The standard way to do this, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product.

Forehead, crown, nape, upper back and ear-coverts bright chestnut ; lores black, continued as a band under the eye and ear-coverts ; wing-coverts, lower back and tertiaries green, the latter tipped with bluish ; rump and upper tail-coverts pale shining blue ; primaries and secondaries green, rufous on the inner webs, and all tipped dusky ; central tail-feathers bluish on the outer, and green on the inner webs ; the others green, margined on the inner web with brown and all tipped dusky ; sides of face, chin and throat yellow ; below this a broad band of chestnut extending to the sides of the neck and meeting the chestnut of the upper plumage ; below this again a short distinct band of black and then an ill-defined band of yellow ; remainder of lower plumage green, tipped with blue, especially on the vent and under tail-coverts.

The remainder of the gas is forced to return in an inner vortex of reduced diameter within the outer vortex.

The remainder of the floor has been resurfaced by subsequent lava flows, producing a relatively flat surface particularly within the inner ring.

The remainder of the rim is significantly eroded and worn, leaving only a ring of rugged ground surrounding the inner floor.

The inner wall is terraced along the southern edge, while the remainder is rougher and less finely featured.

The remainder of the inner wall still displays terraces, although they are worn and rounded due to erosion.

The remainder of the inner rim is a simple slope that runs down to the interior floor.

The remainder of this circular form can be traced out across the interior by an intermittent pattern of lower-albedo surface that is concentric with the inner wall.

The remainder of the inner wall is somewhat irregular in form.

The remainder of the rim is relatively intact, with small craterlets along parts of the inner wall.

The remainder of the rim is worn, with a pair of small craterlets along the northern rim and inner wall.

A number of smaller, worn impact craters lie along the remainder of the rim and inner wall.

The remainder of the inner wall has a slumped in places, forming a steep top section and shelves or terraces of slumped material.

The remainder of the rim and inner wall is somewhat irregular.

There is some slight terracing along the southeast inner wall, but the remainder displays more of a radially grooved appearance.

Around the remainder of the inner wall is some slumping and a few terrace-like features.

A 1997 press release from then Transport Minister Robin Cooper announced that a consortium of Reservoir Bus Company and Dyson's Bus Services were the preferred bidders for the remainder of the Public Transport Corporation's bus routes ( particularly in the inner city ).

After being banished from court, she lived out the remainder of her life at Kimbolton Castle, and died there on 7 January 1536.

In 1530, Catherine was banished from court and spent the remainder of her life ( until her death in 1536 ) alone in an isolated manor home, barred from any contact with Mary ( although her ladies-in-waiting helped the two maintain a secret correspondence ).

Concluding quickly that since a writ of mandamus, by definition, was the correct judicial means to order an official of the United States ( in this case, the Secretary of State ) to do something required of him ( in this case, deliver a commission ), Marshall devotes the remainder of his inquiry at the second part of the question: " Whether it writ can issue from this court.

Case law in tax court and federal courts constitute the remainder of tax law in the United States.

George Aston explains that when Murasaki retired from court she was again associated with Ishiyama-dera: " To this beautiful spot, it is said, Murasaki no Shikibu retired from court life to devote the remainder of her days to literature and religion.

The additions included a fine set of Dutch-gabled buildings backing onto the River Cam, and a ' window-with-nothing-behind-it ' that was designed to solve the problem of connecting the windowed library with the remainder of the court.

After a number of failed marriages throughout the 40s, she spent much of the remainder of her life at the court of her brother Herod Agrippa II, amidst rumors the two were carrying on an incestuous relationship.

For the remainder of 1748 and 1749 Gluck travelled with Mingotti's troupe, contracting a venereal disease from the prima donna and composing the opera La contesa de ' numi for the court at Copenhagen.

After the city abandoned its efforts to punish landlords who could not prove their tenants were lawfully present and narrowed the enforcement mechanism for the remainder of the ordinance, the district court upheld the remainder of the ordinance.

For most of the remainder of Theobald's life he was occupied with ecclesiastical affairs in his diocese, as well as attending the royal court when Henry was in England.

After detention and a short court hearing he was sent back to prison to serve the remainder of his sentence.

In 599 they routed his forces at Dormelles and seized the Dentelin, but they then fell foul of each other and the remainder of their time on the thrones was spent in infighting, often incited by their grandmother Brunhilda, who, angered over her expulsion from Theudebert's court, convinced Theuderic to unseat him and kill him.

Schuyler had already lost his command to Horatio Gates by the time of the court martial, and St. Clair held no more field commands for the remainder of the war.

The court was careful, however, to restrict the decision to the provision relating to discharge, and to express no opinion as to the remainder of the law.

