Assume y = mx, m being the proposed multiple; then r m' and substituting this value for x in the proposed equation, we have yn-l yn-2 mn 2 yn mn + a +b. Assume y=,or x=-, and sub y +&c.=0stitute this value of x in the proposed equation; and we have Or multiplying by mn yn+mayn! + m2 byn_2+ &c.=0; which is an equation, whose roots are equal to the given multiple m of the roots of the proposed equation. Hence we see, that to multiply the roots of an equation by any quantity m, it is only necessary to multiply the several co-efficients; beginning at the first by the terms of the geometrical progression 1, m, m2, m3, m2, &c. observing only that if any term in the original equation be wanting, it must be introduced in its proper place, having zero for its co-efficient; thus Let it be proposed to transform the equation x2+5x3+3x+7=0 to another equation, whose roots are the doubles of these roots. Here, supplying the third term, which is wanting, proposed equat. the equation required. 1 w = 0, That is, we must invert the order of the co-efficients, and divide by the absolute quantity of the pro posed equation; observing here, as in the preceding case, to supply any terms of the equation that may be wanting, by prefixing a cipher to them for a co-efficient. Transform the equation x4+3x3+2x+4=0 to another whose roots shall be the reciprocal of these. Here, supplying the deficient term, and inverting the order of the co-efficients, we have 4+2x+0x2 + 3 x3 + x1 ; x2 + 5x3 +0x2 + 3x+7=0then dividing by 4, and introducing geom. series which is the equation required, the unknown quantity being changed from a to y merely for the sake of distinction. y, we have 2 yt += y3 + y2+ y + 1 quired. 1 =0 3 y + 4 4 =0, as re 4. To change the signs of the roots of an equation. Let an axn −1 + bxn —2+, mial theorem, the second term incxn− s + &c. = 0 be an equation, volves the quantity d in only the the signs of whose roots are to be first degree, the third term in the changed. second degree, the fourth term in the third degree, &c. it follows, that to exterminate the second term, the equation will be a simple one, and may therefore always be effected in rational numbers; but to exterminate the third term, we must solve a quadratic; and to exterminate the fourth, a cubic, and so on. Assume yx, or xy, and substitute this for x in the proposed equation; so shall the roots of the proposed equation be changed as required. Here it is obvious, that only the signs of the odd powers of a will be changed, because the even powers of a negative quantity have the same signs as a positive one; and since, also, no change takes place in the value of an equation, when all the signs of it are changed; we may reduce what has been said above to the following general rule. To change the signs of the roots of an equation, supply such terms as are wanting, and then change the signs of all the even terms, viz. the 2d, 4th, 6th, &c. from the left hand; and the equation will be transformed as required. Let it be required to change the signs of the roots of the equation, 253x4 +7 x2 + 6x + 4 = 0 First, supplying the third term, which is wanting, the equation be conies x53x4+0x3 +7x2+6x+4=0 Changing the alternate terms, and writing y for x, we have y5+ 3y++0y3-7y2+6y-4=0, or y3y4-7y2+ by — 4 =0 the equation sought. 5. To transform an equation to another that shall want the second, or any other term of the equation. Let an+axn−1 + bxn_2 + &c. = 0, be any equation which is to be transformed into another that shall want any required term. Assume ay+d, and substitute this value for a, and we obtain n (n-1) an = yn+ndyn —1+ yn-2+ &c. 2 Thus in the above equation, to exterminate the second term, we have n d2+a (n−1) d + b = 0 -26 d= a2 d2 26 and so on for any other term. TRANSIT, in Astronomy, signi fies the passage of any planet over a fixed star, or the sun; and of the moon in particular, covering or moving over any planet. The transits of Mercury and Venus over the sun's disc are very interesting phenomena, not merely by reason of their rare and singular appearance, but because of their use in determining the parallax of the sun, and thence the real dimensions of the earth's orbit. The transits of Mercury always occur either in May or in Novem ber; but they happen much the most frequently in the latter month. The difference depends upon the position of the elliptic projection of Mercury's orbit upon the plane of the ecliptic. This ellipse is now so placed that it presents to us its perihelion during the winter, and its aphelion during the summer; and, as it is very eccentric, Mercury is much nearer the sun in the month of November than in the month of May. Now, if we consider the luminous cone formed by .the visual rays drawn from the earth to the sun, this cone is conAs from the nature of the bino-tracted in the vicinity of the earth axn-1= ayn-1+a(n-1)d yn 2+ &c. bxn — 2 = while it is enlarged near the sun, the disc of which serves for its base: Mercury ought therefore to cut it more readily when it is near the sun than when it is remote from it; and, consequently, the transits of Mercury ought to occur most frequently in the winter part of the year. Transits of Venus over the sun's disc happen much less frequently than those of Mercury, because Venus is more distant from the sun. The following are all that have occurred or will occur between 1631 and 2110. 1631... Dec. 6. 