To To explain this, let bers correspond or are equal to those of the first T=1— A, tan.° x + A, tan.* x severally. In other words, it is necessary to deterT' = A, tan.x Aztan.8x + A, tan.5 20 mine what relation subsists between the indetermi. .:R= cos.mx.T, nate integers n' and n, neither of which are supR' = cos.mxT'. posed to exceed twice the denominator of m. By changing x into 2n1 + x, the factors 'T, T', determine this, let x = 0, P= 1, T=1, T = 0. Hence of the second members of cos.inn'r = cos.mnt, sin.mn'm = sin.mnt, undergo no change, since these arcs have the .in' = n. same tangent, and since T, T', include no powers These integers are therefore always equal, and the except integral powers of tan.x, they can have each formulæ become but one value for an arc, whose sine and cosine are cos.m/nx + x)=P(Tcos.mna—T'sin.mnr).... [10]. given. The first factor cos.mx has, however, as many different values as there are units in the de- sin.m(na + x)=P(Tsin.mna + T'cos.mnr)....[u]. nominator of m, of which two, at most, can be real, Whether the odd or even integers are to be suband all the others must be imaginary. On the stituted for n in these formulæ, and whether x is to other hand, for an arc whose sine and cosine are be taken with + or -, is to be determined by the given, and which is of the form 2nt + X, n being signs of sin.x and cos.x, which are supposed io be any integer, the first members of these equations given. If cos.x be positive, the values of n are to have as many different values as there are units in be selected from the series the denominator of m, and all these values are real. 0, 2, 4, 6, ; Thus the two members of the equations are incon- if it be negative, they are to be selected from sistent. It is not difficult to perceive that this absurdity 1, 3, 5,.... has arisen from the false assumption that the real If sin.x be positive, x is to be taken with +, and and imaginary parts of the second member of [8] if negative, with – In all cases, however, the were R and ✓ – 1R'. We shall find upon con- coefficient P in the second members is to be consisideration that neither of these quantities are alto- dered as an abstract number independent of any gether real, or altogether imaginary, but that each sign. of them is composed partly of real and partly of If m be an integer, the formulæ are reduced to imaginary quantities, and is of the form a + the forms ✓ - 1.6. cos. mx = cos.mxT, cos.mxT', In the formula which have hitherto been taken to be general for all cos.mx + 7 – 1 sin.mx = cos.mx/T + ✓ - IT'), values of m. let the absolute, real, or arithmetical value of cos.mx, (cos.x being considered merely as a num To develope the cosine or sine of a multiple arc in ber,) be P. It is plain that its several algebraical descending powers of the cosine of the simple arc. values will be expressed by the formula P(+1). And since This problem was investigated by Euler, and (+1) = = cos.mnt tv i sin.mnt, subsequently by Lagrange, and both obtained the same result although they proceeded on different .::cos.mx = P(cos.mnt + V - Isin.mnr), principles and by different methods. The series the indeterminate integer n being even when cos.x which were the results of their investigations, and is positive, and odd when it is negative. which have, even to the present time, been received Making this substitution in the former equation, as general and exact, are the following, and in place of X, substituting the general formula n'm + x for all arcs having the same cosine, in mm-3). 2cos.mx=(2y)"_m(2y)--2+ (2y) which the sign + is used when n' is even, and when it is odd, we obtain m'm--4)(m--5) m{m--5) (m-6)(m—7)(24)* cos.m(n'* + x) + V - I sin.m(nr + x) + = P(Tcos.mnt - T'sin.mnr) 1.2.3 1.2.3.4 + ✓ - 1.P(Tsin.mna + T'cos.mnr). Here the real and imaginary parts are separated on m.m+3 each side, and equating them, we have +(2y).