arificient. The public buildings consist of three t churches, an Episcopalian church, a court-house, olbooth. The last is a handsome modern building eth a fine tower terminated by a very elegant spire. The pcentral school-house, situated upon the Green of Muirtown, is also a fine building, and comprises a large public hall, with six spacious apartments for the accommodation of the different classes and for the library and philosophical apparatus. Inverness is the centre of the custom-house district, which extends from the mouth of the Spey to Dornoch Frith on the east coast, and from Assynt Point to Ardnamurchan on the west. A striking alteration has of late taken place in the trade of grain; within fifteen years about 8000 to 10,000 bolls of oatmeal used to be imported annually into Inverness; while now from 4000 to 5000 bolls of oats are exported from its piers. The foreign annual imports into Inverness consist of from 400 to 600 tons of hemp, and three or four cargoes of timber or Archangel tar.' (New Statistical Account of Scotland.) There is no compulsory assessment for the support of the poor, who are provided for by special quarterly collections, by several charitable mortifications at the disposal of the magistrates, and from other sources. A short account of schools, which are numerous and upon the whole well conducted, is given in the article INVERNESS-SHIRE. The population of the burgh and parish of Inverness in 1831 was 14,324. The people in Inverness speak very good English: the tradition is that they learned it from Cromwell's soldiers. The climate of Inverness is much milder than might be supposed from its northern position in the island. Its mean annual temperature is about 47°, while that of the neighbourhood of London is about 48.5°, and that of London 50-5°. The mean annual quantity of rain which falls at Inverness is about 26-21 inches. This borough unites with Fortrose, Nairn, and Forres in returning one member to parliament.

chulish and in the bed of the river three miles south of Fort William. Sandstone is also frequently met with. The beds of the stratified rocks are usually highly inclined to the horizon, approaching almost to the vertical, but the dip varies. Their general direction is from south-west to north-east. The two principal mountains are Ben Nevis and Mealfourvounie. The former, which is separated from the Grampians by the desolate tract called the Moor of Rannoch, is composed of porphyry and granite, and rises 4374 feet above the level of the sea, being the highest mountain in Great Britain. It is easily ascended on the western side; and at about the height of 1500 feet the prospect, till then confined, opens to the south-west and discovers the Paps of Jura and several of the Hebrides. Above the altitude of 2000 feet there is no vegetation, and on the north-east side of the mountain, near its summit, the snow lies throughout the year. Mealfourvounie, which rises 2730 feet above the sea-level, is composed of a conglomerate rock and stratified sandstone, the latter of which is of so hard a texture as to be used for the pavements of the streets of Inverness. Some veins of lead and silver have been discovered in several parts of the county, and also iron ore in small quantities, but we are not aware that mines have hitherto been worked to any extent. The soil is for the most part light and sandy, with a subsoil of gravel or clay; but in the neighbourhood of the town of Inverness it is enriched by a fine loam deposited by the waters of the adjoining frith.

