"Tetrahedral co-ordinates" are a system of quadriplanar co-ordinates, the fundamental planes being the faces of a tetrahedron, and the co-ordinates the perpendicular distances of the point from the faces, a positive sign being given if the point be between the face and the opposite vertex, and a negative sign if not.
If (u, v, w, t) be the co-ordinates of any point, then the relation u+v-Fw-fit=R, where R is a constant, invariably holds.
In the potential curves of the diagram the ordinates represent the hourly values expressed - as in Tables II.
If the errors of the rectangular co-ordinates of these lines are known, the problem of determining the co-ordinates of any star-image on the plate becomes reduced to the comparatively simple one of interpolating the co-ordinates of the star relative to the sides of the 5 mm.
That if any distortion of the film takes place during development the same distortion will be communicated both to the star-images and to the reseau-lines, and consequently its effect will be eliminated from the resulting star co-ordinates, except in so far as the distortion within the 5 mm.
This form of micrometer is of course capable of giving results of high precision, but the drawback is that the process involves a minimum of six pointings and the entering of six screw-head readings in order to measure the two co-ordinates of the star.
The absolute freedom of the derived co-ordinates from the effects of wear of the screws in the mean of measures made in reversed positions of the plate.
At the head of the financial organization of France, and exercising a general jurisdiction, is the minister of finance, who co-ordinates in one general budget the separate budgets prepared by his colleagues and assigns to each ministerial department the sums necessary for its expenses.
The ordinates of the curve give the strain in cwts., and the abscissae the distance in miles measured from the Canso end; as the strain is proportional to the depth, 18 cwts.
Kepler's Problem, namely, that of finding the co-ordinates of a planet at a given time, which is equivalent - given the mean anomaly - to that of determining the true anomaly, was solved approximately by Kepler, and more completely by Wallis, Newton and others.
Newlands in England, that if they are arranged in the order of their atomic weights they fall into groups in which similar chemical and physical properties are repeated at periodic intervals; and in particular he showed that if the atomic weights are plotted as ordinates and the atomic volumes as abscissae, the curve obtained presents a series of maxima and minima, the most electro-positive elements appearing at the peaks of the curve in the order of their atomic weights.
7 represents the specific volumes of mixtures of ammonium and potassium sulphates; the ordinates re presenting specific volumes, and the abscissae the per centage composition of the mixture.
9 illustrates the first case: the ordinates represent specific volumes, and the abscissae denote the composition of isomorphous mixtures of ammonium and potassium dihydrogen phosphates, which mutually take one another up to the extent of 20% to form homogeneous crystals.
By plotting the specific volumes of these mixed crystals as ordinates, it is found that they fall on two lines, the upper corresponding to the orthorhombic crystals, the lower to the monoclinic. From this we may conclude that these salts are isodimorphous: the upper line represents isomorphous crystals of stable orthorhombic magnesium sulphate and unstable orthorhombic ferrous sulphate, the lower line isomor phous crystals of stable monoclinic ferrous sulphate and unstable monoclinic magnesium sulphate.
For instance, those of a ternary form involve two classes which may be geometrically interpreted as point and line co-ordinates in a plane; those of a quaternary form involve three classes which may be geometrically interpreted as point, line and plane coordinates in space.
Restricted Substitutions We may regard the factors of a binary n ip equated to zero as denoting n straight lines through the origin, the co-ordinates being Cartesian and the axes inclined at any angle.
Or, instead of looking upon a linear substitution as replacing a pencil of lines by a projectively corresponding pencil retaining the same axes of co-ordinates, we may look upon the substitution as changing the axes of co-ordinates retaining the same pencil.
As new axes of co-ordinates we may take any other pair of lines through the origin, and for the X, Y corresponding to x, y any new constant multiples of the sines of the angles which the line makes with the new axes.
Steinmetz's formula may be tested by taking a series of hysteresis curves between different limits of B,' measuring their areas by a pianimeter, and plotting the logarithms of these divided by 47r as ordinates against logarithms of the corresponding maximum values of B as abscissae.
Curves of magnetization (which express the relation of I to H) have a close resemblance to those of induction; and, indeed, since B = H+47r1, and 47rI (except in extreme fields) greatly exceeds H in numerical value, we may generally, without serious error, put I = B /47r, and transform curves of induction into curves of magnetization by merely altering the scale to which the ordinates are referred.
The assemblage of ordinates NP is then the graph of Y.
In particular, the equality or inequality of values of two functions is more readily grasped by comparison of the lengths of the ordinates of the graphs than by inspection of the relative positions of their bounding lines.
In the first class come equations in a single unknown; here the function which is equated to zero is the Y whose values for different values of X are traced, and the solution of the equation is the determination of the points where the ordinates of the graph are zero.
May be called the co-ordinates of q.
In the applications of the calculus the co-ordinates of a quaternion are usually assumed to be numerical; when they are complex, the quaternion is further distinguished by Hamilton as a biquaternion.
Taking co-ordinates in the plane of the screen with the centre of the wave as origin, let us represent M by, n, and P (where dS is situated) by x, y, z.