Little is known of his early life other than that he was a choirboy at Arras in 1525 ; later in life he had a succession of posts in Arras, Tours and Tournai, before going to Spain to be master of the Flemish chapel ( capilla flamenca ) at the court of Philip II, where he stayed for the remainder of his life.

However the City was fought in court as its pursued acquisition of the Tropicana shopping center on the South-west corner, the SJRA abandoned acquisition of the entire shopping center and settled for acquisition of certain portions and the redevelopment of the remainder in conjunction with the existing owners.

After his defeat, he served the remainder of his partial term and left the court in 1915.

In court Gorton " carried himself so mutinously and seditiously " towards both magistrates and ministers that he was sentenced to find sureties for his good behavior during the remainder of his tenure in Plymouth, and given 14 days to be gone from the colony.

For the remainder of his life he was employed by the French court in one capacity or another, often writing music for state occasions — weddings, coronations, papal elections, births and deaths.

Although Mackenzie was completely exonerated at the court of inquiry and at a subsequent court martial ( by a split vote ), the controversial incident ( known as the " Somers Affair ") colored the remainder of his life.

The youthful and ambitious Zenón de Somodevilla had been created Marquis of Ensenada in 1736 for his successful diplomacy after the War of the Polish Succession, and seven years afterwards, in 1743, he was made Philip ( and Elizabeth's ) favorite at court, and for the remainder of Philip's reign, Ensenada effectively ruled Spain.

His parents talked seriously and lengthily to their own doctor and to a specialist at the University Hospital -- Mr. McKinley was entitled to a discount for members of his family -- and it was decided it would be best for him to take the remainder of the term off, spend a lot of time in bed and, for the rest, do pretty much as he chose -- provided, of course, he chose to do nothing too exciting or too debilitating.

It is difficult to tabulate exactly what was meant in each individual situation, but the conclusion may be drawn that 21 towns do not assess movable personal property, and of the remainder only certain types are valued for tax purposes.

Doubleday was humiliated by this snub and held a lasting grudge against Meade, but he returned to division command and fought well for the remainder of the battle.

During the remainder of Nero's reign, Agrippina's grave was not covered or enclosed.

The remainder of flight day four was spent making observations and preparing for activation of the Lunar Module, undocking, and landing the next day.

In 1939 Grothendieck went to France and lived in various camps for displaced persons with his mother, first at the Camp de Rieucros, and subsequently lived for the remainder of the war in the village of Le Chambon-sur-Lignon, where he was sheltered and hidden in local boarding-houses or pensions.

LF Eric Byrnes was on the 60-day disabled list from late June, with a torn left hamstring, and was out for the remainder of the season.

It was through Pacini that Salieri gained the attention of the composer Florian Leopold Gassmann, who, impressed with his talents and concerned for his future, took the young orphan to Vienna where he personally directed and paid for the remainder of his musical education.

In the remainder of the war in the Low Countries he held high commands, and was present at the Battle of Fleurus.

" It concluded, among other things, that Head Start programs designed to boost African-American IQ scores had failed, and that this was likely never to be remedied, largely because, in Jensen's estimation, 80 % of the variance in IQ in the population studied was the result of genetic factors and the remainder was due to environmental influences.

The Beowulf manuscript was transcribed from an original by two scribes, one of whom wrote the first 1939 lines and a second who wrote the remainder, so the poem up to line 1939 is in one handwriting, whilst the rest of the poem is in another.

" If the remainder of the Conservative Party could muster the electoral support necessary to form a government, then Disraeli was now guaranteed high office.

In 1974 the remainder of the Master Plan was canceled, leaving Boston with a severely overstressed expressway system.

For the remainder of his life, although he was passionately devoted to Hungary, its people and its culture, he never felt much loyalty to the government or its official establishments.

The remainder of the squadron was scattered ; the ships of the line sheltered at San Pietro Island off Sardinia, while the frigates were blown to the west and failed to return.

The effect on the strategic situation in the Mediterranean was immediate, reversing the balance of the conflict and giving the British control at sea that they maintained for the remainder of the war.

Bombardier was one of the companies that took over British Rail's Research and Development facilities after privatisation ( the remainder largely being absorbed into AEA Technology and Alstom ).

The maneuver was aborted, the remainder of the demonstration canceled and all aircraft landed safely.

The battle was over within hours, with the Dutch surrendering Fort Nassau-after some subterfuge-and within days the remainder of the Banda Islands.

During the Middle Ages, most of the remainder of Europe was Christianized, with Christians also being a sometimes large religious minority in the Middle East, North Africa, Ethiopia and parts of India.

Thus he sought a position in astronomy, and in 1807 was appointed Professor of Astronomy and Director of the astronomical observatory in Göttingen, a post he held for the remainder of his life.