Farms, Estates, and Agriculture.-In 1808 the landed property of this county was divided among 83 proprietors, viz.7 estates of the valued rents of 3000l. per annum (Scotch ); 6 from 1000l. to 30007.; 23 from 4007. to 1000l.; 33 from 1007. to 4007.; and 14 under 1007. From that period to the present time we believe the above distribution has not undergone any material alteration. Formerly there were INVERNESS-SHIRE, a maritime county of Scotland, a great number of small arable farms only a few acres in bounded on the north by Ross-shire, on the south by the extent, but these have much decreased since the introducshires of Perth and Argyle, on the east by those of Nairn, tion of sheep farming. What remain of them are usuElgin, Banff, and Aberdeen, and on the west by the At-ally let from year to year, but the larger farms are lantic Ocean; the mainland is comprised between 56° 40′ let on lease, varying from seven to nineteen years. The and 57° 36' N. lat., and between 3° 50′ and 5° 50′ W. long. farm-houses erected within the last forty years by the from Greenwich. Its greatest length from north-east to wealthier class of store farmers are for the most part well south-west is 88 miles, and its greatest width from north-constructed, but the dwellings of the cottagers and poorer west to south-east nearly 55 miles. According to Mr. tenants are described as being in every respect comfortless M'Culloch (Statistical Account of the British Empire the and mean. (McCulloch's British Empire, vol. i., p. 310.) entire county contains 4245 square miles, or 2,716,800 acres, The attention of the farmers is chiefly directed to the of which the mainland occupies 1,943,920, and its islands rearing of sheep and cattle. The sheep are mostly of the 773,760; the former having 84,480, and the latter 37,760 Cheviot and Linton breeds, and the stock at the present acres of water. It comprehends various districts, particu- time is estimated at 120,000; the stock of cattle is supposed larly that of Badenoch on the south-east, where it borders to amount to 40,000 or 45,000, and is chiefly of the Skye upon Perth and Aberdeenshire; Lochaber on the south ad- or Kyloe breed. In the month of July a fair for the sale of joining Argyleshire; Glenelg on the north-west bordering sheep is held annually at the town of Inverness, where, upon the ocean; besides many inferior districts, such as upon an average, 100,000 sheep and as many stones of Glengary, Glen Morrison, Glenshiel, &c. It also compre- wool are bought up for the southern markets. The labourers hends a considerable portion of the Hebrides, or Western and farm-servants generally live on potatoes with milk, and Isles, including the Isles of Skye, Harris, Benbecula, North oats and barleymeal prepared in various ways, to which the and South Uist, Barra, &c. [HEBRIDES.] This county, wealthier tradesmen are able to add fish and butcher's which is extremely mountainous, is intersected by innumer- meat. The usual rate of ploughmen's and farm-servants' able lakes and rivers, and is divided into two nearly equal wages is 87. in money and six bolls of meal, with liberty to parts by the deep valley of Glenmore, which runs in a di- plant as much ground with potatoes as they can manure, rection from Fort William on the south-west to the town and female labour is commonly reckoned at two-thirds that of Inverness on the north-east. This county forms a large of men. The fields are frequently enclosed, and within the part of the Highlands of Scotland, and the general descrip- last twenty years a great deal of waste land has been tion of its geographical features cannot well be separated drained and reclaimed, and much ground planted; but from that of the division of the island to which it belongs. none of any consequence has been irrigated or embanked. [GREAT BRITAIN, p. 402.] By far the greater part of The average rent of cultivated ground varies from 17. to the surface is covered with heath, but a good deal of the 27. 10s. the acre, but in the immediate vicinity of the town heathy ground is arable, and a considerable extent of it has of Inverness it is as high as 57. to 77. the acre. been brought into cultivation during the present century. The population in 1831, according to the population returns for that year, was 94,797, of which 44,510 were males, and 60,287 females. The valued rent at the same period was 73,1887. Scotch, but the annual value of real property in 1815 was 185,5651. The county sends one member to parliament. [INVERNESS.]

Forests.-The fir woods in Glenmore and those of Strathspey in the adjoining county of Elgin are supposed to be more extensive than all the other natural woods in Scotland together. Glen Morrison, which opens into Glenmore, also contains much fine timber. In the parish of Kilmalie alone, near Fort William, it is estimated that there are about 14,000 acres covered with trees. Those which grow Geology and Mineralogy.-The prevailing rocks are of naturally are the oak, fir, birch, ash, mountain ash, holly, the primary class, having a highly crystalline structure, and elm, hazel, and the Scotch poplar. Those which are being entirely destitute of organic remains. Gneiss is planted are the larch, spruce, silver fir, beech, plane, perhaps the most abundant, but huge masses of granite and fruit trees. In these forests and the neighbourand of the oldest trap or porphyritic rocks are met with in ing mountains the herds of red and roe-deer roam in safety the Grampians and the mountains of Glencoe and Ben in recesses almost impenetrable to man. The alpine and Nevis. Limestone is found in several districts, and ap-common hare and other game are also abundant. proaches to the nature of marble, particularly near Balla

Manufactures.-Formerly a good deal of hemp, worsted,

and linen yarn was made in this county, but this has greatly The parochial schools throughout the county at ad wept declined since the establishment of the large manufactories ous and increasing, and the reader will find a very beanie of the south. At the present time there is a hemp manu- tory description of their present state under the head or factory employing nearly 300 hands, and a woollen factory several parishes in the above-cited work. for the weaving of coarse clothing and Highland plaids and tartan. The produce of the former is principally exported to the London market and to the East and West Indies.

