If ABCD is a tetrahedron of reference, any point P in space is determined by an equation of the form (a+13+ - y+5) P = aA+sB +yC +SD: a, a, y, b are, in fact, equivalent to a set of homogeneous coordinates of P. For constructions in a fixed plane three points of reference are sufficient.
It is remarkable that Mobius employs the symbols AB, ABC, Abcd In Their Ordinary Geometrical Sense As Lengths, Areas And Volumes, Except That He Distinguishes Their Sign; Thus Ab = Ba, Abc= Acb, And So On.
Any closed path or figure, such as ABCD, represents a complete cycle or series of operations, in the course of which the substance is restored to its original state with respect to temperature, intrinsic energy and other properties.
A cycle such as ABCD enclosed by parts of two isothermals, BC, AD, and two adiabatics, AB, CD, is the simplest form of cycle for theoretical purposes, since all the heat absorbed, H', is taken in during the process represented by one isothermal at the temperature o', and all the heat rejected, H", is given out during the process represented by the other at the temperature 0".
The area ABCD, representing the work, W, per cycle, is the difference (H' - H") of the quantities of heat absorbed and rejected at the temperatures 0 and 0".
Then by relations (2) the heat, H, absorbed in the isothermal change BC, is to the work, W, done in the cycle ABCD in the ratio of o to (o' - o").
Of temperature (o' - o") is small, the figure ABCD may be regarded as a parallelogram, and its area W as equal to the rectangle BE XEC. This is accurately true in the limit when (0' - 0") is infinitesimal, but in practice it is necessary to measure specific heats, &c., over finite ranges of temperature, and the error involved is generally negligible if the range does not exceed a few degrees.
Then (as the difference of two triangles) area ABCD - (h cot 2 (h cot -a)2 2(cot ¢- cot 4) 2(cot +cot 0) (ii) If 0=0, this becomes tan area (h + a tan 0) 2 - a tan 0.
At the points ABCD there is no displacement, and the line AD through these points is called the axis.
On the whole the air S within ABCD neither gains nor g D loses momentum, so that on the whole it receives as much through AB as it gives up to CD.
In this curve ABCD are nodes.
Let ABCD be drawn at such level that the areas above and below it are equal; then ABCD is the axis of the curve.
Take AB equal to one-fourth of the given line; on AB describe a square ABCD; join AC; in AC produced find, by a known process, a point C 1 such that, when C 1 B 1 is drawn perpendicular to AB produced and C 1 D 1 perpendicular to BC produced, the rectangle BC,.
3) be the sun, ABCD the earth's orbit, and s the true position of a star.
Every star, therefore, describes an apparent orbit, which, if the line joining the sun and the star be perpendicular to the plane Abcd, will be exactly similar to that of the earth, i.e.
The same holds for the four points B, C, D, E and so on; but since a parabola is uniquely determined by the direction of its axis and by three points on the curve, the successive parabolas ABCD, BCDE, CDEF ...
Let ABCD be any quadrilateral formed of jointed links.
As a simple example, take the case of a light frame, whose bars form the slides of a rhombus ABCD with the diagonal BD, suspended from A and carrying a weight W at C; and let it be required to find the stress in BD.
Again, if G be the mass-centre of four particles a, $, 7, situate at the vertices of a tetrahedron ABCD, we find a: ~ :~: tet GBCD: tetUGCDA: tetGDAB: tetGABC, and by suitable determination of the ratios on the left hand we can make G assume any assigned position in space.
If a+$+y+~=O, G is at infinity; if a = fi =~ =~, G bisects the lines joining the middle points of opposite edges of the tetrahedron ABCD; if a: ~: 7: = M3CD: z~CDA: ~DAB: L~ABC, G is at the centre of the inscribed sphere.
2 let ABCD be the beam of a scale-beam, Z the 1.
Then (de)(abc) = (abde)c+(cade)b+(bcde)a = (abce)d - (abcd)e, (ab) (AB) = (aA) (bB) - (aB) (bA) abic = (alc) b - (bjc)a, (ablcd) = (ajc) (bjd) - (af d) (bIc).