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 due on arrears of ground-annuals. Like other real burdens, ground-annuals may now be freely assigned and conveyed (Conveyancing (Scotland) Act 1874, s. 30).

The term “ground rent” in the English sense does not seem to be generally used in the United States, but is applied in Pennsylvania to a kind of tenure, created by a grant in fee simple, the grantor reserving to himself and his heirs a certain rent, which is the interest of the money value of the land. These “ground rents” are real estate, and, in cases of intestacy, go to the heir. They are rent services and not rent charges—the statute Quia Emptores never having been in force in Pennsylvania, and are subject to all the incidents of such rents (see ). The grantee of such a “ground rent” may mortgage, sell, or otherwise dispose of the grant as he pleases; and while the rent is paid the land cannot be sold or the value of the improvements lost.

A ground rent being a freehold estate, created by deed and perpetual in duration, no presumption could, at common law, arise from lapse of time, that it had been released. But now, by statute (Act of 27th of April 1855, s. 7), a presumption of release or extinguishment is created where no payment, claim or demand has been made for the rent, nor any declaration or acknowledgment of its existence made or given by the owner of the premises subject to it, for the period of 21 years. Ground rents were formerly irredeemable after a certain time. But the creation of irredeemable ground rents is now forbidden (Pennsylvania Act 7 Assembly, 22nd of April 1850).

For English Law see Foa, Landlord and Tenant (3rd ed., London, 1901); Scots Law, Bell’s Principles (10th ed., Edinburgh, 1899); American Law, Bouvier, Law Dict. (Boston and London, 1897).

GROUNDSEL (Ger. Kreuzkraut; Fr. seneçon), Senecio vulgaris, an annual, glabrous, or more or less woolly plant of the natural order Compositae, having a branched succulent stem 6 to 15 in. in height, pinnatifid irregularly and coarsely-toothed leaves, and small cylindrical heads of yellow tubular florets enveloped in an involucre of numerous narrow bracts; the ribbed fruit bears a soft, feathery, hoary tuft of hairs (pappus). The plant is indigenous to Europe, whence it has been introduced into all temperate climates. It is a troublesome weed, flowering throughout the year, and propagating itself rapidly by means of its light feathery fruits; it has its use, however, as a food for cage-birds. Senecio Jacobaea, ragwort, is a showy plant with heads of bright yellow flowers, common in pastures and by roadsides. The genus Senecio is a very large one, widely distributed in temperate and cold climates. The British species are all herbs, but the genus also includes shrubs and even arborescent forms, which are characteristic features of the vegetation of the higher levels on the mountains of tropical Africa. Many species of the genus are handsome florists’ plants. The groundsel tree, Baccharis halimifolia, a native of the North American sea-coast from Massachusetts southward, is a Composite shrub, attaining 6 to 12 ft. in height, and having angular branches, obovate or oblong-cuneate, somewhat scurfy leaves, and flowers larger than but similar to those of common groundsel. The long white pappus of the female plant renders it a conspicuous object in autumn. The groundsel tree has been cultivated in British gardens since 1683.

The Old English word, represented by “groundsel,” appears in two forms, grundeswylige and gundæswelgiæ; of the first form the accepted derivation is from grund, ground, and swelgau, to swallow; a weed of such rapid growth would not inaptly be styled a “ground-swallower.” If the form without the r be genuine, the word might mean “pus-absorber” (O.E. gund, filth, matter), with reference to its use in poultices for abscesses and the like.

 GROUND-SQUIRREL, one of the names for a group of (chiefly) North American striped terrestrial squirrel-like rodents, more generally known as chipmunks. They are closely allied to squirrels, from which they are distinguished by the possession of cheek-pouches for the storage of food. The sides, or the sides and back, are marked with light stripes bordered by dark bands; the ears are small, and without tufts; and the tail is relatively short. With the exception of one Siberian species (Tamias asiaticus), ground-squirrels are confined to North America, where they are represented by a large number of species and races, all referable to the genus Tamias. In North America ground-squirrels are migratory, and may be abundant in a district one year, and absent the next. They feed on nuts, beechmast, corn and roots, and also on grubs. With the assistance of their cheek-pouches they accumulate large supplies of food for the winter, during which season they lie dormant in holes. Although generally keeping to the ground, when hunted they take to trees, which they climb in search of food. One of the longest known American species is T. striatus.

 GROUPS THEORY OF. The conception of an operation to be carried out on some object or set of objects underlies all mathematical science. Thus in elementary arithmetic there are the fundamental operations of the addition and the multiplication of integers; in algebra a linear transformation is an operation which may be carried out on any set of variables; while in geometry a translation, a rotation, or a projective transformation are operations which may be carried out on any figure.

In speaking of an operation, an object or a set of objects to which it may be applied is postulated; and the operation may, and generally will, have no meaning except in regard to such a set of objects. If two operations, which can be performed on the same set of objects, are such that, when carried out in succession on any possible object, the result, whichever operation is performed first, is to produce no change in the object, then each of the operations is spoken of as a definite operation, and each of them is called the inverse of the other. Thus the operations which consist in replacing x by nx and by x/n respectively, in any rational function of x, are definite inverse operations, if n is any assigned number except zero. On the contrary, the operation of replacing x by an assigned number in any rational function of x is not, in the present sense, although it leads to a unique result, a definite operation; there is in fact no unique inverse operation corresponding to it. It is to be noticed that the question whether an operation is a definite operation or no may depend on the range of the objects on which it operates. For example, the operations of squaring and extracting the square root are definite inverse operations if the objects are restricted to be real positive numbers, but not otherwise.

If O, O′, O&Prime;, ... is the totality of the objects on which a definite operation S and its inverse S′ may be carried out, and if the result of carrying out S on O is represented by O·S, then O·S·S′, O·S′·S, and O are the same object whatever object of the set O may be. This will be represented by the equations SS′ = S′S = 1. Now O·S·S′ has a meaning only if O·S is an object on which S′ may be performed. Hence whatever object of the set O may be, both O·S and O·S′ belong to the set. Similarly O·S·S, O·S·S·S, ... are objects of the set. These will be represented by O·S2, O·S3, ... Suppose now that T is another definite operation with the same set of objects as S, and that T′ is its inverse operation. Then O·S·T is a definite operation of the set, and therefore the result of carrying out S and then T on the set of objects is some operation U with a unique result. Represent by U′ the result of carrying out T′ and then S′. Then O·UU′ = O·S·T·T′·S′ = O·SS′ = O, and O·U′U = O·T′·S′·S·T = O·T′T = O, whatever object O may be. Hence UU′ = U′U = 1; and U, U′ are definite inverse operations.

If S, U, V are definite operations, and if S′ is the inverse of S, then

Let S, T, U, ... be a set of definite operations, capable of being carried out on a common object or set of objects, and let the set contain—

(i.) the operation ST, S and T being any two operations of the set;

(ii.) the inverse operation of S, S being any operation of the set; the set of operations is then called a group.

The number of operations in a group may be either finite or infinite. When it is finite, the number is called the order of the group,