Using a logarithmic spiral shape results in a constant angle between the cam and the rock at each contact point ; this constant angle is designed to always provide the necessary friction to hold a cam in equilibrium.

Using a known electric field, Millikan and Fletcher could determine even the charge on oil droplets in mechanical equilibrium.

Using these values and the equality conditions, the concentrations of the three species, chromate, CrO < sub > 4 </ sub >< sup > 2 -</ sup >, hydrogen chromate, HCrO < sub > 4 </ sub >< sup >-</ sup > and dichromate, Cr < sub > 2 </ sub > O < sub > 7 </ sub >< sup > 2 -</ sup > can be calculated, for various values of pH, by means of the equilibrium expressions.

Using the aggregate price index the equilibrium condition becomes

Using notable sociological and anthropological field studies as the grounding for such general ideas, the book makes a persuasive case for treating groups as social systems that can be analyzed in terms of a verbal analogue of the mathematical method of studying equilibrium and stability of systems.

Using the Henderson-Hasselbalch equation we get an equilibrium expression between the acid and conjugate base in terms of the log of the ratio of the acid to conjugate base ( the salt of the acid ).

Using this approximation, Einstein reproduced the incorrect Newtonian value for the deflection of light in 1909.

Using the relaxation time approximation for the Boltzmann equation and assuming steady-state conditions, the phonon thermal conductivity λ < sub > L </ sub > can be determined.

Using the approximation

Using the approximation and using Stirling's approximation for the factorials gives

Using the approximation

Using another approximation ( for example, the trapezoidal rule with twice as many points ), it is possible to take a suitable weighted average and eliminate another error term.

Using Stirling's approximation of the factorial one can establish the following asymptotic result:

Using the usual approximation for the speed of light in free space c < sub > 0 </ sub > = 3 × 10 < sup > 8 </ sup > m / s gives the frequently used expression:

Using Stirling's approximation for the Gamma function which omits terms of less than order N, the entropy for large N becomes:

Using the Stirling approximation again for large N, ln ( N!

Using a common approximation of ohms, we get

Using this approximation, the equation near becomes a differential equation:

Using the results from either Maxwell-Boltzmann statistics, Bose-Einstein statistics or Fermi-Dirac statistics, and considering the limit of a very large box, the Thomas-Fermi approximation is used to express the degeneracy of the energy states as a differential, and summations over states as integrals.

Using a continuum approximation, the number of

Using Stirling's approximation for the gamma function, and ignoring factors that disappear in the logarithm after taking N large,

Using this approximation, it can be shown that around 95 % of this distribution's probability lies within 2 standard deviations of the mean.

Using the WKB approximation, equations ( 5 ) and ( 7 ), we obtain:

Using the steady-state approximation, the rate of propagation becomes

Using Edgerton's method, the fast-closing action is obtained from the blackening of a window by exploding a series of parallel lead wires.

Using the F-distribution is a natural candidate because the test statistic is the ratio of two scaled sums of squares each of which follows a scaled chi-squared distribution.

Using visual imagery of a soroban, one can arrive at the answer in the same time as, or even faster than, is possible with a physical instrument.

Using a byte-oriented approach, it is possible to combine the < tt > SubBytes </ tt >, < tt > ShiftRows </ tt >, and < tt > MixColumns </ tt > steps into a single round operation.

Using two anchors set approximately 45 ° apart, or wider angles up to 90 °, from the bow is a strong mooring for facing into strong winds.

Using the same techniques they calculated the length of the Nile as, which is longer than previous estimates but still shorter than the Amazon.

Using the biblical Book of Revelation as a point of departure, Caesar Antichrist presents a parallel world of extreme formal symbolism in which Christ is resurrected not as an agent of spirituality but as an agent of the Roman Empire that seeks to dominate spirituality.

Using archery to take game animals is known as bow hunting.

Using a bow and arrow to take fish is known as bow fishing.

Using this criterion of falsifiability, astrology is a pseudoscience.

Using the formula, the overall speed boost is 1 / 0. 4575

Using this binding mechanism, an antibody can tag a microbe or an infected cell for attack by other parts of the immune system, or can neutralize its target directly ( for example, by blocking a part of a microbe that is essential for its invasion and survival ).

* Using a suitable catalyst ( e. g. Wilkinson's catalyst ), it is possible to reduce just one of the double bonds of an allene.

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

Using an existing language to bootstrap a new language is one way to solve the " chicken or the egg " causality dilemma.

Using ballpoint pens to create artwork is not without limitations.

Using the familiar cartesian basis, a vector A is written ;

Using programs or proofs of bounded lengths, it is possible to construct an analogue of the Berry expression in a formal mathematical language, as has been done by Gregory Chaitin.

Using terms from formal language theory, the precise mathematical definition of this concept is as follows: Let S and T be two finite sets, called the source and target alphabets, respectively.

Using numeric references that refer to permanently undefined characters and control characters is forbidden, with the exception of the linefeed, tab, and carriage return characters.

Using a three-digit Julian day number saves one byte of computer storage over a two-digit month plus two-digit day, for example, " January 17 " is 017 in Julian versus 0117 in month-day format.

Using color this way, the darkness and intensity ( or value ) of the color is evaluated by the eye as a measure of intensity or concentration.