1 + {√(x − 1 }2 = 2, √(1+a − 1) = r. We need do no more than rame addition and subtraction, multiplication and division, raising of powers and extraction of roots, as pairs of inverse operations.

(New Statistical Account of Scotland; Playfair's Description of Scotland; Beauties of Scotland; Society's Map of Scotland; Parliamentary Reports on the Caledonian Canal, 1803-4-5-6; Parliamentary Papers, &c.) Inland Navigation and Roads.-The Glenmore, or Great INVERSE, INVERSION. Any two operations of alglen of Albion,' as it is sometimes called, which stretches gebra are said to be inverse when one of them undoes, so to across the county from Fort William to the town of Inver-speak, the effect of the other; so that if both be successively ness, is partially covered by three lakes, Loch Lochy, Loch performed upon the same quantity, the result is that quanOich, and Loch Ness, which lie nearly in a straight line tity itself. For instance, the operations implied in 1+22 between the above-mentioned limits. Their aggregate length and (x-1) are inverse to one another; for is 37 miles 4 yards, and the entire distance between Fort William and re verness is 59 miles and 1628 yards. In 1802 Mr. Telford as appointed by the commissioners of the treasury to make a survey of these lakes and of the adjoining country preparatory to the cutting of a canal. His report was made in the following year, and the works were in full operation in 1805, but the whole line of navigation was not opened till the latter end of the year 1822. The expense of constructing the Caledonian Canal, as it is called, was defrayed by government. That part of the navigation which is not upon the lakes is 22 miles and 1628 yards in length: it is 50 feet wide at the bottom and 15 feet deep, though the original plan contemplated a depth of 20 feet. Loch Oich is the summit-level of the canal, and its elevation is 94 feet above the level of the sea on the east coast, at high water and ordinary spring tides. The entire cost was 986,924/., to which must be added a portion of the annual expenditure since the opening of the navigation, which has hitherto exceeded the produce of tonnage dues. The tonnage dues on vessels, whether laden or unladen, is one farthing per ton per mile, and produced in 1829 a revenue of 25751.; but the expenditure during the same year amounted to 45737., so that this canal promises to be but an unprofitable speculation. Its chief effect, as regards the town of Inverness, has hitherto been the commencement and gradual formation of a direct intercourse with the great western marts of Glasgow and Liverpool, and, through them, with the manufacturing districts with which these cities are so closely connected.'" (New Statis. Acct.) The roads are under the management of the Parliamentary Commissioners for Highland Roads -a body appointed for opening the communication by land about the same time that the Caledonian Canal was projected. They are said to be maintained in a state of most efficient repair, the expense being defrayed partly by government and partly by contributions from the county proprietors. The principal rivers are the Spey, Ness, and Beauly; in all of which there are, where a and B are different, or having absolutely valuable salmon fisheries, more particularly upon the Spey and Ness.

