Alternate forms

One complication is the matter of units while the Systeme International d Unites (SI) requires additional and sometimes awkward constants, its broad use requires attention [1]. Hence, while we present the derivation in the cgs/esu system, we show alternative forms appropriate to the SI system in Tables V-1 and V-2. 

An alternative form of exact nonrelativistic vibration-rotation Hamiltonian for triatomic molecules (ABC) is that used by Handy, Carter (HC), and... 

Expressions (2.7) and (2.8) differ by a factor 311/8, or about 1.18, and this is typical of the order of error associated with simple momentum transfer arguments. The flux relation (2.5) can also be written in the alternative form... 

The right hand sides of these equations may be expressed in a number of alternative forms. Perhaps the most compact is obtained by writing c = p /H.T, Then... 

Several variants of the conjugate gradients method have been proposed. The formulatior given in Equation (5.7) is the original Fletcher-Reeves algorithm. Polak and Ribiere proposed an alternative form for the scalar constant 7) ... 

An alternative form of the Henderson-Hasselbalch equation for acetic acid is... 

If the temperature is below the critical temperature of the gas, the alternative form... 

Illustration showing an alternative form of the method of standard additions. In this case a sample containing the analyte is spiked with a known volume of a standard solution of analyte without further diluting either the sample or the spiked sample. 

An alternative form of this equation is the van Deemter equation... 

Equation (6.26) is an important result which can be expressed in several alternate forms ... 

A small amount of particleboard is made with a fire-retardant treatment for use in locations where codes require this material, as in some offices and elevators. Particleboards receive overlay and finishing treatments with ease. Wood veneers, melamine overlays, printed paper overlays, vinyl overlays, foils, and direct grain printing can all be done quite simply. A small amount of particleboard is also made in the form of shaped, molded articles such as furniture parts, paper roU plugs, bmsh bases, and even toilet seats. There is another small increment of particleboard made by the extmsion process. These products are made in small captive operations owned by furniture manufacturers which consume all of this production in their furniture. The extmsion process differs from conventional flat-pressed particleboard in that the wood furnish is forced between two stationary heated surfaces. The mats are formed from one edge and this edge is alternately formed and pushed between the heated platens, which are maintained at a distance equal to the thickness of board produced. This is an old, slow, small-scale process, but is stiU in use in at least one location. 

This form of Darcy s law is appHcable only to saturated flow. As discussed earlier, there are distinctions between the state of soil water in the saturated and unsaturated regions. These distinctions lead to an alternative form of Darcy s law for the case of unsaturated flow (2,5). 

Once this process has been completed for each of the essential elements, patent claims maybe drafted which cover the invention. These claims will cover, in the broadest sense, only those elements of the invention which are essential. Narrower, more focused claims, however, should also be included within the patent appHcation. These claims may focus on aspects of the invention that the appHcant beHeves are preferred, or may otherwise represent essential aspects of any commercial product that will stem from the invention. Finally, claims should also be drafted to cover alternative forms of the invention. Such alternative forms of the invention may not necessarily be considered to be preferred commercially, but they may present an area where a competitor could attempt to engineer "around" the invention. 

An alternative form of the fundamental property relation given by equation 60 is provided by the mathematical identity of equation 166 ... 

The preceding two equations may be used to write the gradient equation for the countercurrent gas centrifuge in an alternative form. If the ratio of the actual gas circulation rate in the centrifuge to the circulation rate that rninimizes the stage length T/Tq is designated by m, then equation 74 may be rewritten... 

An alternative form of Eq. (4-101) follows by introduction of the activity coefficient through Eq. (4-103) ... 

Equations (4-140) and (4-141) may sometimes be advantageously expressed in alternative form through use of mean heat capacities ... 

An alternative form of the virial equation expresses Z as an expansion in powers of pressure about the real-gas state at zero pressure (zero density) ... 

Multiplication of the Redlich/Kwong equation (Eq. [4-220]) by V/RT leads to its expression in alternative form ... 

Lapple (in Periy, Chemical Engineers Handbook, 3d ed., McGraw-Hill, New York, 1950) presents an alternative form of Eq. (17-10) in which the gas-viscosity term is explicit instead of being incorporated into the resistance coefficients ... 

An alternative form of spiro nomenclature more common in Europe than in America is similar to that used in examples (64)-(66), but the order of citation of components is based on size, and the term spiro appears between the component names (see lUPAC Blue Book B-79MI10200) Rule B-11). Thus (64) would be called 3H-indole-S-spiro-l -cyclopenta-2, 4 -diene, and (63) would be cyclohexanespiro-4 -[l,3]dithian-2 -spirocyclopentane. 

For the special cases in which the normality conditions may be solved for k, it is possible to find alternate forms of the constitutive equations. Using (5.59) in (5.58) and (5.62) in (5.61) and dividing the first resulting equation by the second... 

Fig. 23.12. Shear banding, an alternative form of polymer plasticity which appears in compression.

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