In the potential curves of the diagram the ordinates represent the hourly values expressed - as in Tables II.
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.
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.
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.
When the series is theoretically continuous, the theoretical graph will be a continuous figure of which the lines actually drawn are ordinates.
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.
The second class of cases comprises equations involving two unknowns; here we have to deal with two graphs, and the solution of the equation is the determination of their common ordinates.
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.
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.
are called the ordinates of the points A, B, C,.
the area is half the sum of the products obtained by, multiplying each ordinate by the distance between the two adjacent ordinates.
Then, if we take ordinates Kb, Lg, Mc, Nd, Pf, equal to B'B, GG', C'C, D'D, FF', the figure abgcdfe will be the equivalent trapezoid, and any ordinate drawn from the base to the a LM N P e X top of this trapezoid will be equal to the portion of this ordinate (produced) which falls within the original figure.
Either or both of the bounding ordinates may be zero; the top, in that case, meets the base at that extremity.
The sides of the trapezette are the " bounding ordinates "; their abscissae being xo and xo+H, where H is the breadth of the trapezette.
from a line drawn through 0 parallel to the ordinates) is equal to the mean distance (§ 32) of the trapezette from this axis; moments with regard to the central ordinate are therefore sometimes described in statistics as " moments about the mean."
The data are then either the bounding ordinates uo, ui, ...
the ordinates from the centres of their bases.
In some cases the data for a trapezette or a briquette are not only certain ordinates within or on the boundary of the figure, but also others forming the continuation of the series outside the figure.
(b) - 49(a), where 4)(x) is any function of x, by [c P(x)]; the area of the trapezette whose bounding ordinates are uo and u m may then be denoted by [Ax.
If this is done for every possible value of x, there will be a series of ordinates tracing out a trapezette with base along OX.
The volume comprised between the cross-section whose area is S and a consecutive cross-section at distance 0 from it is ultimately SO, when B is indefinitely small; and the area between the corresponding ordinates of the trapezette is (S/1).
The top is then a parabola whose axis is at right angles to the base; and the area can therefore (§ 34) be expressed in terms of the two bounding ordinates and the midordinate.
If instead of uo, u 1, and u 2, we have four ordinates uo, ul, u2, and u 3, so that m = 3, it can be shown that area = 8h(uo + 3/41 + 3u2 - Fu3).
Generally, if the area of a trapezette for which u is an algebraical function of x of degree 2n is given correctly by an expression which is a linear function of values of u representing ordinates placed symmetrically about the mid-ordinate of the trapezette (with or without this mid-ordinate), the same expression will give the area of a trapezette for which u is an algebraical function of x of degree 2n + 1.
When u is of degree 4 or 5 in x, we require at least five ordinates.
There are two classes of cases, according as m is even or odd; it will be convenient to consider them first for those cases in which the data are the bounding ordinates of the strips.
(i) If m is even, ug m will be onejof the given ordinates, and we can express h 2 u, m, 4 u" m, ...
(ii) If m is odd, the given ordinates are uo, ...
Hence, for the case of a parabola, we can express the area in terms of the bounding ordinates of two strips, but, if we use mid-ordinates, we require three strips; so that, in each case, three ordinates are required.
The question then arises whether, by removing the limitation as to the position of the ordinates, we can reduce their number.
It follows that, by taking two ordinates in a certain position with regard to the bounding ordinates, the area of any parabolic trapezette whose top passes through their extremities can be expressed in terms of these ordinates and of the breadth of the trapezette.
This is a particular case of a general theorem, due to Gauss, that, if u is an algebraical function of x of degree 2p or 2p + I, the area can be expressed in terms of p -}- i ordinates taken in suitable positions.
To extend these methods to a briquette, where the ordinate u is an algebraical function of x and y, the axes of x and of y being parallel to the sides of the base, we consider that the area of a section at distance x from the plane x = o is expressed in terms of the ordinates in which it intersects the series of planes, parallel to y=o, through the given ordinates of the briquette (§ 44); and that the area of the section is then represented by the ordinate of a trapezette.
Suppose, for instance, that u is of degree not exceeding 3 in x, and of degree not exceeding 3 in y, that it contains terms in x3y3, x 3 y 2, x2y3, &c.; and suppose that the edges parallel to which x and y are measured are of lengths 2h and 3k, the briquette being divided into six elements by the plane x=xo+h and the planes y = yo+k, y = yo+2k, and that the 12 ordinates forming the edges of these six elements are given.
