Arcs. 2m - 1)*+a The developments respecting multiple arcs may be divided into two distinct classes. The first in2(m-1)++x 2(m-1)*+ cludes all series in which the sine or cosine of a Z- -(cos. VIsin. multiple are is expressed in powers of those of the simple arc, and the second those in which a power If each pair of simple factors be united by mul- of the sine or cosine of a simple arc is expressed in tiplication, we shall have the proposed formula re- a series of sines or cosines of its multiples; to the solved into real quadratic factors, former we shall devote the present section, reserv22m — 2zm cos.X +1= ing the latter for the following one. The series in powers of the sine, cosine, &c. may (22z cos. +1 (22z cos. be either ascending or descending, and accordingly 47+x x(22-22 cos. the several problems into which our analysis re+ 1)(zo - 22 cos. 67 +20 solves itself may be enumerated as follow: sin.mx cos. X. 20. sin.mx 2 in ascending powers of x(zo - 2z cos. 2(m-1) + + 1). cos.mx sin. X. 3o. sin.mx / in ascending powers of It appears, therefore, that in general all the simple factors are imaginary, and the quadratic real. cos.mx tan. X. This theorem was discovered by Moivre, and pre 4o. cos.mx 2 in descending powers of viously another similar to it, and deducible from it, sin.mx cos.x. was discovered by Cotes. In the last, let x = 0, 5o. sin.mx ) in descending powers of and we have cos.mx sin.x. 27 22m2zm + 1 = (x - 2z+1)x(22—2z cos. +1) To develope cos.mx in a series of ascending powers of C08.X. , x(zo--2Z cos. +1) Let cos x:= y, and let = =+ V –1, m m x (zo - 2z cos. 2(M — 1)* + 1). The principles which have been established in this section supply a very elegant geometrical construction, representing the sums of the squares, cubes, &c. of the roots of a quadratic equation of the form z-pz+1=0, in which p is not > 2. A5 A4 Let a circle be described with a radius equal to unity, A2A and let APA,= 2, A1 and let the arcs A,A,,A,A,,A,A3, AOL р be equal. It is obvious that But 1 2cos.mx = 2m + 2m If then zm be obtained in ascending powers of y and z-m deduced from it by changing the sign of m, we shall thence obtain 2cos.mx in a series of the required form.' Let zm = u= A + Any + A,yo + Agys +.. The solution of the question will be effected if the values of the coefficients of this series can be ob. tained without introducing any condition which restricts the generality of the problem. Let the series assumed to express u be twice dif. ferentiated, and the results will be A3 mu dy vy—1 m zm=(-1)?? du however, fail to determine the first two coefficients A., A,. To find these, let y = 0 in the series for u du dou and and also in the values = 2A,+2.3A2y+3.4A_yo+.... dy dy u=zm=(y + y2-1)m, Also, let du u = (y + Vy — 1)" be twice successively differentiated, and the results are and equating the results, we obtain A. =(-1)=(-1)", A,=ml v1)m-= m( - 1)7", mou=0, whence we find dy dyo dy dy du A, ( which divided by gives 2 m d’u du (y - mot = 0. Jy • A = + 2. 3. 4 dy dy' (mo—1°) (m?—3°). m(-1)"=?, ferentiating the assumed series, be substituted in Ag=+ 2.3.4.5 the last equation, and let the result be arranged ac mo (mo — 2°) (m? — 4*)(1) cording to the ascending powers of y. We shall A6 thus obtain the following series: 2.3.4.5.6 + Hence we find + [A,(mo—9) + 4.5A, ]y, yemo (mo— 2o)(mo—49) +[A.(mo-16) + 5.6A"]y*. 1.2.3.4 1.2.3.4.5.6 m” (mo 9°)(° – 4°)(mo-6°). - .} 1.2.3.4.5.6.7.8 + {An–2[m?—(1-2)*]+(n-1)(n)A, lyn—2 (m - 1) (m2—12)(mo—59) m-1) y + ys 1.2.3 1.2.3.4.5 (mo--1°) (m2—34) (m2-5°) » 0. y + } 1. 2. 3. 4. 