then cos. k=0, ..=x0; but when x=0, we evidently P and by changing into, (as in (2),) x into y, we have y = z cos. 2 or y=z sin. 6, (7); by adding the squares of (6) and (7), we have (since cos.28+sin.28=1,) x2+y3=z2, (8); hence it is evident that the resultant is represented in direction and quantity by the diagonal of the rectangle whose adjacents sides denote the components x and y. Suppose now, that the directions of x and y include any angle a: let z denote their resultant, the angle which its direction makes with that of x; then by resolving y in the direction of x, and adding x, we have x+y cos. a= the components resolved in the direction of x, but z cos. = the resultant resolved in the same direction, .'. x+y cos. a=z cos. ; and by resolving the components, and the resultant in a direction perpendicular to that of x, we have y sin. a= z sin.; by adding the squares of these equations we have x2+ 2xy cos. a+y1=z: hence the resultant is represented in direction and quantity by the diagonal of the parallelogram whose adjacent sides denote the components x and y. Again, let three forces x, y, z be applied to M, in such a manner that the direction of each is at right angles to the directions of the other two let r denote their resultant, whose direction makes the angles a, b, c, with the directions of x, y, z severally; then we shall have x=r cos. a, y=r cos. b, z=r cos. c; whose squares, when added, give (since cos.a+cos. 'b+cos. 2c=1,) x2+y2+z2=r2; hence the resultant is represented in direction and quantity by the diagonal of the rectangular parallelopiped, whose adjacent sides denote the components x, y, z. Let us now suppose that any number of forces acting in any directions, are applied to M, to determine the quantity and direction of their resultant. Draw any three rectangular axes denoted by x, y, z, through M; and let r, r', &c. denote the forces, a, b, c, the angles which the direction of r makes with the directions of x, y, z respectively, and let a', b', c' denote the corresponding angles for r', and so on; let R denote the resultant, A, B, C severally, the angles which its direction makes with the directions of x, y, z. Then by resolvVOL. XXVI.-No. 2. 40 ing the components and the resultant in the directions of x, y, z respectively, we shall have r cos. a + r'cos. a' +, &c. = R cos. A' r cos. b + r' cos. b'+, &c. = R cos. B, r cos. c+r' cos. c'+, &c. = R cos. C, (9); whose squares, when added, give (r cos. a+r' cos. a, +, &c.) + (r cos. b + r' cos. b'+, &c.)2 + (r cos. c + r' cos. c'+, &c.) =R', (10); hence R being found, the angles A, B, C are easily found by (9); and it may be remarked that the known rules for the algebraic signs of the cosines must be observed. If M is free, and the conditions of equilibrium are required, then we must have R=0, .. the first members of (9) being each put =0, will be the conditions required; if the first members of (9) are each identically =0, then R=0, and M will not be affected by the forces. If M is not free, but is pressed by the forces against any line or surface, it will be necessary that R should be at right angles to the line or surface, so that it may be destroyed by the reaction. We will now consider the subject after the manner of La Place, at pp. 4, 5, Vol. I. of the Mecanique Celeste. Let the two forces x and y, whose directions form a right angle, be applied as before to M, also let z denote the resultant, ◊ the angle which its direction makes with that of x, then Р - the angle which 2 its direction makes with that of y. Hence we shall have (1) and (2) P in the same manner as before; if y=0, 8=0, .'. ¢ 2 P Φ =0; if y is indefinitely small relative to z, will be indefinite 2 ly small, and may be denoted by de; hence representing the value of y in this case by dy, we shall have by (2) dy P 2 - kd, by neglecting quantities of the orders de2, de3, &c. as is evident by the method of indeterminate coefficients, or Taylor's are each resolved into two forces x', x', and y', y'; x', y' acting in the direction of z, and x", y" perpendicular to it; then we evidently must have +y'=z, (2′), x"=y′′, (3′). P Since x forms the angle with x', and - with ", and that y 2 P makes the angles-6, 6, with y' and y', we have by (1) and (2) 2 = P 0, 2 ·=z or x2+y2=z2, (4′); which shows that the resultant is represented in quantity by the diagonal of the rectangle whose adjacent sides denote the components. We will now find the direction of the resultant. Let x and y become r+dr, y+dy, and let z' be their resultant which makes the indefinitely small angle d with z. Let x, y denote the values of x+dr, y+dy, resolved at right angles to z, and suppose that isy, then it is evident that 2' is between z and ; by (1) and (2) It is evident that x-y=z' resolved at right angles to z,.. by x-y' (2) TM'—"'—。 (P—da) or by (1′), and substituting the value of the orders dz2, dz3, &c. we may use z for z', .:. by substituting the value of z2 from (4′), we shall have integral gives = tan. (k+c), or by (4') x=z cos. (kê+c); where x=z cos. ; which shows that the direction of the resultant is the same as that of the diagonal of the rectangle whose adjacent sides denote the components. We will now find the direction of the resultant in another manner. Suppose then, that x is changed to x+x', but that y is the same as before; let z' be the resultant of these forces, the angle which its direction makes with that of x; then it is manifest that is 0, and that the angle formed by the directions of z' and z. Put s—s'=v, (a), then by resolving 2' in the direction of z by (1), we shall have 'p(v) the resultant resolved in the direction of z; but '()=x' resolved in the same direction, and by adding z, we have the components reduced to the same direction; hence z'q(v)=z+ x'q(4), (b). x y = n, Put=m, x+x' = =m', n', (c), then by (4′), m2+n2=1, m'2+n'2 = 1, (d); by (c),x=mz, x+x'=m'z', y=nz=n'z' ; hence 'nm' — mn'' n m=q(^), hence z+x′q(0)=(n'(1— m2)+ nmm') n' ≈, but by (1),~= =n2,) = (n'n + mm')”—=—= (mm′+nn′)z′; hence by (6), we shall have n =mm'+nn', (e). It is evident that 8 and are independent of each other, and that m, n are functions of without ', and that m', n' are functions of only; hence by putting dp(v) dv ing the partial differentials of (e) relative to and ', we shall have '(r) × ds we have by reduction, dm' -n= n', (g). Since the first mem dł ber of (g) is a function of 8 only, and the second of ', it is evident that each member k denote the constant, and since by (d) n=√1 – m2, we shall have by multiplying by dê, = =kd3, whose integral gives m === cos. (k+c), c = the correction; hence (as before,) we shall have x=z cos. 8. ART. VI.—Notices on Thermo-Electricity and Electro-Magnetism, in a letter to the editor, from Prof. JOHN P. EMMET, dated, University of Virginia, May 8th, 1834. SIR, I have been induced to offer the following brief observations, and to request their publication, in order that I may have it in my power to make a timely correction of a statement made in my former communication "upon caloric, as a cause of voltaic currents." This opportunity will likewise enable me to announce a very interesting law of thermo-magnetism, which I altogether omitted to notice in that communication. I shall also, in conclusion, be able to offer to the medical portion of your readers, a notice respecting my form of the coil-magnet, which I think promises fair to become a substitute for the leyden jar and common electrical machine, in all such cases as require the sanative agency of the latter instrument. The results of my communication, above referred to, and which may be found in Vol. xxv, No. 2, of this Journal, were obtained by means of a galvanometer, delicate it is true, but far from being perfect; and which did not indicate currents of low intensity. Shortly after the manuscript was forwarded, I constructed a multiplier of excessive delicacy, and which, in all its details, exactly resembles and may be understood from the instrument which I see described in the last number of your Journal. I was not a little struck by the coincidence, and pleased to see the notice.* The object which I had in view, was to give the maximum effect with the smallest current, and, as there is always a great loss of power when the coil of the multiplier is extensive, I limited the wire to a few turns over and under a couple of connected needles, rendered perfectly astatic. With a view also, of applying the current as advantageously as possible to the needles, the coil, instead of being wound upon the same spot, as usual, was spread out, laterally, so as to form a kind of box within which the lower needle traversed. By this arrangement, the tangential magnetic force of the voltaic current was applied close to the extremities of the needles in every portion of their revolution. The needles were suspended by raw silk, and the whole instrument included within the glass frame of a Coulomb's balance of torsion. The delicacy of this multiplier is so great that a declination of 90° may be obtained * By Dr. Locke of Cincinnati; see Vol. xxvi, p. 103. |