In the case of a trapezette, for instance, the data are the magnitudes of certain ordinates; the problem of interpolation is to determine the values of intermediate ordinates, while that of mensuration is to determine the area of the figure of which these are the ordinates.
It is only necessary to consider the trapezette and the briquette, since the cases which occur in practice can be reduced to one or other of these forms. In each case the data are the values of certain equidistant ordinates, as described in §§ 43-45.
The first, which is the best known but is of limited application, consists in replacing each successive portion of the figure by another figure whose ordinate is an algebraical function of x or of x and y, and expressing the area or volume of this latter figure (exactly or approximately) in terms of the given ordinates.
The second consists in taking a comparatively simple expression obtained in this way, and introducing corrections which involve the values of ordinates at or near the boundaries of the figure.
um, the figure formed by joining the tops of these ordinates is a trapezoid whose area is h(Iuo -}- ui+u2 -I-
u m _ 4, we can form a series of trapezia by drawing the tangents at the extremities of these ordinates; the sum of the areas of these trapezia will be h(u 4 .+u 2 +...
If we write CI for the chordal area obtained by taking ordinates at intervals Zh, then T i =2CI-C I.
- The extension of this method consists in dividing the trapezette into minor trapezettes, each consisting of two or more strips, and replacing each of these minor trapezettes by a new figure, whose ordinate v is an algebraical function of x; this function being c h osen so that the new figure shall coincide with the original figure so far as the given ordinates are concerned.
This means that, if the minor trapezette consists of k strips, v will be of degree k or k - I in x, according as the data are the bounding ordinates or the mid-ordinates.
(i) Suppose that the bounding ordinates are given, and that m is a multiple of 2.
Suppose, for instance, that m=6, and that we consider the trapezette as a whole; the data being the bounding ordinates.
2um) Now, if p is any factor of m, there is a series of equidistant ordinates uo, up, 142p, um - p, um; and the chordal area as determined by these ordinates is ph (2uo + up + u2p +.
The preceding methods, though apparently simple, are open to various objections in practice, such as the following: (i) The assignment of different coefficients of different ordinates, and even the selection of ordinates for the purpose of finding C3, &c. (§ 70), is troublesome.
(ii) This assignment of different coefficients means that different weights are given to different ordinates; and the relative weights may not agree with the relative accuracies of measurement.
(iii) Different formulae have to be adopted for different values of m; the method is therefore unsuitable for the construction of a table giving successive values of the area up to successive ordinates.
76 If we know not only the ordinates uo, ul,.
, but also a sufficient number of the ordinates obtained by continuing the series outside the trapezette, at both extremities, we can use central-difference formulae, which are by far the most convenient.
- The above methods can be applied, as in §§ 59 and 60, to finding the moments of a trapezette, when the data are a series of ordinates.
are given, we have only to find the area of a trapezette whose ordinates are xo P uo, x 1 'u 1, x2 P u2,
There is, however, a certain set of cases, occurring in statistics, in which the data are not a series of ordinates, but the areas A I, A I,.
A m _ I of the strips bounded by the consecutive ordinates uo, 72 1, ...
as if they were ordinates placed at the points for which i x=x l, ...
In cases other than those described in § 82, the pth moment with regard to the axis of u is given by Pp = XPrA where A is the total area of the original trapezette, and S 2 _ 1 is the area of a trapezette whose ordinates at successive distances h, beginning and ending with the bounding ordinates, are o, x1P -1A, x2 P-1 (AI+AI),.
In order to find the corrections in respect of the terms shown in square brackets in the formulae of § 75, certain ordinates other than those used for C 1 or T I are sometimes found specially.
i g h oo hsuvii where T is the area of a trapezette whose ordinates at successive distances h are o, Ali' (x i), (Al +A4), g 5' (x2), (Az +A 2 + ...
If the conditions are such that the methods of § 61 cannot be used, or are undesirable as giving too much weight to particular ordinates, it is best to proceed in the manner indicated at the end of § 48; i.e.
to find the areas of one set of parallel sections, and treat these as the ordinates of a trapezette whose area will be the volume of the briquette.
The ordinates of this new briquette at the points of intersection of x =x 0, x = xi,.
When the sequence of differences is not such as to enable any of the foregoing methods to be applied, it is sometimes possible to amplify the data by measurement of intermediate ordinates, and then apply a suitable method to the amplified series.
The formula applied can then be either Simpson's rule or a rule based on Gauss's theorem for two ordinates (§ 56).
Fourier's theorem asserts that such a curve may be built up by the superposition, or addition of ordinates, of a series of sine curves of wave-lengths X, IX, 3A, 4A...
