# How to use *Ordinates* in a sentence

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.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.When the series is theoretically continuous, the theoretical graph will be a continuous figure of which the lines actually drawn are

**ordinates**.AdvertisementIn 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**.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.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.AdvertisementEither 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.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.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).AdvertisementThe 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.AdvertisementHence, 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.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.AdvertisementIn 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.If we write CI for the chordal area obtained by taking

**ordinates**at intervals Zh, then T i =2CI-C I.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**.Suppose, for instance, that m=6, and that we consider the trapezette as a whole; the data being the bounding

**ordinates**.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,.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.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.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.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.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.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.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.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.Watts indicator-diagram, by the area cut off by the

**ordinates**x = x0, x = Xi.