-m+m( 2y)----2+ 3/2y)--m-4, 1.2 cos.m/nr+ x) = P(Tcos.mnr-T'sin.mn,), sin.m(n's + x)=P(Tsin.mnt + T'cos.mnr). m(m+4) (m+5)(2y)--m-6 m(m+5) (m+6) (m+3)(2y)-----8 + Each member of these equations is susceptible of 1.2.3 1.2.3.4 as many different values as there are units in the + denominator of m. But it remains still to be determined which of the values of the second mem [12]. sin.mx = m- 1.2 --$ (2y)m--6 + + . . . where y=cos. X. The series for sin.mx was deduced from this by differentiation. Poinsot has examined the analysis by which these When y=l, the second members of these equations results were obtained, and shown that it is fallacious, and that the results themselves are false. become equal severally to A, B, 1C, 1.2.3. D..... To render this refutation intelligible, it would be Let the values of the function U and its successive necessary to detail the process by which Euler and differential coefficients when y=1 be called Y, Y', Lagrange established the formulæ, which would Y", Y'', &c; we have hence lead to investigations unsuited to the purposes of IS the present treatise. As, however, the results of A=Y 2m + om 22 Lagrange have been hitherto universally received 1 as correct, it is proper to make the reader aware of B= Y'= the fact of their having been proved erroneous. 22 We shall confine ourselves here to the investiga C=Y" tion of the true development of cos.mx and sin.mx. Let p=cos. x and q=sin.x We have then D=Y!!! 24 cos.mx=pm(1–4,99 2) -3) :) - 3m(-+ where 1, A,, A,, . ... are the coefficients of the binomial series, m being the exponent. We have E=Y"=m(m--1)(m—2)(m—3) (2m-4+0m-4) q* = 1—p', q* = 1 — 2p+ p*, . 25 Let these values be substituted for q°,9, &c. and —6m(m--1) (m-2) (2m-3—0m-3)+15m(m-1) (2m-2 +0m-2) let the results be arranged according to the descending powers of p, and we have &c. &c. cos.mx=Ap—Bpm-2 +_cp + &c. In these analytical expressions for the coeffi1.2.3 cienis of the sought series, it is necessary to prewhere serve the terms om, om--1, om--2, &c. because each of A=1+ A, + A + A. these powers of o become either unity, 0, or infiB = A, + 2A, + 3A6 + 4A, + 5A,0 nite, according as the exponent of the power is = 0, C = A, + 3A + 6A, + 10A10 +... positive or negative. 1 The true development, therefore, of cos.mx in deD=A6 + 4A: + 10A,0 + 20A 12 scending powers of cos.x or p, the angle x being 1.2.3 supposed less than a right angle, and only considering a single value of cos.mx relative to the arc } { 2-l_0") }; 2 {m_m—1)(2m-2-tom-2)—m( 2m=l_om-1)} b{ — +A. – 15m(2m-1on-1)}, &c. m-4 1 m--6 DP X, is 1.2 The law by which these coefficients are formed is cos.mx=Ypm-Y'pm--2+1Y"pm-?, Y"" pm--67.... evident, but it is necessary to obtain finite expressions for them as functions of m. For this pur If m be a positive integer, this series will be fipose, let us suppose that the successive terms of nite, since all the terms beyond a certain term will the first coefficient A were multiplied by the suc :0, and it will thus give the exact value of cos.mr. cessive powers of an arbitrary quantity y, so that it Thus when m=0, or m=1, we find that the first coeffibecomes cient only has a finite value, and all the others = 0. 1 + A2y + A 47° + A6y3 t.. For m=2 and m= 3, the first two coefficients are fim(m—1) orl+ y + m{m-1)(m—2)(m—3) - go+.... nite, and all the rest = 0. For m = 4, m = 5, there 1.2.3.4 are three terms finite, and all the rest equal nothing; But this last is equivalent to and in general, if m be an even integer, the number (1 + y) + (1 - Vy)" of finite terms is " +1, and if it be odd, m.21. =U; But if m be a fraction, the series never termiso that U becomes equal to A when y El. It is nates, and the coefficients only continue finite as not difficult to perceive that the other coefficients long as the exponent of 0 which occurs in them is are what the successive differential coefficients of not negative. After this happens, all the succeed. U taken with respect to y as a variable become ing coefficients are infinite. Thus, if m be a fracwhen y = 1. We have tion between 0 and I the first coefficient alone is fiU=1+ A,y + Any? + A6y + .... nite, and all the rest infinite. If m be between i dU and 2, the first two coefficients are finite, and all the = A, + 2A y + 3A y + 4A .y .... rest infinite, and so on. If m be a fraction between I d’U n -- 1 and n, the first n terms are finite and all the = A + 3A6y + 6A yo +.... rest infinite. The series, therefore, in these cases 2 dy 2 dy Also, in is useless and absurd, and the same happens when ... 