=cos.mx.T', bers correspond or are equal to those of the first These integers are therefore always equal, and the cos.m(n+x)=P(Tcos.mn-T'sin.mn).... [10]. .... undergo no change, since these arcs have the sistent. = P(Tcos.mn - T'sin.mn) +✔― 1.P(Tsin.mn + T'cos.mn*). Here the real and imaginary parts are separated on each side, and equating them, we have cos.m(n'+x) = P(Tcos.mn - T'sin. mn), T'sin.mn,), sin.m(n'+x) = P(Tsin.mn + T'cos.mn). Each member of these equations is susceptible of 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 Whether the odd or even integers are to be substituted for n in these formulæ, and whether a is to be taken with+or, is to be determined by the signs of sin.x and cos.x, which are supposed to be given. If cos.x be positive, the values of n are to be selected from the series 0, 2, 4, 6,. . . . ; 1, 3, 5,.... If sin. be positive, a is to be taken with +, and if negative, with. In all cases, however, the coefficient P in the second members is to be considered as an abstract number independent of any sign. If m be an integer, the formulæ are reduced to the forms cos.mx = cos.mxT, sin.mx cos.mxT', which have hitherto been taken to be general for all values of m. To develope the cosine or sine of a multiple arc in This problem was investigated by Euler, and 2cos.mx=(2y)”—m(2y)m--2+ + + 1.2.3 + m(m-3), ፃ--4 1.2 m(m▬▬5) (m—6) (m—7)(2y)" 1.2.3.4 m--8 m(m + 4) (m+5) (2y)--m--6+ m(m+5) (m+6) (m+7) (2y)--m--8 m(m+5)(m+6) (m+7)(2y)-m--8 1.2.3 + + 1.2.3.4 [12]. where y=cos.x. The series for sin.mx was deduced from this by differentiation. Poinsot has examined the analysis by which these results were obtained, and shown that it is fallacious, and that the results themselves are false. To render this refutation intelligible, it would be necessary to detail the process by which Euler and Lagrange established the formulæ, which would lead to investigations unsuited to the purposes of the present treatise. As, however, the results of Lagrange have been hitherto universally received as correct, it is proper to make the reader aware of the fact of their having been proved erroneous. We shall confine ourselves here to the investigation of the true development of cos.mx and sin.mx. Let p=cos.x and q=sin.x We have then where 1, A,, A2, . are the coefficients of the binomial series, m being the exponent. We have 24 {m(m — 1) (m — 2) (2m—3—— (m-3) - 3m(m—1) (2m-2+0m−2) + 3m(2m--1 m-1)} m-1)}, 23 { m(m−−1) (m—2) (m—3) (2m−1 +0m−1) 25 In these analytical expressions for the coefficients of the sought series, it is necessary to preserve the terms om, Om--1, Om--2, &c. because each of these powers of O become either unity, 0, or infinite, according as the exponent of the power is = 0, positive or negative. The true development, therefore, of cos. mx in descending powers of cos.x or p, the angle x being supposed less than a right angle, and only considering a single value of cos.mx relative to the arc x, is Cos.mx= = 1 =Ypm—Y1pm--2+4Y"pm--4—Y""' pm--6+•••• 2.3 nite, since all the terms beyond a certain term will If m be a positive integer, this series will be fi= 0, and it will thus give the exact value of cos.mx. Thus when m=0, or m=1, we find that the first coefficient only has a finite value, and all the others = 0. For m=2 and m = 3, the first two coefficients are finite, and all the rest = 0. For m 4, m = 5, there are three terms finite, and all the rest equal nothing; and in general, if m be an even integer, the number of finite terms is m+1. +1, and if it be odd, 2 m 2 = But if m be a fraction, the series never terminates, and the coefficients only continue finite as long as the exponent of 0 which occurs in them is not negative. After this happens, all the succeeding coefficients are infinite. Thus, if m be a frac tion between 0 and 1 the first coefficient alone is finite, and all the rest infinite. If m be between 1 and 2, the first two coefficients are finite, and all the rest infinite, and so on. If m be a fraction between 1 and n, the first n terms are finite and all the rest infinite. The series, therefore, in these cases n sin.mx sin.c is useless and absurd, and the same happens when m is negative. From whence we may conclude, that the development of the cosine of a multiple arc in descending powers of that of the simple arc is never possible, except when the coefficient of the multiple is a positive integer; and in this case, since the number of terms is finite, the series is nothing more than the series already obtained in ascending powers, the order of the terms being reversed. So that, in effect, the only case in which the development by descending powers is possible, it is useless. It is worthy of remark, that in the analytical expression for the coefficients A, B, C, &c. if the powers om, om-1, Om-2, &c. be neglected, the coefficients will be exactly those of the series [12], which has been hitherto considered exact.. Whence may be seen the reason why this series gives false values for cos.mx, and also why in the particular case in which m is an integer the value resulting from it will be exact if we retain in it only the positive powers of p, for that is in effect rejecting all that part of the true development which becomes =0. 