5.6.7 To find the series for z-m, it is only necessary to change the sign of m in the result which has just -mP-A,, Az= been obtained. Since neither of the series in this 2.3 m-4 result contains any odd power of m, this change m-9 Aq= Ag, produces no other effect than to change the sign of the coefficient of the second parenthesis. Let the and that in the second S', and we have A, (n-1) =(-1)7.5 + m{ — 1)"7.S, Hence we obtain the following conditions: z-m=(-1).s+m(-1) --! S'; m-1 2.3 since—m(-1) m( - 1)-"? A4=+ Hence, by addition we obtain, (m?—1) (m?—9)A,, A,=+ zm +3-=[(-1) +(-1)7]s+[(-1))+(-1)-=3ms' , mo (mo—4) (m?–16).A., A6 ... 2cos.mx = [(-1)+(-1)]S +[(-1)";} 2.3.4.5.6. +(-1)]mS.... [1], The law of which is evident. These conditions, which is the development sought. m! m' (m? -22) + . 2 Ag=m? 4.5 ma m 'The form of the coefficients of this formula may be changed. cos.m(2n'r + x) = cos. m( 4n 7 17).S. We have + cosm--1) (4n + ! )* .ms', (cos.x+vI sin.x)m=cos.m(2n3+x)+V-1 sin.m(2n7+x), where both members have the same number of n being any positive integer. Let x = 1, :;: values, and where the values of the indeterminate integers, n', n are supposed to be less than the (-1)=cos.fm(4n+1++ -sin. Im( 4n +1 ), denominator of m. It still remains, however, to show the values of m=cos. łm( 4n+1)-V-lsin.}m( 4n+1)T, each member which correspond respectively to .:(-1)+(-1)+"= 2cos.fm(4n+1)*, those of the other. Since the value of each mem ber changes by ascribing different values to the in(-1)"}+(-1)} = 2cos. }(m— 1)(49+1) 7. tegers n' and n, this question only amounts to the Hence the series for cos.mx becomes determination of the relation between any two corcos.mx = cos.fm(4n+1)*.S+cos.(m—1)(4n+1)=.mS'... [27. responding values of these integers. Let x -17, and therefore S= 1, S' = 0. llence In this formula n is an indeterminate integer for each value of which the second member has two cos.m(2n'r + 1) = cos.im( 4n + 1), values corresponding to the double sign +. The or cos.fm( 4n' + 1)* =cos.žm( 4n+ 1), successive terms of the series Since n and n' are not supposed to receive any value 0, 1, 2, 3, ... greater than the denominator of m (for all ihe vabeing substituted for n in cos. {m(4n+17, it will lues of the cosine after that would only be repetisuccessively assume different values until the num tions of former values), this last condition can only ber substituted for n is equal to the denominator of be satisfied by m; for this value of n the value of cos. m (4n+1) n=n'. cos.m(2n7+x) = cos. m( 4n+1).S + cos. (m--1) (4n+1). . mS' [3] To develope sin. mx in ascending powers of cos.X. of as many different values as there are units in the By subtracting the value of z-m from that of denominator of m, and no more. In like manner zm, the result being disengaged from the imaginary cosm( 4n-1)7 is susceptible of the same number symbols becomes of values, and therefore the coefficient of S is sus. sin.m(2n7+ x) = sin. žm( 4n+1)+S + sin.}(m — 1) ceptible of twice as many values as there are units in the denominator of m, and a like observation (4n+1)*. mS'. [4]. applies to the coefficient of mS'. All the observations in the last proposition are Since S and S' involve no functions of x except equally applicable here. When the denominator of cos.x, the change of x into 2n7 + x makes no m is an odd integer there are always two values of change in their value, and it follows, therefore, that an angle x whose cosine is given, which are such for a given value of cos.x the second member of that sin.mx will be expressed by only one of the [2] is susceptible of twice as many values as there two series in [4]. are units in the denominator of m. It is therefore Another form for the development of sin.mx in necessary to show how cos.mx can have several ascending powers of the cos.x may be established corresponding values to a given value