1874... Dec. 8. 1639... Dec. 4. 1882 ... Dec. 6. 1761... June 5. 2004... June 7. 1769... June 3.2109... Dec. 10. Now the chief use of these conjunctions is accurately to deter. mine the sun's distance from the earth, or his parallax, which astronomers have in vain attempted to find by various other methods; for the minuteness of the angles required easily eludes the nicest instruments. But in observing the ingress of Venus into the sun, and her egress from the same, the space of time between the moments of the internal contacts, observed to a second of time, that is, to of a second, or 4 of an arch, may be obtained by the assistance of a moderate telescope and a pendalum clock, that is consistent with itself exactly for 6 or 8 hours. Now, from two such observations rightly made in proper places, the distance of the sun within a 500th part may be certainly concluded, &c. TRANSIT Instrument, in Astronomy, is a telescope fixed at right angles to a horizontal axis; this axis being so supported that the line of collimation may move in the plane of the meridian. The axis, to the middle of which the telescope is fixed, should gradually taper toward its ends, and terminate in cylinders well turned and smoothed; and a proper weight or balance is put on the tube, so that it may stand at any elevation when the axis rests on the supporters. Two upright posts of wood or stone, firmly fixed at a proper distance, are to sustain the sup porters to this instrument; these supporters are two thick brass plates, having well smoothed angular notches in their upper ends, to receive the cylindrical arms of the axis; each of the notched plates is contrived to be moveable by a screw, which slides them upon the surfaces of two other plates immoveably fixed to the two upright posts: one plate mov ing in a vertical direction, and the other horizontally, they adjust the telescope to the planes of the horizon and meridian; to the plane of the horizon, by a spirit level hung in a position parallel to the axis, and to the plane of the meridian, in the following manner. Observe the times, by the clock, when a circumpolar star, seen through this instrument, transits both above and below the pole; then, if the times of describing the eastern and western parts of its circuit be equal, the telescope is then in the plane of the meridian; otherwise, the notched plates must be gently moved till the time of the star's revolution is bisected by both the upper and lower transits, taking care at the same time that the axis keeps its horizontal position. When the telescope is thus adjusted, a mark must be set up, or made, at a considerable distance (the greater the better) in the horizontal direction of the intersection of the cross wires, and in a place where it can be illuminated in the night-time by a lautern near it, which mark, being on a fixed object, will serve at all times afterwards to examine the position of the telescope, by first adjusting the transverse axis by the level. To adjust a clock by the sun's transit over the meridian, note the times by the clock, when the preceding and following edges of the sun's limb touch the cross wires: the difference between the middle time and 12 hours, shows how much the mean, or clock time, is faster or slower than the apparent or solar time, for that day; to which the equation of time being applied, it will show the time of mean noon for that day, by which the clock may be adjusted. TRANSMISSION, in Optics, isum be bisected, and the adjacent used to denote the passage of the points of bisection be joined, it will rays of light through transparent form a parallelogram, the sides of bodies. which are parallel to the corresTRANSPARENCY, or TRANSLU-ponding diagonals of the trapeziCENCY, in Physics, a quality in um; and the area of the former is certain bodies, by which they give equal to half the area of the latpassage to the rays of light. ter. TRANSPOSITION, in Algebra, is the bringing any term of an equation over to the other side of it, in which case the sign of the quantity thus transposed must be changed, viz. from plus to minus, or from minus to plus. For example, if a+x=b, then transposing a and changing its sign, we have xb-a; and if, on the other hand, we had a =b, then by transposition we have x = a + b. = 5. The sum of the four lines formed by the intersection of the two diagonals of a trapezium, is less than the sum of any other four lines drawn from a point within the trapezium to its four angles. 6. The sum of the opposite angles of a trapezium, which can be inscribed in a circle, is equal to two right angles, and if one side of such a figure be produced, the external angle is equal to the inThis operation is performed internal and opposite angle. order to bring all the known terms to one side of the equation, and all those that are unknown on the other side, whereby the value of these last became determined. TRANSVERSE Axis, or Diameter, in the Conic Sections, is the first or principal diameter passing through both the foci of the curve. In the ellipse the transverse is the longest of all the diameters, in the hyperbola it is the shortest, and in the parabola it is like all the other diameters, infinite in length. TRAPEZIUM, in Geometry, a plane figure contained under four right lines, of which neither of the opposite sides are parallel. If a quadrilateral have both its pairs of opposite sides parallel, it is called a parallelogram; if only one of its pairs of opposite sides are parallel, it is called a trapezoid; and when neither pair are parallel, it is called a trapezium. The trapezium has several remarkable properties, of which the following are the most important: 1. Any three sides of a trapezium are greater than the fourth side. 2. The sum of the four inward angles of a trapezium is equal to four right angles. 