2mcos.mx=e te mx-1 ( m2)xV--1 (m~4)XV-1 multiple is a positive integer; and in this case, 2mcos.mx = e + Ae t.... since the number of terms is finite, the series is where nothing more than the series already obtained in 1, A, B, C, .. ascending powers, the order of the terms being re are the coefficients of the binomial series. versed. So that, in effect, the only case in which the development by descending powers is possible, Eliminating e by the general formula, it is useless. MX-1 It is worthy of remark, that in the analytical ex cos.mx + V-1 sin.mx = e pression for the coefficients A, B, 1C, &c. if the we obtain powers om, om-1, om--2, &c. be neglected, the coefficients will be exactly those of the series [12], 2mcos.mx = cos.mx + Acos.(m—2)x+ Bcos.(m~4)+ which has been hitherto considered exact. +-1[sin. m.c+Asin.(m—2)x+ Bsin.(m—4)x+ .. ]. Whence may be seen the reason why this series Let the first series be Px, and the second Qr, and gives false values for cos.mx, and also why in the we have particular case in which m is an integer the value (2cos.x) = Px +iQx. resulting from it will be exact if we retain in it only the positive powers of p, for that is in effect that case (2cos.x)m must have at least one real va Let cos.x be first supposed to be positive, and in rejecting all that part of the true development lue. Let this be X, and all its other values will be which becomes = 0. found by multiplying X by the values of (1)m. They To develope sin. mx in descending powers of cos.X. are, therefore, all expressed by the formula X (cos. 2mnrt - I sin. 2mnr), To effect this, it is only necessary to differentiaten being any integer not exceeding the denominator the series (12]. This being done, and the result of m. divided by 2m, and observing that dy=dcos.c = - sin. xdx, we obtain (2cos.x)m= Px + - 1Qxı sin.mx =(24)m-4m—2)(24)m--3 + (M—3)(m-4) (2y)m--5 no change is made in the first member by changing sin. 30 1.2 3c into 2n7 + x, and therefore This development, like the last, is only possible (2cos.xm = P (2cos. mm = P2n=+*+/-iQ2n+« when m is an integer. Hence +[1 descending powers of the sign of the simple arc. Equating the real and imaginary parts of this equation, we find 1 the two series being expressed by M and M', and X = · [2]. being understood to express sin.x, instead of cos.x, sin.2mnt we shall have Hence it appears that the real and positive value Xof (2 cos. 2) can be indifferently expressed, either in a series of powers of the cosines or sines of the one another only in the constant coefficients. Between the two series thus found, there subsists a constant relation: In this case, as in the former, m must be an integer. cos. 2mns ; by which it appears that these series have a conOf the Development of a Power of the Sine or Cosine stant ratio, whatever be the value ascribed to x, of an Arc in a series of Sines or Cosines of its from 0 to Multiples. 2 To develope cos.mX in a series of cosines or sines If n = 0, we obtain by [2] of multiples of x. which is therefore perfectly general, provided x be We have supposed less than , and X confined to the real 2cos.x = xv-te-, and positive value of (2cos.x)m. VOL. XVIII. PART I. N -2, and 1 P2ng+x, cos. 2mn P2no-tx X = Px2 n = 1 . 