The operation of inversion is the solution of an equation, and vice versa. Let it be required to find the operation inverse to or. Assume oxy, and find x in terms of y; say ay, then (y)=y, or and are inverse operations. Thus if 2 x = y, x = 1 ± √y + 1, and either of the two, 1+ √ x + 1, or 1 +1, is inverse to x2 - 2x. It thus appears that a function may have more than one inverse function, and there are functions which have an infinite number; but there is a distinction by which one may be separated from all the rest. Let the Greek letters in this article be all functional symbols, or marks of operations to be performed, and let them come before the subject of operation, the quantity, or y, &c., in the order in which they are to be performed. Thus apr denotes the result of performing the operation upon r, and then the operation a upon 2. Now let = give x=42, where x is an unambiguous operation, and z is, generally speaking, ambiguous, or presenting several different forms. Then and are inverse operations, and ≈≈, and we might suppose at first that pr; that is to say, we might imagine that destroys as well as that destroys. But since is ambiguous, it may be that only one or more of the forms of will satisfy xx, and not all: and that this will be the case with one is obvious, while we can show that it cannot happen with more than one. For though the same operation, performed on different functions, may produce the same function, yet different operations, performed on the same function, must produce different functions. If then a and ẞ be different forms of 4, we have a x = x and Br=; but we cannot have both ax = x and ẞ 4x the same form and value in both equations.

From all the inverses of a function px, then, we separate that one, ax, which gives both par=x and apx=x, and call it the convertible inverse. Its symbol is 1, so that means that operation which satisfies both the the preceding example 1+(x+1) is the convertible equations 4'xx and p-x=x. [EXPONENT.] In inverse of a 2x: for 1+ No (x2 − 2 x + 1) = 1 + x − 1 =x. But 1 √x2 - 2x + 1 gives 1 − (x − 1) or 2 − x; and we call this an inconvertible inverse. Every function which has more than one inverse is not only a function of x, but the same function of other functions of r. Let ar be an inconvertible inverse of; then a pr is not a, let it be r. Then pax being x, ax is px, or x is pr, so that or is the same function of which it is of r. Thus in the preceding example x2 is the same function of 2 r which it is of x; or x3 − 2 x = (2 − x)2 — 2 (2 − x).

Education, Schools, &c.-Upon the formation, in the year 1818, of the Society for Educating the Poor in the Highlands,' a central or model school was erected on a large scale in the town of Inverness. This establishment, as might be expected, has already proved highly beneficial to the poor of the town and suburbs, and is upon an average attended by about 300 scholars. The same society supports twelve other similar institutions of less extent in the more remote and thinly peopled part of the Highlands, and from its first establishment in 1818 to the 30th of September, 1834, its total expenditure amounted to 80237. The Raining School, founded in 1747 by Mr. John Raining of Norwich, and endowed by that gentleman with the sum of 1000l., is superintended by two well qualified teachers, having salaries of 48.. and 407. per year, together with a house and garden. The number of pupils is usually about 250. In addition to these, there are several private elementary schools, besides Sabbath evening-schools for religious instruction, which are attended by a very considerable numWe have then this theorem: every function has as many ber of children. 'Inverness, thus already more amply different forms as inverses, and all these forms can be made provided than many other towns with the means of education by writing different functions of instead of x in the original and improvement, has been further enriched by a munifi-function; and each inverse of the function is the converticent bequest of 10,000l. 3 per cent consols, left by the late Rev. Dr. Bell, the ingenious author of the Madras System of Education, and committed by him to the charge of the magistrates of Inverness, who contemplate, we understand, erecting another large charity-school, and relieving the Education Society of the burthen of supporting their central or model school on the Green of Muirtown.' (New Statistical Account.)

2x

ble inverse to one of its forms, and an inconvertible inverse to all the rest. Thus 1-(x + 1), which is an inconvertible inverse to -2x, is the convertible inverse of (2 — x)1 - 2 (2x): for

√ ( (2 − x)2 − 2 (2 − x) + 1) = 1 −(2−x− 1) = x.

The way to make the convertible inverse of a given func

The angular error thus committed is only a small portion

The following figure represents an ellipse with its evolute. of the angle made by the tangents at P and A.

Whenever the two arcs adjacent to a normal (or perpen- parts, the first two of which have been long kno nd wept dicular to the tangent) of the involute are equal and similar, the third, which contains the peculiar distinction there is a cusp in the evolute; and the evolute generally method, is due to Mr. Horner.* recedes without limit as we approach a point of contrary flexure in the involute.