90 Y A R V A R A similar equation holds if the lower limit be restricted by another surface. By aid of these equations the arbitrary functions involved in the solution of (53) are determinate. For example, to find the surface such that // fulS shall be a minimum, (IS being au element of the co-ordinates x, y, z. Here T^fe?" Hence the eriua - p rf / 1
5 ) + jT,( ~T
} 2 + j dyWl+ + 3 / surface and /j. a given function of the co-ordinates x, y, z. Here tf P= = fe" Hence the eriua - T tion N- dP dQ -J --- r dx dy d to - _ _ ~ P q 11 I/ du. r,+ r, = ( cosa~/ : R 11 fj. dx
- ) / dx dy n du. da ^ ::: + cos7 T dy d where R and R are the principal radii of curvature of the surface at, the point, and a, /3, y the angles which the normal to the surface makes with the co-ordinate axes. Again, if we suppose the surface limited by two fixed surfaces, the equation V+P(p -p) + Q(q -q) = Q reduces to l+p/ + ^ = ........................ (59). This shows that the surface intersects each of its bounding surfaces orthogonally. As a particular case of (58) we infer that the surface of minimum area is in general such that its principal radii of cur vature at every point are equal and of opposite signs. A remark able investigation of the equation of this surface was given by Bonnet in Liouville s Journal, 1860. Again, in the surface of given area and of maximum volume we have F= z - a/l +p* + q". The equation X ; ---- , = readily leads in this case to + . = a. dx dy R R 30. The principles and method of the calculus of varia tions are largely employed in all branches of mathematics, both pure and applied. Want of space prevents our enter ing into any account of such application. The reader will liud the history of this calculus fully detailed in Todhunter s treatise, already mentioned. The most important recent work in English is Mr CarlPs Calculus of Variations, New York and London, 1885. (B. w.) VARICOSE VEINS. See SURGERY, vol. xxii. p. 684. VARLEY, CORNELIUS (1781-1873), water-colour painter, a younger brother of John Varley (see below), was born at Hackney, London, on 21st November 1781. He was educated by his uncle, a philosophical instrument maker, and under him acquired a knowledge of the natural sciences; but about 1800 he joined his brother in a tour through Wales, and began the study of art. He was soon engaged in teaching drawing. From 1803 till 1859 he was an occasional exhibitor in the Royal Academy ; and he also contributed regularly to the displays of the Water-Colour Society, of which, in 1803, he was one of the founders, and of which he continued a member till 1821. His works, which are not numerous, consist mainly of carefully finished classical subjects, with architec ture and figures. He published a series of etchings of Boats and other Craft on the River Thames, and during his life as an artist he continued deeply interested in scientific pursuits. For his improvements in the camera lucida, the camera obscura, and the microscope lie received the Isis gold medal of the Society of Arts ; and at the International Exhibition of 1851 he gained a medal for his invention of the graphic telescope. He died at Hamp- stead on 2nd October 1873. VARLEY, JOHN (1778-1842), water-colour painter, was born at Hackney, London, on 17th August 1778. His father, a man of scientific attainments and tutor in the family of Lord Stanhope, discouraged his leanings towards art, and placed him under a silversmith. But on his parent s death Varley escaped from this uncongenial em ployment, and, after working with a portrait painter, en gaged himself at the age of sixteen to an architectural draughtsman, who took him on a provincial tour to sketch the principal buildings in the towns they visited. His spare hours were employed in sketching from nature, and in the evenings he was permitted, like Turner and Girtin, to study in the house of Dr Munro. In 1798 he exhibited his first work, a View of Peterborough Cathedral, in the Royal Academy. In 1799 he visited North Wales, and in its Avild mountain scenery found the subjects best suited to his brush. He returned to the same district in 1800, and again in 1802, and the impressions then received powerfully influenced the whole course of his art. In 1804 he became a foundation member of the Water-Colour Society, and contributed over forty works to its first ex hibition. He had married in the previous year ; and, in order to provide for the wants of an increasing family, he was obliged to produce for the dealers much work of a slight and commonplace character. He also taught draw ing, and some of his pupils, such as John Linnell and William Hunt, afterwards became celebrated. He was a firm believer in astrology, skilful in casting horoscopes ; and some curious instances were related of the truth of his predictions. It was at his house that his friend William Blake sketched his celebrated Visionary Heads. Varley died at London on 17th November 1842. Yarley s landscapes are graceful and solemn in feeling, and simple and broad in treatment, being worked with a full brush and pure fresh transparent tints, usually without any admixture of body- colour. Though his works are rather mannered and conventional, they are well considered and excellent in composition. Some of his earlier water-colours, including his Views of the Thames, were painted upon the spot, and possess greater individuality than his later productions, which are mainly compositions of mountain and lake scenery, produced without direct reference to nature. Among his literary works are Zodiacal Physiology, 1828 ; Observations on Colour and Sketch-ing from Nature, 1830 ; A Practical Treatise on Perspective, and Principles of Landscape Design for Young Artists. VARNA, a fortified town and seaport of Bulgaria, in 43 12 N. lat. and 27 56 E. long., about midway between the Danube delta and the Bosphorus, lies on the north side of the Bay of Varna, at the opening of the Pravadi valley, 5 miles wide and skirted by hills on both sides. The town, which covers the slope of one of these hills, facing seawards, comprises Greek, Bulgarian, Turkish, Jewish, and Gipsy quarters, with a total population (1888) of 25,256 (8449 Bulgarians, 7569 Turks, 5423 Greeks). It has many finely built houses and shops. Its principal buildings are the gymnasium, a hospital, the barracks, and several churches and mosques. Amongst the industrial establishments are soap factories, breweries, distilleries, and tanneries. More than 3,740,000 gallons of wine are made annually within the department of Varna. The town is well supplied with good water, and is not unhealthy. Although open to the south and south-east, the bay affords safe anchorage, being sheltered from the north and north-east winds. Through the Pravadi it communicates with Lake Devno, which, if connected, as has been proposed, by a navigable canal with the bay, would form one of the best harbours in the Euxine. Varna is the chief outlet for the grain and agricultural produce of Bulgaria for Constantinople and the west of Europe, the export of grain in 1887 amounting in value to 244,210 (total exports, 349,030) and the total imports to 580,300. It lies 48 miles east of Shumla (Shumna), with which it is connected by the main Bulgarian railway, running thence to the Danube at Rustchuk, these three fortresses forming with Silistria (on the Danube) the Bul garian "quadrilateral." The old stone walls of Varna have been destroyed to make room for earth-fortifications. Yarna has been identified with the ancient Milesian colony of Odessus on the coast of Mresia Inferior. It figures largely in the history of more recent times, and close by was fought in 1444 the battle in which Amurath (Murad) II. slew the Hungarian King Ladislaus, and routed his forces commanded by the renowned John Hunyady. Yarna was occupied in 1828 by the Russians, in 1854 by the allies, who here organized the invasion of the Crimea, and in 1877 by the Egyptian troops sinnmoned to the defence of Turkey against the Russians. The following year it was ceded to the principality of Bulgaria. It has long been the seat of a Greek metropolitan and
since 1870 of a Bulgarian bishon.