2mcos.mx = cos.mx + Acos. (m—2)x+ Bcos.(m—4)x + +-1[sin.mx+Asin.(m-2)x + Bsin. (m-4)x+ Let the first series be Pr, and the second Qr, and we have (2cos.x) = Px + √−1Qx. that case (2cos.x)m must have at least one real va- n being any integer not exceeding the denominator of m. Also, in (2cos.x)m Px +✔―1Qx, no change is made in the first member by changing x into 2n + x, and therefore (2cos.xm=P, Hence Xcos.2mnx+✔—1Xsin. 2mnr=P '2nx+x+√−1Q2nx+x · · [ 1 Equating the real and imaginary parts of this equation, we find 1 cos. 2mnT -P2nx+x, X = 1 sin.2mn Hence it appears that the real and positive value Xof (2 cos.x)m can be indifferently expressed, either in a series of powers of the cosines or sines of the multiples of x, and that the two series differ from one another only in the constant coefficients. Between the two series thus found, there subsists a constant relation: by which it appears that these series have a constant ratio, whatever be the value ascribed to x, from 0 to If n = 2 = 0, we obtain by [2] X = Pr 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. Hence n =n'=1, which is therefore the only value of n which can satisfy the proposed condition. Hence, if m be a fraction with an odd denominator (n'), we have X=+P being used when m' is even, and when odd. But if m be a fraction with an even denominator, there is no arc (2n+1) which can render cos.m (2n + 1) = ±1; and, consequently, no arc 2mn + for which the series Pr can become equal to the real value of (2cos.x)m. By the formulæ [3], [4], it follows that when cos. is negative, the real and positive value of (2cos.x)m may be expressed either in a series of sines or cosines of the multiples of x, and that the two developments differ only in the coefficients; and finally, that their ratio is the same for all va3л lues of x between and π 2 2 The development which has been thus obtained gives the value of the mth power of the cosine of an arc in a series of cosines or sines of its multiples. Similar series for the mth power of the sine may be obtained in a similar way. By expanding (2sin.x)^{√=1]»=(e*V=I_e~*√−1)", and eliminating e by the formula, cos.mx±√−1 sin.mx = e+mx√—ī ̧ we obtain and let (2sin.x)TM(√—)m=cos.mx—Acos.(m—2)x+Bcos.(m—4)x—. but m' being prime to n' measures I. Let,i,... 2n + 1 = in'. +✔―1[sin.mx—Asin. (m—2)x+Bsin(m—4x)—...] the series be called Pr and Q, we have m (2sin.x)(−1) = P2 + √ − 1Q2. This formula being treated in a manner similar to that for (2cos.x)m, will give similar results. TRINCOMALEE. See CEYLON, Vol. V. p. 576. TRINIDAD, an island on the east coast of South America, within ten or eleven miles of the continent. It is of an irregular square form, about thirty leagues long, and from two to ten leagues broad. It is the largest and finest of the Leeward Islands, abounding with the noblest forests, and possessing a soil fitted for the growth of every article of West India produce. The northern part of the island is covered with a ridge of mountains which end with point Galera, and seem to have been a continuation of the Parian Mountains on the continent. They consist of gneiss, and mica slate, with great masses of quartz, and of compact bluish grey limestone. For many leagues to the south, there is little else than a thick fertile argillaceous soil, without a stone or a single pebble. The purest streams issue from both sides of these mountains, and form on the south the river Caroni; which is navigable by canoes and floats for some distance into the interior. The ridge of the Montserrat hills begin at L'Ebranche on the east side, and stretch in a south westerly direction. The forests contain the finest wood for ship building and ornamental purposes; among which are the red cedar, and a great variety of palms. The articles of West India produce which are extensively cultivated are, sugar, cocoa, indigo, coffee, tobacco, cinnamon, and cloves. There are extensive savannahs on the island, on which quantities of cattle, horses, and mules are fed in common. Game abounds in the woods, particularly