With the reduced state variables, i. e. V < sub > r </ sub >= V < sub > m </ sub >/ V < sub > c </ sub >, P < sub > r </ sub >= P / P < sub > c </ sub > and T < sub > r </ sub >= T / T < sub > c </ sub >, the reduced form of the Van der Waals equation can be formulated:

* With r between 0 and 1, the population will eventually die, independent of the initial population.

* With r between 1 and 2, the population will quickly approach the value

* With r between 2 and 3, the population will also eventually approach the same value

* With r between 3 and ( approximately 3. 45 ), from almost all initial conditions the population will approach permanent oscillations between two values.

* With r between 3. 45 and 3. 54 ( approximately ), from almost all initial conditions the population will approach permanent oscillations among four values.

* With r increasing beyond 3. 54, from almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, etc.

Consider now the acceleration due to the sphere of mass M experienced by a particle in the vicinity of the body of mass m. With R as the distance from the center of M to the center of m, let ∆ r be the ( relatively small ) distance of the particle from the center of the body of mass m. For simplicity, distances are first considered only in the direction pointing towards or away from the sphere of mass M. If the body of mass m is itself a sphere of radius ∆ r, then the new particle considered may be located on its surface, at a distance ( R ± ∆ r ) from the centre of the sphere of mass M, and ∆ r may be taken as positive where the particle's distance from M is greater than R. Leaving aside whatever gravitational acceleration may be experienced by the particle towards m on account of ms own mass, we have the acceleration on the particle due to gravitational force towards M as:

With the aid of the eunuch Zheng Zhong ( d. 107 CE ), Emperor He ( r. 88 – 105 CE ) had Empress Dowager Dou ( d. 97 CE ) put under house arrest and her clan stripped of power.

With the beginning of the Byzantine offensives in the East and the Balkans in the 10th century, especially under the warrior-emperors Nikephoros II ( r. 963 – 969 ), John I Tzimiskes ( r. 969 – 976 ) and Basil II ( r. 976 – 1025 ), newly gained territories were also incorporated into themes, although these were generally smaller than the original themes established in the 7th and 8th centuries.

As derived above, the ordinary generating function for the sequence of squares is With r = 1, α = 0, β = 3, A ( x ) = 0, and B ( x ) = x ( x + 1 ), we can verify that the squares grow as expected, like the squares:

With r = 1 / 4, α = 1, β = − 1 / 2, A ( x ) = 1 / 2, and B ( x ) = − 1 / 2, we can conclude that, for the Catalan numbers,

* Eric Uyttewaal: < cite > Dynamic Scheduling With Microsoft ( r ) Project 2000: The Book By and For Professionals </ cite >, ISBN 0-9708276-0-1

With a large-area emitter, which contains many individual emission sites, A < sub > r </ sub > will nearly always be very very much less than the " macroscopic " geometrical area ( A < sub > M </ sub >) of the emitter as observed visually ( see below ).

With the death of Theophilos, the young Michael III ( r. 842 – 867 ) ascended the throne.

With r, θ, and z being constant over time, this simplifies to

With some change of symbols ( and units ): r → J, R → − E and the material constant E < sup >− 2 </ sup > → 4π ε < sub > r </ sub > ε < sub > 0 </ sub > these equations take the familiar form:

With his head erect, he approached, not glancing at us, and passed by with his clear eyes raised and fixed straight ahead.

With hinged, fixed or free end support conditions the deflected shape in neutral equilibrium of an initially straight column with uniform cross section throughout its length always follows a partial or composite sinusoidal curve shape, and the critical load is given by

With a population of about 1. 3 million, Estonia has about 1. 7 million cellular telephones, 0. 5 million fixed phones, and 0. 8 million internet users ( as of 2006 ).

With the sharp growth of economies in the 20th century, and increasing foreign exchange, the world's gold reserves and their trading market have become a small fraction of all markets and fixed exchange rates of currencies to gold were no longer sustained.

With video machines, the fixed payout values are multiplied by the number of coins per line that is being bet.

With beamed propulsion one can leave the power-source stationary on the ground, and directly ( or via a heat exchanger ) heat propellant on the spacecraft with a maser or a laser beam from a fixed installation.

With a fixed working frequency it is large by night and may even disappear by day.

With a fixed spool, spin cast reels can cast lighter lures than bait cast reels, although friction of the nose cone against the unspooling line slightly reduces casting distance compared to spinning reels.

With a fixed working frequency, large changes in ionospheric conditions may create skip zones at night.

With Lady Croom, for whom Mrs Chater is a " harlot ", Septimus delicately admits that " her passion is not as fixed " as a suitor might wish.

With the motion of the beam this revolved around, and turned, the ' sun ' a second rotating cog fixed to and which turned the drive shaft.

With fixed wing UAVs, payloads can also be attached to the airframe, but aerodynamics of the aircraft with the payload must be assessed.

With the exception of Sessions in the first series and Stiles and Mochrie starting with the seventh and eighth series ( respectively ), there were no fixed regulars on the show, though there were many recurring regulars ( including Fry ).

With the addition of fixed cameras for the Active Traffic Management system, the number of cameras on the Highways Agency's CCTV network is likely to increase significantly over the next few years.

With strong signals of fixed frequency and high amplitude, the amount of bias needed is reduced.

With the time equipment fixed, Scaroth uses it to travel back 400 million years in hopes to stop the explosion of his ship.

With this form of lift, the grip of an aerial tramway cabin is fixed onto the propulsion rope and cannot be decoupled from it during operations.

****** 421. 121. 311 With fixed stopped lower end – ( apparently non-existent ).

****** 421. 221. 311 With fixed stopped lower end.

With bayonets fixed, the Americans marched towards redoubt # 10.

With the deflector lowered about halfway into the jetstream, forward and reverse thrust are equal so the boat maintains a fixed position, but steering is still available to allow the vessel to turn on the spot-something which is impossible with a conventional single propeller.

The north span of the Three Nations Crossing, seen from Cornwall's east end. With a fleet of 24 buses, Cornwall Transit serves the community 6 days per week ( excluding holidays ) on fixed routes and supplementary " rush hour " routes.

With Meiji Restoration and the transfer of the imperial capital from Kyoto to Tokyo, Kansai dialect became fixed in position as a non-standard local dialect.

With fixed assets worth slightly more than £ 43. 5m and land insured for approximately £ 72m, Fitzwilliam is also about average in terms of college wealth.

With fixed shutter speed, aperture and focus, it continued in the tradition of Kodak's earlier Brownie cameras, providing a simple snapshot camera anyone could use.