3. The diagonals of a trapezium divide it into four proportional triangles. 7. Also, in this case, the rectangle of its two diagonals is equal to the sum of the rectangles of its opposite sides. TRAPEZOID, a quadrilateral figure, having two of its opposite sides parallel; the area of which is equal to half a parallelogram, whose base is equal to the sum of the two parallel sides, and its altitude equal to the perpendicular distance between them. TRAVERSE, in Navigation, is the variation or alteration of a ship's course, occasioned by the shifting of the winds, currents, &c. or a traverse may be defined to be a compound course, consisting of several courses and distances. TRIANGLE, in Geometry, a figure bounded by three sides, and consequently containing three an gles, whence it derives its name. Triangles are of different kinds, as plane or rectilinear, spherical, and curvilinear. A Plane or Rectilinear TRIANGLE, is that which is bounded by right lines. A Spherical TRIANGLE, is that which is bounded by three arcs of great circles of the sphere. A Curvilinear TRIANGLE, is that which is bounded by any three curve lines. And each of these three distinct classes of triangles receive other 4. If the four sides of a trapezi-distinguishing denominations, ac cording to the relation of their sides and angles. These Of Plane TRIANGLES. are distinguished as follows: 1. An Equilateral TRIANGLE, is that which has its three sides equal, and consequently its angles, whence it is sometimes also called an equiangular triangle. 2. An Isosceles TRIANGLE, is that which has only two of its sides equal. The angles at the base of an isosceles triangle are equal to each other. 3. A Scalene or Oblique TRIANGLE, is that which has no two of its sides equal. This is termed a scalene triangle with reference to its sides; and oblique with regard to its angles. Triangles also receive other denominations with reference to their angles. 4. A Right-angled TRIANGLE, is that which has a right angle. Here the side opposite the right angle is called the hypothenuse, and the other two sides the base ́and perpendicular, or sometimes the legs. 5. An Oblique-angled TRIANGLE, is that which has not a right an gle; and is either acute or obtuse. 6. An Acute angled TRIANGLE, is that which has three acute angles. 7. An Obtuse-angled TRIANGLE, is that which has one obtuse angle. 8. Similar TRIANGLES, are such as have the angles of the one equal to the angles of the other, each to each. Some of the principal properties of plane triangles are as follow: 1. The greater side of a triangle is opposite the greater angle, and the less side opposite the less angle. 2. Any side of a triangle is less than the sum, but greater than the difference of the other two sides. 3. The sum of the three angles of a triangle is equal to two right angles; and the external angle formed by producing one of its sides is equal to the sum of the two internal and opposite angles. 4. In a right-angled triangle the square of the hypothenuse is equal to the sum of the squares of the other two sides. 5. Or more generally, if a, b, and c, represent the three sides of a triangle, and C the angle contained by a and b; then if far + b2 C= 90° a2 + ab + b2= c2, the C=120° a2- ab+b2 = c2, the C 60°. 6. In any triangle the rectangle of any two sides is equal to the rectangle of the perpendicular on the third side, and the diameter of the circumscribing circle. = 7. The square of a line bisecting any angle of a triangle, together with the rectangle of the segments of the opposite sides, is equal to the rectangle of the two sides, including the bisected angle. 8. Triangles, when their bases are equal, are to each other as their altitudes; and when their altitudes are equal they are to each other as their bases; and when neither are equal they are to each other in the compound ratio of their bases and altitudes. 9. Similar triangles have their like sides proportional, and their areas are to each other as the squares of the like sides. 10. The line which bisects any angle of a triangle divides the op posite side into two segments, which are in proportion to the adjacent sides. 11. The line which is drawn parallel to one side of a triangle divides the other two sides proportionally. 12. The three lines which bisect the three angles of a triangle intersect in one common point, as also do the three lines which bisect the three sides perpendicularly; the former point being the centre of the inscribed circle, and the latter the centre of the circumscribing one. 13. If perpendiculars be let fall from the three angles of a triangle to the opposite sides, they will intersect in one common point; as will also the three lines drawn from the three angles to the middle of the opposite side, which latter point is the centre of gravi ty of the triangle. 14. In a right-angled triangle, the perpendicular let fall from the right angle to the opposite side or base, is a mean proportional be. |