mon electrometer is composed of a j pithball supported on a wire, and traversing a graduated arch. ELECTROPHORUS is a machine consisting of two plates, one of which is a resinous electric, and the other metallic; and when the former is once excited by a peculiar application of the latter, the machine will furnish electricity for a considerable time, from which circumstance it derives its name. ELIMINATION, in Analysis, that operation by means of which all the unknown quantities except one are exterminated out of an equation, whence the value of that one becomes determined, and hence by substitution the value of all the others. This is a subject which in all its generality involves a variety of cases, and numerous and tedious investigations,which of course cannot be treated of in this place: but we shall give one or two of the most simple cases. Let there be proposed the two following equations of the first degree: ax+by+c=0 In order to eliminate one of the unknown quantities, for example y, the 1st equation is multiplied by e, and the 2d by b, so that the coefficient of y in both may be equal, then we have eax+bey+ec = 0 bdx + bey+bf = 0 Subtracting the second of these equations from the first, we have (ea-bd) x + ec- -bf=0 bfec ea-bd whence again, a== Having thus obtained the value of x, that of y may be found either by substituting this value of a in either of the original equations, or by multiplying the first of them by d, and the second by e, and then subtracting as above, in either way we find y= cd-af ae-bd which solutions are general for all equations of two unknown qualities of the first degree, and may therefore be considered as formulæ of solution in such cases, + big -ifc And by means of those general formulæ, we have the complete solution of all equations of the first degree, containing three unknown quantities. Another method isthe following: Let u, x, y, z, &c. represent the unknown quantities, and let the number of them, as also the num ber of the equations, be n. Let a, b, c, d, &c. be the respective co-efficients in the first equa. tion. al, b', c', d', those of the second, a", b, c, d", those of the third, and so on. And let us conceive the known term in each equation to be ef fected by some unknown quantity, which may be represented by t. Form the product uxyzt, of all the unknowns, written in any order at pleasure; only when this order is once determined on in any case,the same must be observed throughout, unknown for its co-efficient in the Change now successively each first equation, and observe to change the sign of each even term, and this result is called the first line. Change in this first line each unknown for its co-efficient in the se cond equation,observing, as above, to change the sign of each even term, and this result is called the second line. Change again in this second line, each unknown for its co-efficient in Each unknown quantity will be expressed by a fraction, of which the numerator is the co-efficient of the same unknown letter in the last or nth line, and the general denominator will be the co-efficient corresponding with the unknown quantity t, which was introduced in the beginning of the operation. This will be understood from the following example: Let a x + by + c = 0 ax + by + c = 0 Introducing t these equations become, ax + by + c t = o a x + by + d t = o and form the product ayt. Now change x into a, y into b, t into c, and change the sign of those terms which stand in an even place; thus we have ELLIPSE, one of the conic sections, formed by the intersection of a plane and cone when the plane makes a less angle with the base than that formed by the base and the side of the cone. The word is derived from ixλrns, defective, and is thus denominated, because the square of the ordinate in this figure is always less than the rectangle of the parameters and abscisses. There are three ways in which we may define an ellipse; viz. 1. As being produced by the intersection of a plane and cone, as we find it treated of by Apollonius and all the ancients. 2. According to its description in plano, as it is treated of by several of the moderns. And 3. As being generated by the motion of a variable line or ordinate, along another line or directrix. Properties of the Ellipse.