to observe that when this condition is fulfilled in the generalized system of measurement the parallel lines so obtained enjoy many of the properties of ordinary parallel lines. The perpendicular distance between such a pair of parallels is constant, and the angles which they make with any common transversal are equal.
It will be shown in a moment that any pair of straight lines which intersect the same two generators of the same system on the fundamental quadric are parallel in this new sense. The fact is that in the degraded circumstances of ordinary geometry two quite different conceptions have become confused, A pair of lines which intersect on the fundamental quadric and a pair of lines which intersect the same pair of generators of the same kind on the fundamental quadric are quite different conceptions; but when the fundamental quadric degrades to the ordinary infinity then the conceptions coalesce, and each of them is merely a pair of parallel lines in the ordinary sense of the word. The ordinary properties of parallel lines have all their analogues in the generalized geometry, but these analogues are distributed between the two original sources of parallels. Clifford proposes to retain the word “parallel” in non-Euclidean space for that conception which exhibits the more remarkable properties of ordinary space, and defines as follows:—
Straight lines which intersect the same two generators of the same system on the fundamental quadric are parallel.
Let X and Y be two rectilinear generators of the fundamental quadric belonging to the same system, and let A and B be two straight lines which intersect both X and Y. Since AX and AY are tangent planes their poles must lie on X and Y respectively, and therefore A′ (and B′), the polar of A (and B), must intersect both X and Y. The anharmonic ratio of the four points in which X intersects AB, A′B′ respectively is equal to that of the tangent planes drawn at the points where X intersects A′B′, AB respectively ; and, as all these tangent planes contain X, their anharmonic ratio must be equal to that in which they are cut by the line Y. It hence follows that the lines X and Y are divided equianharmonically by the four rays A, B, A′, B′, and therefore the four rays A, B, A′, B′ must be all generators of the same system on an hyperboloid. An infinite number of transversals can therefore be drawn to intersect these four rays, that is to say, an infinite number of common perpendiculars can be drawn to the two rays A and B, and it is easy to show that the lengths of all these perpendiculars are equal.
Clifford has proved the very remarkable theorem that rotations of equal amplitude about two conjugate polars have simply the effect of translating every point operated on through equal distances along parallel lines. This property leads to most important consequences, but it would lead us too far to enter into the subject at present.
A memoir by the present writer on the extension of the theory of screws to space of this description will be found in the Transactions of the Royal Irish Academy, vol. xxvii. pp. 157–184.
Units of Measurement.—A most excellent account of the units employed in scientific measurements will be found in Professor J. D. Everett's Units and Physical Constants, London, 1879. We shall here only give a very brief out line of this branch of the general theory of measurement, referring inquirers to Everett's volume for further details.
Most of the quantities for which measurements are needed can be ultimately expressed by means of (1) a definite length, (2) a definite mass, or (3) a definite interval of time.
It is very important that the units thus referred to should be chosen judiciously, and it must be admitted that the units ordinarily used do not fulfil the conditions which a well-chosen system of units should fulfil. The most scientific system is beyond doubt that which has been suggested by the units committee of the British Association. In this system the unit of length is the centimetre, the unit of mass is the gramme, and the unit of time is the second, and the system is therefore often referred to for brevity as the C.G.S. system. The unit of force is termed the dyne, and it is defined to be the force which, acting upon a gramme of matter for a second, generates a velocity of a centimetre per second. The unit of work is the work done by this force working through a centimetre, and this unit is termed the erg. The unit of power is the power of doing work at the rate of one erg per second, and the power of an engine can be specified in ergs per second. By the prefixes deca, hecto, kilo, mega, we can express a magnitude equal to the unit multiplied by 10, 100, 1000, or 1,000,000 respectively. On the other hand the prefixes deci, centi, milli, micro, signify the units divided by 10, 100, 1000, or 1,000,000 respectively.
For comparison with the ordinary units the following statements will be useful. The weight of a gramme at any part of the earth's surface is about 980 dynes, or rather less than a kilodyne. The weight of a kilogramme is rather less than a megadyne, being about 980,000 dynes.
The application of these units to electrical and many other measurements will be found in Professor Everett's book already referred to. On the general principles of appliances for measurement, see a paper by Clifford in the Handbook to the Special Loan Collection of Scientific Apparatus, 1876, pp. 55–59, reprinted in Clifford's Mathematical Papers, pp. 419–23.
MEATH, a maritime county of Ireland, in the province of Leinster, is bounded E. by the Irish Sea, S.E. by Dublin, S. by Kildare and King s county, W. by West- meath, 1ST. W. by Cavan and Monaghan, and N.E. by Louth. Its greatest length north and south is about 40 miles, and its breadth east and west about 45 miles. The total area comprises 578,247 acres, or 904 square miles.
The county forms part of the great limestone plain that occupies the central portion of Ireland. In some districts the surface is variegated by hills and swells, which to the west reach a considerable elevation, although the general features of a fine champaign country are never lost. The coast, which is low and shelving, extends to about 10 miles, but there is no harbour of importance. The Boyne, whose banks are specially beautiful, enters the county at its south-west extremity, and flowing north-east to Drogheda divides it into two almost equal parts. At Navan it receives the Blackwater, which flows south-west from Cavan. The Boyne is navigable for barges as far as Navan, where a canal is carried to Trim. The Eoyal Canal passes along the southern boundary of the county to Dublin. There are no lakes of importance.
Climate and Agriculture.—The climate is genial and favourable for all kinds of crops, there being less rain than even in the neighbouring counties. The principal substratum is limestone, but there are some districts of clay slate. Except a small portion occupied by the Bog of Allen, the county is very verdant and fertile. The soil is principally a rich deep loam resting on limestone gravel, but varies from a strong clayey loam to a light sandy gravel.
The total number of holdings in 1881 was 11,867, of which 1632 were less than 1 acre in extent, and 4300 between 5 and 15 acres. Only 93 were above 500 acres. According to the agricultural statistics for 1881, the area of arable land was 532,708 acres, or 92 4 per cent, of the whole area of the county, while 9599 acres were under plantations, 11,260 bog and marsh, and 201 barren mountain land. Of the arable land, 60,411 were under tillage, 85,893 meadow and clover, and 386,374 pasture. The following table shows the area under the different crops in 1855 and 1882:—
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Wheat. 18,764 2,783 Oats. Other Cereals. Potatoes. Turnips. Other Green Crops. Flax. Meadow and Clover. Total. 1855 1882 8(5,831 31,202 4.876 1,577 19,235 12,071 9,904 6,724 4,005 3,109 26(5 59 64,646 82,572 208.527 140,097
Horses between 1855 and 1882 have diminished from 23,310 to 15,316, an average of 2 9 to every 100 acres under cultivation, the average for Ireland being 3 2. Cattle in 1855 numbered 135,485, and in 1882 had increased to 176,121, an average of 33 to every 100 acres under cultivation, the average for Ireland being 26 "2. Sheep in 1855 numbered 170,582, and in 1882 only 146,749, although in 1880 they were 174,573. Pigs in 1882 numbered 19,709, goats 6398, and poultry 318,968.
According to the latest return the land was divided among 1322 proprietors, who possessed 577,846 acres, with a total annual value of 544,550. The average size of the properties was 436 acres, and the average annual value per acre was 18s. lOd. The following fou~ proprietors possessed more than 10,000 acres each : viz., Earl of Darnley, 21,858 acres; J. L. Naper, 18,863 ; Marquis of Laiis- downe, 12,995 ; and Lord Athlumney, 10,213.
Manufactures.—Almost the sole industry of the county is