The form of (3) shows immediately that, if a and b be altered, the co-ordinates of any characteristic point in the pattern vary as a-'- and b-1.
If x and y be co-ordinates in the plane of the wave-surface, the axis of y being parallel to the lines of the grating, and the origin corresponding to the centre of the beam, we may take as an approximate equation to the wave-surface -- -} z =+Bxy 2, +ax 3 13x2 2pp p y+-yxy2-?-Sy3+..
With sufficient approximation we may regard BQ and b as rectangular co-ordinates of Q.
Fitzgerald), of exhibiting as a curve the relationship between C and S, considered as the rectangular co-ordinates (x, y) of a point.
The origin of co-ordinates 0 corresponds to v = 0; and the asymptotic points J, J', round which the curve revolves in an ever-closing spiral, correspond to v= =co .
The co-ordinates of J, J' being (- z, - z), I 2 is 2; and the phase is, period in arrear of that of the element at 0.
Let x, y, z be the co-ordinates of any particle of the medium in its natural state, and, 7 7, the displacements of the same particle at the end of time t, measured in the directions of the three axes respectively.
(b2V2 + n2) (a2 - b 2) = - z It will now be convenient to introduce the quantities a l, a 2', 7731 which express the rotations of the elements of the medium round axes parallel to those of co-ordinates, in accordance with the equations Ty - 1 = dz ' 3= - dy 2 = dx - In terms of these we obtain from (7), by differentiation and subtraction, (b 2 v 2 + n 2) 7,3 = 0 (b 2 0 2 +n 2) .r i = dZ/dy (b 2 v 2 +n 2)', , 2 = - dZ/dx The first of equations (9) gives 3 = 0 (10) For al we have ?1= 47rb2, f dy e Y tkr dx dy dz
(This solution may be verified in the same manner as Poisson's theorem, in which k=o.) We will now introduce the supposition that the force Z acts only within a small space of volume T, situated at (x, y, z), and for simplicity suppose that it is at the origin of co-ordinates that the rotations are to be estimated.
Thus if the plane is normal to Or, the resultant thrust R =f fpdxdy, (r) and the co-ordinates x, y of the C.P. are given by xR = f f xpdxdy, yR = f f ypdxdy.
Quadrator, squarer), in mathematics, a curve having ordinates which are a measure of the area (or quadrature) of another curve.
"In the beginning of my mathematical studies, when I was perusing the works of the celebrated Dr Wallis, and considering the series by the interpolation of which he exhibits the area of the circle and hyperbola (for instance, in this series of curves whose common base 0 or axis is x, and the ordinates respectively (I -xx)l, (i (I &c), I perceived that if the areas of the alternate curves, which are x, x 3x 3, x &c., could be interpolated, we should obtain the areas of the intermediate ones, the first of which (I -xx) 1 is the area of the circle.
We can then draw a continuous surface through the summits of all these ordinates, and so obtain a freezing-point surface, or liquidus; points above this surface will correspond to wholly liquid alloys.
Let us apply the above theorem to the case of a small parallelepipedon or rectangular prism having sides dx, dy, dz respectively, its centre having co-ordinates (x, y, z).
Its angular points have then co-ordinates (x t Zdx, y t Zdy, z * zdz).
Since the potential rises proportionately to the quantity in the conductor, the ends of these ordinates will lie on a straight line and define a triangle whose base line is a length equal to the total quantity Q and V height a length equal to the final potential V.
This is mathematically expressed by the statement that dE is an exact differential of a function of the co-ordinates defining the state of the body, which can be integrated between limits without reference to the relation representing the path along which the variations are taken.
Observing that F is a function of the co-ordinates expressing the state of the substance, we obtain for the variation of S with pressure at constant temperature, dS/dp (0 const) '=' 2 F/dedp =-0d 2 v/d0 2 (p const) (12) If the heat supplied to a substance which is expanding reversibly and doing external work, pdv, is equal to the external work done, the intrinsic energy, E, remains constant.
(33) The state of a substance may be defined by means of the temperature and entropy as co-ordinates, instead of employing the pressure and volume as in the indicator diagram.
If the system is supposed to obey the conservation of energy and to move solely under its own internal forces, the changes in the co-ordinates and momenta can be found from the Hamiltonian equations aE aE qr = 49 - 1 57., gr where q r denotes dg r ldt, &c., and E is the total energy expressed as a function of pi, qi,.
Since the values of the co-ordinates and momenta at any instant during the motion may be treated as " initial " values, it is clear that the " extension " of the range must remain constant throughout the whole motion.
This result at once disposes of the possibility of all the systems acquiring any common characteristic in the course of their motion through a tendency for their co-ordinates or momenta to concentrate about any particular set, or series of sets, of values.
Let us imagine that the systems had the initial values of their co-ordinates and momenta so arranged that the number of systems for which the co-ordinates and momenta were within a given range was proportional simply to the extension of the range.
Ow are any momenta or functions of the co-ordinates and momenta or co-ordinates alone which are subject only to the condition that they do not enter into the coefficients a 1, a 2, &c.