of cos.x. by differentiating the series found for cos.mx. By The angle x being changed into 2n'r +x, n' being this process we obtain an integer, makes no change in cos.x, but changes ds cos.mx into cos.m(2n'r+x), which has twice as msin.m(2nr +x)=-cos. m( 4n+1) many values as there are units in the denominator dx of m. Hence the formula [2] will be more gene ds' rally and correctly expressed thus, cos.:(m — 1) (41 + 1)7. m dx * In clearing the formula [1] of imaginary quantities, Lagrange has fallen into an error which was lately detected by Poinsot, and the difficulty explained as above. Lagrange's mistake arose from assuming that (= 1)m)=cos.m7+/-1 sin. ma, which is evidently erroneous, since the first member has as many different values as there are units in the denominator of m, and the second member has but one value. lle forgot to take into account, that while the change of « into 2ntx produces no change on (cos.ctv1 sincm, it does produce a change on cos.mx+ -1 sin:mz. In fact, without this consideration, Moivre's formula itself is involved in the absurdity of one member having a greater number of different values than the other. VOL. XVIII. Part I. M* -y + -y& dy 1 m--1 m2 sin.x, m Let y ds m? m”(mo— 2o)(m?— 4")ys 2 visin.mx = [(1)-(1)"] Q+v=1[(1)*** 1.2.3 1.2.3.4.5 +(1) mQ dS' - 12 (m2 — 12)(m? — 32). -g+. Ly =R' It will be observed, that by changing x into dy 1.2 1.2.3.4 218 + x, no change is made on the series Q and Q'; dy but there is a change made upon the first member da of each equation. The coefficients of Q and Q ds ds' have exactly as many different values as the first members of the equation. This is a circumstance which has been hitherto overlooked.t msin.m(2n + 2 sin.x[cos. į m ( 4n + 1)m.m. R The above formulæ can be cleared of imaginary quantities by the usual method, (1) = = cos.nmt tv--I sin.nmi, (1) cos.n(m — 1)* + V-1 sin.n(m-1), under the formula no + x, x being taken with the sign + when n is even, and when n is odd. Hence the formulæ become z= V1-y2 + yv - 1, cos.m/nr+x)=cos.nmr.Q-sin.n(m-1).mQ'-[6.] 11-y2 + y - 1,); zm = sin.m(n+x)=sin.nmr.Q+cos.n(m--1)..mQ':[7.] To develope the sine and cosine of a multiple arc in a series of ascending powers of the tangent of the simple arc. By developing the formula cos.mx + - I sin.mx = (cos.x + -] sin..c)m; by the binominal theorem we shall obtain 2m = A. cos.mx + - 1 sin. mx = R +IR'. [8.] ув where R represents the sum of the odd, and R' of the even terms of the development, and therefore R=cos.m2-A, cos.m--2xsin. *x + A cos. M--4xsin.4x—. m2 — 1? (m? 12)(m2 32) + +A, {y -y + -....}: R'=A,cos.m-lxsin.x--Azcos.m--3xsin.3x+A, COS.7--5rsin.5x—... 1.2.3 1.2.3.4.5 where A,, A2, A3, . represent the coefficients dizm) making y = 0 in the two values of zm and ed binomial, whose exponent is m. dy As each side of the equation [8] consists partly and equating the results, which gives* of real and partly of imaginary quantities, it is A. = (1) %, A, = m v1(1)";} equivalent to two distinct equations, between each The value of z-m may be deduced from that of separately. If we consider R composed exclusivezm by changing the sign of m. Hence, if the series ly of real, and - IR' of imaginary quantities, which enter these values be Q, Q', we obtain we should therefore have z= (1)Q+ –1(1)"+mQ', cos.mx = R. sin.mx = = (1).Q-V-1(1)– ","mQ', These formulæ, which were first published by ...2 cos.mx=[(1)+ (1) 7] Q+v1(1)"! John Bernoulli in the Leipsic Acts, 1701, have been, even to the present day, considered as exact and + (1)".*]mQ, general. This, however, is not the case. { Lagrange, and all the mathematicians after him, have fallen into an error in the determination of these coefficients. † Poinsot, 1825. |