Rectangular co-ordinates

It is easily shown that, if corresponding values of v and p are represented in a rectangular co-ordinate system, the elasticity at any point on the curve is equal to the length of the p-axis intercepted between the tangent at that point and the horizontal through the point (Fig. 1). 

Consider an infinite tube whose axis is parallel to the axis Ox of a rectangular co-ordinate system Oxyz- Let S denote the domain occupied by the interior of the tube in the plane Oyz, and let s be its area and the curve T its perimeter. In steady uniform flow the velocity u is everywhere in the direction Ox and is a function of y and z given by... 

Choosing a suitable unit, so as to multiply all the previous lengths by l /2j Lowenherz s mode of representation may thus be reduced to a rectangular co-ordinate system, the quantities of the single salts, in gram-molecules (or equivalents), being represented on the horizontal axes, and on the vertical axis the sum of these quantities. 

A system is said to be singly or multiply periodic if the variables just defined can be found such that each rectangular co-ordinate is periodic m the quantities Wk i.e. can be represented as a Fourier series... 

Graphic Representation.—Hitherto the concentrations of the components have been represented by means of rectangular co-ordinates, although the numerical relationships have been expressed in two... 

The location of this point is simplified by dividing the normals from each of the corners on the opposite side into ten or one hundred parts, and drawing through these divisions lines at right angles to the normal and parallel to the sides of the triangle. A network of rhombo-hedra is thus obtained, and the position of any point can be read off in practically the same manner as in the case of rectangular co-ordinates. Thus the point P in Fig. 87 represents a ternary mixture of the composition A = 0 5, B = 0 3, C = o 2 the perpendiculars Pa,... 

If we again employ rectangular co-ordinates for the graphic repre-... 

We shall assume that we are dealing with the aqueous solution of two salts which can give rise to a double salt, in which case we can represent the solubility relations in a system of rectangular co-ordinates. 

A rotation of the rectangular co-ordinate system (a , y, z) involves a linear transformation of the co-ordinates with constant coefficients. The momenta transform then contravariantly. In this case, where the coefficients aik defining the rotation fulfil the conditions... 

If the energy radiated in the course of one period is small, the damping may for the present be neglected. For a system of one degree of freedom, such as we consider here, the rectangular co-ordinates of the charged particles will be periodic functions of... 

We shall now examine more closely the manner in which the action variables alter when, in the case of vx—vy, the co-ordinate system is rotated. Let the action variables correspond to the rectangular co-ordinates x, y, and the... 

In tho case of tho spatial oscillator with vx vy, we could rotate the coordinate system arbitrarily about the z-axis without destroying tho property of separation in x, y, z co-ordinates. We obtained, in the various co-ordinate systems, different Jz s and Jv s. Further, rectangular co-ordinates are not the only ones for which the oscillator may be treated by the separation method. 

It will be seen from the equations (19) that Jr and have a completely different meaning from the quantities Jx and. lv, derived by separation in rectangular co-ordinates J, for example, is now 2 i times the angular momentum about the z-axis. J z, however, has the same meaning as before also, the factor of v, namely 2.Jr has the same significance as tho former (it is Ijv times the energy of an oscillator for which J, is... 

The restriction of Jr and individually by such conditions would, on the other hand, lead to quantum motions altogether different from those arising from the corresponding restriction of Jx and Jv in the case of a certain rectangular co-ordinate system. 

If we transform to the rectangular co-ordinates , rj, , where is perpendicular to the orbital plane, we find for the components of the electric moment p expressions of the form... 

In order to make clear the geometrical significance of u, we introduce rectangular co-ordinates , rj such that the f-axis is the major axis of the orbit and the origin is the centre of force, Z (fig. 13), thus ... 

In the ease of the Kepler motions the Fourier series for the rectangular co-ordinates , rj and for the distance r are comparatively easy to find. Noting that rja and ja are even functions, and rjja an uneven function of u, and therefore also of wlt we can put... 

If we choose the z-axis of a rectangular co-ordinate system as the direction of the field, the energy function becomes... 

The rectangular co-ordinates of the system are thus periodic functions of the wk° s, as well as of the wka in other words, a periodicity parallelepiped in the wfc°-space will be transformed into another in the lo -space. Apart from an arbitrary integral linear transformation of the wk s among themselves with the determinant 1, we have, therefore,... 

Fig. 4.6. Establishment of the rectangular co-ordinate origin on the first bond for rotational isomeric state calculations bond 2 is trans to the hypothetical bond 0 the x-direction is along bond 1 the y-direction has a positive intercept on the projection of bond 0 the z-direction completes a right-handed Cartesian system (after Flory, 1969).

Another exponential relationship common to reactor engineering is that between conversion and the ratio of the reactor (or catalyst) volume, V, and the volumetric flow rate, Q. This ratio is referred to as the contact time, t. Maleic anhydride (MA) is an important chemical intermediate that is formed by the partial oxidation of butane over a vanadium pyrophosphate. Figure 2.14 plots -butane conversion, X, against contact time, r—collected in an ideal micro-reactor on ordinary rectangular co-ordinates. Since the shape of the data is concave down, we might assume that the relationship is a power law with a negative exponent. However, the data points do not fall on the trend line drawn in Figure 2.15. 

Rectangular co-ordinates - where the dimensions are taken relative to the datums at right angles to each other, i.e. the general pattern is rectangular. This is the method shown in Figs 3.2 and 3.4. 

The possibility of error is less with rectangular co-ordinates, and the polar co-ordinate dimensions shown in Fig. 3.1 could be redrawn as rectangular co-ordinates as shown in Fig. 3.5. 

Phase Diagrams Using Rectangular Co-ordinates (Findlay,... 

In examining the emulsifier-solvent mixtures different approaches are possible but in order that our results should be directly comparable with the results obtained with the commercial scintillation fluids we first adopted the same method of rectangular co-ordinates, using the emulsifier-solvent ratio recommended for general use. The information obtained in this way showed that a concentration of 0.5% NaOH in water was likely to yield useful information when a triangular phase diagram of the toluene-Triton X-lOO-aq.NaOH system was examined. By this means the prismatic phase diagram for this system was examined by means of two mutually perpendicular plane sections that both yielded useful information. 

Fig. 2. The G.I.E. Chiomaticity Diagram (rectangular co-ordinates). This diagram constitutes a transformation of Fig. 16 with rectangular co-ordinates in terms of x, y, the C.I.E. trichromatic coefficients derived from the original X Y Z trisUmulus values.

The most popular of these, used with CNC machines, is the rectangular co-ordinate system which relates directly to the axis motions at... 

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