1 Q2n + . The second formula of [2] gives sin.2m.na 0,::: Since 2n + 1 is odd, both i and n' must be odd. But since n is supposed not to exceed n', i must be =1. This fails in giving any value of X, but shows Hence that Q=0 n'-1 for all values of x from 0 to +7 which is therefore the only value of n which can If the cos.x be negative, let (2cos.x)m be express satisfy the proposed condition. ed thus, Hence, if m be a fraction with an odd denomina. (-2cos.x)m (2cos. 2)(-1)= x(-1)m. tor (n'), we have But since X = + P" Q(—1 (-1)m=cos.m(2n + 1)* +- 1 sin.m(2n +1)*. (n’—137+x P{n:—157=z=0, + being used when m' is even, and when odd. Hence xcos.m(2n + 1)* + -- 1x sin.m(2n + 1* But if m be a fraction with an even denominator, there is no arc (2n+1), which can render cos.m = P2n7+z+1Q2na tx' (2n + 1) = +1; and, consequently, no arc 2mni By equating the real and imaginary parts, we find + x for which the series Pu can become equal to the real value of (2cos.x). X P2nx+x • . [3], By the formulæ [3], [4], it follows that when cos.m(2n + 1)T cos.x is negative, the real and positive value of X= (2cos.x)m may be expressed either in a series of sin.m(2n + 1) sines or cosines of the multiples of x, and that the In which the integer n is susceptible of any value two developments differ only in the coefficients; from 0 to the denominator of m. and finally, that their ratio is the same for all vaIf n = 0, we have lues of x between 5 and 2 The development which has been thus obtained which give developments of the real value of (2cos. gives the value of the mth power of the cosine of XC)" when cos.x is negative. an arc in a series of cosines or sines of its multiFrom this it appears that Qr is not = 0 as in the ples. Similar series for the mth power of the sine former case, where cos.x was supposed positive. may be obtained in a similar way. But although Q2mn2+x may not = 0 when n = 0, By expanding yet there may be some other value of n, which will XV-1 render this series To discover this, let it be -1)= and eliminating e by the formula, cos.mx+v=1 sin.me = c5mxv-T, That these conditions be fulfilled, it is necessary we obtain m' that m(2n + 1) be an integer. Let m = and let (2sin. x)(V)m=cos.mx-Acos.(m—2)x+Bcos.(m—4)2– n' tv-l[sin.mx— Asin.(m—2)x+ Bsin(m—4x)—...] I be any integer, :: the series be called P, and Qx, we have m'(2n + 1) = In'; but m' being prime to n' measures I. Let (2sin.x)(-1) = Px+v- iQr. This formula being treated in a manner similar to 2n + 1 = in'. that for (2cos.x)m, will give similar results. 37 COS. mr 0. m Let=i, TRINCOMALEE. See Ceylor, Vol. V. p. 576. of bituminous scoriæ, vitrified sand, and earth, all TRINIDAD, an island on the east coast of South cemented together. In some places only beds of America, within ten or eleven miles of the conti- cinders are found; a strong sulphureous smell is felt nent. It is of an irregular square form, about on approaching this cape, and it prevails in many thirty leagues long, and from two to ten leagues parts of the ground to the distance of eight or ten broad. miles from it. The bituminous plain occupies the It is the largest and finest of the Leeward Isl- highest part of the point of land which shelves into ands, abounding with the noblest forests, and pos- the sea, and the plain is separated from the sea by sessing a soil fitted for the growth of every arti- a margin of wood which surrounds it. At first it cle of West India produce. The northern part of resembles a lake of water, and in hot and dry weathe island is covered with a ridge of mountains ther it is actually liquid to the depth of about an which end with point Galera, and seem to have inch. It is of a circular form, and about three been a continuation of the Parian Mountains on the miles in circuit, bounded on the north and west by continent. They consist of gneiss, and mica slate, the sea, on the south by a rocky eminence of porcewith great masses of quartz, and of compact bluish lain jasper, and on the east by the usual argillaceous grey limestone. For many leagues to the south, soil of ihe country. Its more common consistence there is little else than a thick fertile argillaceous and appearance are those of pit coal, but of a greysoil, without a stone or a single pebble. The purest ish colour. Dr Anderson regards it as the bitustreams issue from both sides of these mountains, men asphaltum of Linnæus. It is ductile by a genand form on the south the river Caroni; which is tle heat, and when mixed with grease or common navigable by canoes and floats for some distance pitch, it is much used for the bottoms of ships, and into the interior. The ridge of the Montserrat is regarded as a