deer, the lap, the cuenca, a species of wild hog of exquisite flavour, and a variety of other kinds. Among the birds are the wild turkey, the raimier, and the parrot. The coasts abound in various kinds of fish. The mangrove oyster, which breeds on the branches of the trees of this name, are very abundant. Lobsters, crabs, shrimps, and prawns are plentiful. The climate is considered to be as healthy as in any part of the New World. The winter or rainy season begins in June, and terminates in October. The fine season begins in November, when the cold north-east winds from North America give freshness to the air. Among the healthiest and mildest parts of the island are the vallies of Sta Anna, Maraval, Diego, Martin, Aricagua, and the heights of St Joseph to the north-west, and the vallies on the north coast. In spring, the thermometer stands in the day at 80° of Fahrenheit. During the night it descends to 60° and sometimes to 50° in places of moderate elevation. Among the objects of particular interest in this island, are the pitch or tar lake of Brea, and the Mud volcanoes. The pitch lake or plane is situated on the leeward side of the island, on a point of land extending about two miles into the sea, and opposite to the Parian Mountains on the continent. The headland is about fifty feet above the sea, and is the highest point on this side of the island. When seen from the sea it resembles a dark scoriaceous mass, but when more closely examined, it is found to consist of bituminous scoriæ, vitrified sand, and earth, all cemented together. In some places only beds of cinders are found; a strong sulphureous smell is felt on approaching this cape, and it prevails in many parts of the ground to the distance of eight or ten miles from it. The bituminous plain occupies the highest part of the point of land which shelves into the sea, and the plain is separated from the sea by a margin of wood which surrounds it. At first it resembles a lake of water, and in hot and dry weather it is actually liquid to the depth of about an inch. It is of a circular form, and about three miles in circuit, bounded on the north and west by the sea, on the south by a rocky eminence of porcelain jasper, and on the east by the usual argillaceous soil of the country. Its more common consistence and appearance are those of pit coal, but of a greyish colour. Dr Anderson regards it as the bitumen asphaltum of Linnæus. It is ductile by a genthe heat, and when mixed with grease or common pitch, it is much used for the bottoms of ships, and is regarded as a preservative against the Teredo navalis, which is so destructive to ships on all the coast of Guaiana. The Spanish government found out its value, and intended to have made use of it in their naval yards. Alli According to Dr Nugent, who has more recently examined this lake, the bituminous substance is asphaltum, which is traversed by numerous and sometimes deep crevices filled with good water, which the inhabitants use, and in which mullet, and other species of fish are often caught. Its surface had the colour of ashes, and did not adhere to the foot, from which, however, it received a partial impression, but not so as to be dangerous to walk over it. Pieces of unaltered wood were found enveloped in the pitch. The lake contains many islets covered with long grass and shrubs, which are the haunts of birds of the most exquisite plumage, as the pools are of snipe and plover. gators are said to abound here, but Dr. Nugent did not see any of them. The asphaltum is sometimes black and hard, with a dull conchoidal fracture, but in general it may be easily cut, and its interior appears oily and vesicular. At a candle it melts like sealing wax, and burns with a flame which soon ceases when removed. It acquires fluidity when mixed with oil, butter or tar, and is used as pitch. Dr. Nugent found petroleum perfectly fluid in one part of the lake, and Captain Mallet, in his topographical sketch of the island, observes that "near Cape La Brea, a little to the south-west, is a gulf or vortex, which in stormy weather gushes out, raising the water five or six feet, and covering the surface for a considerable space with petroleum or tar." He adds, that on the east coast on the Bay of Mayaro, there is another gulf similar to the former, which in March and June detonates like thunder, having some flame with a thick black smoke, which vanishes immediately. In about twenty-four hours afterwards, there is found along the shore of the bay a quantity of bitumen about three or four inches thick. Captain Mallet likewise quotes Gumilla, as stating in his description |