By Euler's criterion, which had been discovered earlier and was known to Legendre, these two definitions are equivalent.

By Euler's rotation theorem, we may replace the vector with where is a 3x3 rotation matrix and is the position of the particle at some fixed point in time, say t = 0.

By Euler's formula,

By treating Euler's problem as a Liouville dynamical system, the exact solution can be expressed in terms of elliptic integrals.

By varying the formula, this curve may be moved forward or backward along the coordinates to produce any desired compression strength / density ratio.

By the continuous formula,

By using the formula

By doing this to the formula above, we find:

By using logarithmic identities to transform the formula, the multiplications can be expressed as a sum and the power as a multiplication.

By sorting formulas according to the number of atoms of each element present in the formula according to these rules, with differences in earlier elements or numbers being treated as more significant than differences in any later element or number — like sorting text strings into lexicographical order — it is possible to collate chemical formulas into what is known as Hill system order.

By degrees, however, Justinian came to understand that the formula at issue not only appeared orthodox, but might also serve as a conciliatory measure toward the Monophysites, and he made a vain attempt to do this in the religious conference with the followers of Severus of Antioch, in 533.

By the end of the 5th dynasty, the formula in all tombs becomes " An offering the king gives and Osiris ".

By most accounts, Coca-Cola was India's leading soft drink until 1977 when it left India after a new government ordered The Coca-Cola Company to turn over its secret formula for Coke and dilute its stake in its Indian unit as required by the Foreign Exchange Regulation Act ( FERA ).

By contrast, ext2 and other Berkeley FFS-like file systems of that time simply used a fixed formula for computing inode locations, hence limiting the number of files they may contain.

By the same formula we can find the geostationary-type orbit of an object in relation to Mars ( this type of orbit above is referred to as an areostationary orbit if it is above Mars ).

The canonical model is a model of L, as every L-MCS contains all theorems of L. By Zorn's lemma, each L-consistent set is contained in an L-MCS, in particular every formula unprovable in L has a counterexample in the canonical model.

By comparison, prediction in frequentist statistics often involves finding an optimum point estimate of the parameter ( s ) — e. g. by maximum likelihood or maximum a posteriori estimation ( MAP ) — and then plugging this estimate into the formula for the distribution of a data point.

By using this formula as distance, Euclidean space ( or even any inner product space ) becomes a metric space.

By the time of Diocletian, however, this two-stage process had largely disappeared, and the Praetor would either hear the whole case in person or appoint a delegate ( a iudex pedaneus ), taking steps for the enforcement of the decision ; the formula was replaced by an informal system of pleadings.

By convention, the formula is always true, regardless of the formula P ( x ); see vacuous truth.

By mid-century, Robert Wallace acquired the formula for nickel silver and established with Samuel Simpson, R. Wallace & Company the forerunner of Wallace Silversmiths.

By repeated applications of the half-angle formula

By Heron's formula, the area of the triangle is

By choosing a = 0, this formula gives both an integral and a series representation for a function that interpolates the harmonic numbers and extends a definition to the complex plane.

By substituting this formula for current into one or both factors of current in Joule's law, the power dissipated can be written in the equivalent forms:

By inserting different prices into the formula, you will obtain a number of break even points, one for each possible price charged.

By counting the number of stalls and urinals I attempted to form a loose estimate of how many men the hall would hold at one time.

By applying this general principle, a great number of complex compounds of osmium, ruthenium, iridium, and rhenium, with triphenylphosphine, triphenylarsine, and triphenylstibine have been obtained in this laboratory during the past few years.

By 1907 the number of undergraduates had risen to 1,107.

By its inclusion the order has grown enormously in number of species.

By making the mass number of soldiers blend within the landscape / painting, it shows that he believed that the usage and depiction of landscape was just as significant as a historical event, such as a war.

By the 18th century an increasing number of new plants had arrived in Europe from newly discovered countries and the European colonies worldwide and a larger number of plants became available for study.

By measuring the mean squared displacement over a time interval along with the universal gas constant, the temperature, the viscosity, and the particle radius, Avogadro's number can be determined.

By this point their number of MPs was severely depleted.

By now the rebel forces were said to have numbered 230, 000, however, this number should be treated with scepticism — Dio's account is known only from a late epitome, and ancient sources commonly exaggerate enemy numbers.

By the late 1950s, the once-prosperous port area of downtown Manhattan was occupied by a number of dilapidated shipping piers, casualties of the rise of container shipping which drove sea traffic to Port Elizabeth, New Jersey.

By attempting to predict which branch ( or path ) a conditional instruction will take, the CPU can minimize the number of times that the entire pipeline must wait until a conditional instruction is completed.

By the mid-20th century, the number of examples of people skipping stages, such as going from hunter-gatherers to post-industrial service occupations in one generation, were so numerous that 19th-century evolutionism was effectively disproved.

By convention, a calendar year consists of a natural number of days.

By far the most visible and obvious power of many modern central banks is to influence market interest rates ; contrary to popular belief, they rarely " set " rates to a fixed number.

By the end of 2005, the number of telephone main lines in use totaled 7, 851, 649.

By the time of Howard's suicide in 1936, he had written 21 complete stories, 17 of which had been published, as well as a number of unfinished fragments.

By contrast, the album had limited success in the US, only reaching number 83 on the Billboard 200 Album Chart.

By reducing the number of tracks used ( and thus the capacity ), Commodore could further reduce cost-in contrast to the double-density drives used e. g. in IBM PCs of the day which saved 180 KB on one side ( by using a 40-track format ).

By 1968, the number of dance clubs started to decline ; most American clubs either closed or were transformed into clubs featuring live bands.

By connecting to the Forgotten Tribe's settlements ( represented by number tokens ), players may earn cloth tokens when the number token for the Forgotten Tribe's villages are rolled.

By contrast, integrated circuits packed a large number — often millions — of tiny electrical components, mainly transistors, into a small chip around the size of a coin.

By the early 18th century, other publishers began to issue collections of dances as well ; a conservative estimate of the number of dances in the English style published between 1651 and 1810 would run to around 20, 000.

By inserting the energetic definitions of the ionization potential and electron affinity into the Mulliken electronegativity, it is possible to show that the Mulliken chemical potential is a finite difference approximation of the electronic energy with respect to the number of electrons., i. e.,