-1. The squares of the ordinates of the axis, are to each other as the rectangle of the abscisses. Or if one of the ordinates be taken at the centre. As the square of the transverse axis Is to the square of the conjugate, So is the rectangle of the abscisses To the square of their ordinate. From which property is readily drawn the equation of the ellipse; for, make the transverse =t, the conjugate=c, the abscissx, and the ordinate=y, then c2 x (t−x): y3 whence y2=(tx-x2) or y = √(tx—x2) 2. The sum of the two lines drawn from the two foci to any point in the curve is always equal to the transverse axis. From this property is derived the common method of describing the curve mechanically by points, or by a thread, thus: In the transverse axis take any point whatever. Then will the segments of the transverse describe, from the foci as centers, two arcs intersecting each other in a point in the curve. In like manner determine other points. Then with a steady hand a curve-line may be drawn through all the points of intersection, which will be the ellipse required. Or, otherwise, take a thread of the length of the transverse axis, and fix its two ends in the foci. Then carrying a pen or pencil round by the thread, keeping it always stretched, and its point will trace out the ellipse, as is evident from the property above stated. 3. If a tangent be drawn to any point in an ellipse, and two lines drawn from the two foci to the point of contact, these two lines capofthe cylinder moves in a small the machine from acting powerful- amalgam on a piece of leather, seanductor, the other of ane and mereury. If a tie of required. ted the different ELECTRICITY, the name of an tes, as ally damp mark on this subject. Some other ments, adheretion. ame, the con ELECTROMETER, an ating milars;' ment contrived tor measuring yadually diss- quantity and determining the ad prevent lity of electricity. The" 3. Ls it is he monerated › line or le or di -1. The of the the rec nates be ansverse onjugate, the abordinate. is readily e ellipse; et, the 58=x, and -x): y2 ) or x2) e two lines foci to any lways equal is derived the escribing the by points, or axis take any Then will the transverse deoci as centers, ag each other in ve. In like maner points. Then and a curve-line through all the ection, which will equired. e, take a thread of he transverse axis, wo ends in the foci. ng a pen or pencil ne thread, keening it etched, and i out the ellip m the pro a tangent be in an ellipse, wn from the tw int of contact, th will form equal angles with the tangent. It is this property that gives the name foci to these two points, for as opticians find that the angle of incidence is equal to the angle of reflection; it follows from the above property, that rays of light issuing from the one focus, and meeting the curve in every point, will be reflected back into the other focus, and hence these points are denominated foci, or burning points. duct again by 7854, which will be the area required. Area of an Elliptic Segment.— Find the area of the corresponding circular segment, described on the same axis to which the cutting line or base of the segment is per pendicular. Then as this axis is to the other axis, so is the circular segment to that of the ellipse. ELLIPTIC Arc, is any part of the periphery of an ellipse, the length of which is found as follows: {1 + c2 + 4. If there be any number of ellipses described on the same verse, and c the semi-conjugate Let t represent the semi-transtransverse axis, and any ordinate axis of any ellipse, z the distance be drawn so as to meet all those of the ordinate from the centre, ellipses, the tangents to the several then the arc bounded by the ordiellipses at those points will all ter-nate and parallel semi-axis will minate in one common point in the be, prolongment of the transverse axis. 1. Ellip. arc And as this is necessarily true when the ellipse becomes a circle, we have hence an easy method of drawing a tangent to any point in an ellipse, namely, describe a semi-circle upon the transverse, produce the ordinate till it meet this circle, from the point of contact draw a radius, and a perpendicular to it at the point of contact, 2. Arc= will cut the transverse produced in the subtangent. 8t4 c2-4t2 c++ c5 40 t3 112/12 ·z6+ &c. } make =9; then t2 3. Arc= 5. The ordinate in the circle is to the ordinate of the ellipse, as the transverse axis of the ellipse is to its conjugate axis. And if a circle were described on the conjugate axis, and an ordinate drawn 4. Arc as before, then the ordinate of the circle would be to the corresponding ordinate in the ellipse, as the conjugate axis of the ellipse is to its transverse. And hence it follows, that the area of any ellipse is a mean proportional between the area of the circles described on its two axes. generated by the revolution of any ELLIPTIC Spindle, is the solid 6. Every parallelogram circum-chord; the solidity of which may segment of an ellipse about its scribed about an ellipse, at the ex- be found by the following formu tremities of any pair of its conju-læ: Put the perpendicular axe of gate diameters, is equal to the rec- the ellipse: =a, the parallel axe tangle of its two axes. |