The distribution of shear on vertical sections is given by the ordinates of a sloping line.
As the load travels, the shear at the head of the train will be given by the ordinates of a parabola having its vertex at A, and a maximum F max.
travels the reverse way, the shearing force at the head of the train is given by the ordinates of the dotted parabola.
For the position shown the distribution of bending moment due to W 1 is given by ordinates of the triangle 000 A'CB'; that due to W2 by ordin al, W, WW1 W„ ates of A'DB'; and that due to W3 by ordinates of A EB'.
As the loads move over the girder, the points C, D, E describe the parabolas M1, M2, M3 i the middle ordinates of which are 4W 1 1, 4W 2 1, and 4W3l.
be the corresponding ordinates of the influence curve (y = Ff) on the verticals under the loads.
from the left abutment, being the ordinates of the influence curve under the loads, is S = Piyl+P2y2+
This method distorts the curve, so that vertical ordinates of the curve are drawn to a scale b times greater than that of the horizontal ordinates.
Integrating (27) again, (31) y =g(zTt2t 2) = zgt(T -t); and denoting T-t by t', and taking g= 32f/s2,) y =16tt', (32 which is Colonel Sladen's formula, employed in plotting ordinates.
A relation which is of historical interest connects the logarithmic function with the quadrature of the hyperbola, for, by considering the equation of the hyperbola in the form xy=const., it is evident that the area included between the arc of a hyperbola, its nearest asymptote, and two ordinates drawn parallel to the other asymptote from points on the first asymptote distant a and b from their point of intersection, is proportional to log bla.
2, in which the ordinates of the line ABC represent the percentage of pearlite corresponding to each percentage of carbon, and the intercept ED, MN or KF, of any point H, P or L, FIG.
9) can be defined by the co-ordinates (E, n) of this point 0 in an object plane I, at right angles to the axis, and two other co ordinates (x, y), the point in which the ray intersects the entrance pupil, i.e.
von Rohr's Theorie and Geschichte des photographischen Objectivs, the abscissae are focal lengths, and the ordinates wave-lengths; of the latter the Fraunhofer lines used are A' C D Green Hg.
The ordinates give the wave-lengths in, up. The abscissae give fc in 0.01 mm., commencing at fc fF.
The measured lengths are marked off on ordinates erected on an abscissa, along which the times are noted.
The abscissae represent intervals of time, the ordinates the measured lengths of the growing filament.
The enclosed area for each temperature represents the total emission of energy for that temperature, the abscissae are the wavelengths, and the ordinates the corresponding intensities of emission for that wave-length.
It will be seen that the maximum ordinates lie upon the curve A9 = constant dotted in the figure, and so, as the temperature of the ideal body rises, the wave-length of most intense radiation shifts from the infra-red X towards the luminous part of the spectrum.
From this or otherwise it is readily deduced that the ordinates of an ellipse and of the circle described on the major axis are in the ratio of the minor to the major axis.
on this theory, the ordinates also represent the E.1VI.F.
He discovered a simpler method of quadrating parabolas than that of Archimedes, and a method of finding the greatest and the smallest ordinates of curved lines analogous to that of the then unknown differential calculus.
It may be useful to record whether a service enables searches by geographical co- ordinates.
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.
The series of values of X will in general be discontinuous, and the graph will then be made up of a succession of parallel and (usually) equidistant ordinates.
If BN, CP, DQ, FS, GT are the perpen diculars to AE from the angular points, the ordinates NB, PC,..
The data of a trapezette are usually its breadth and either the bounding ordinates or the mid-ordinates of a series of minor trapezettes or strips into which it is divided by ordinates at equal distances.
6 (§ 34) we draw ordinates QD midway between KA and MC, and RE midway between MC and LB, meeting the top in D and E (fig.
The distribution of bending moment is given by the ordinates of a triangle.
The length of a parabolic arc can be obtained by the methods of the infinitesimal calculus; the curve is directly quadrable, the area of any portion between two ordinates being two thirds of the circumscribing parallelogram.
The numerical results of the civil service examinations are reduced so as to conform to a certain symmetrical "frequency-curve," of which the abscissae represent percentages of marks between definite limits and the ordinates the number of candidates obtaining marks between those limits.
Its gradient represents the acceleration, and the area (Jzidl) included between any two ordinates represents the space described in the fnterval between the corresponding instants (see fig.
Watts indicator-diagram, by the area cut off by the ordinates x = x0, x = Xi.
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