preservative against the Teredo hills begin at L’Ebranche on the east side, and navalis, which is so destructive to ships on all the stretch in a south westerly direction. coast of Guaiana. The Spanish government found The forests contain the finest wood for ship out its value, and intended to have made use of it in building and ornamental purposes; among which their naval yards. are the red cedar, and a great variety of palms. According to Dr Nugent, who has more recently The articles of West India produce which are ex examined this lake, the bituminous substance is tensively cultivated are, sugar, cocoa, indigo, cof- asphaltum, which is traversed by numerous and fee, tobacco, cinnamon, and cloves. There are ex sometimes deep crevices filled with good water, tensive savannahs on the island, on which quanti- which the inhabitants use, and in which mullet, and ties of cattle, horses, and mules are fed in common. other species of fish are often caught. Its surface Game abounds in the woods, particularly deer, the had the colour of ashes, and did not adhere to the lap, the cuenca, a species of wild hog of exquisite foot, from which, however, it received a partial flavour, and a variety of other kinds. Among the impression, but not so as to be dangerous to walk birds are the wild turkey, the raimier, and the par over it. Pieces of unaltered wood were found enrot. The coasts abound in various kinds of fish. veloped in the pitch. The lake contains many The mangrove oyster, which breeds on the branches islets covered with long grass and shrubs, which of the trees of this name, are very abundant. Lob are the haunts of birds of the most exquisite plusters, crabs, shrimps, and prawns are plentiful. mage, as the pools are of snipe and plover. Alli The climate is considered to be as healthy as in gators are said to abound here, but Dr. Nugent did any part of the New World. The winter or rainy not see any of them. The asphallum is sometimes season begins in June, and terminates in October. black and hard, with a dull conchoidal fracture, but The fine season begins in November, when the cold in general it may be easily cut, and its interior apnorth-east winds from North America give fresh- pears oily and vesicular. At a candle it melts like ness to the air. Among the healthiest and mildest sealing wax, and burns with a flame which soon parts of the island are the vallies of Sta Anna, ceases when removed. It acquires fluidity when Maraval, Diego, Martin, Aricagua, and the heights mixed with oil, butter or tar, and is used as pitch. of St Joseph to the north-west, and the vallies on Dr. Nugent found petroleum perfectly fluid in one the north coast. In spring, the thermometer stands part of the lake, and Captain Mallet, in his topoin the day at 80° of Fahrenheit. During the night graphical sketch of the island, observes that “ near it descends to 60° and sometimes to 50° in places Cape La Brea, a little to the south-west, is a gulf of moderate elevation. or vortex, which in stormy weather gushes out, Among the objects of particular interest in this raising the water five or six feet, and covering the island, are the pitch or tar lake of Brea, and the surface for a considerable space with petroleum or Mud volcanoes. tar.” He adds, that on the east coast on the Bay The pitch lake or plane is situated on the leeward of Mayaro, there is another gulf similar to the side of the island, on a point of land extending former, which in March and June detonates like about two miles into the sea, and opposite to the thunder, having some flame with a thick black Parian Mountains on the continent. The headland smoke, which vanishes immediately, In about is about fifty feet above the sea, and is the highest twenty-four hours afterwards, there is found along point on this side of the island. When seen from the shore of the bay a quantity of bitumen about the sea it resembles a dark scoriaceous mass, but three or four inches thick. Captain Mallet likewhen more closely examined, it is found to consist wise quotes Gumilla, as stating in his description |