## Integral Sentence Examples

- In 1548 Charles laid before the states a scheme for making the Netherlands an
**integral**part of the empire under the name of the Circle of Burgundy; but the refusal of the German Electors to make his only son Philip king of the Romans led him to abandon the project, which was never renewed. - Hungary, deprived of her ancient constitution, became an
**integral**part of theAustrian empire. - The critical examination of the nature and growth of this compilation has removed much that had formerly caused insuperable difficulties and had quite unnecessarily been made an
**integral**or a relevant part of practical religion. - For about a century the possession of this marchland was disputed by Hungary, Poland and Bohemia, but in 1029 it was finally incorporated with Bohemia, and so became an
**integral**part of the German empire.. - Arithmetic, algebra, and the infinitesimal calculus, are sciences directly concerned with
**integral**numbers, rational (or fractional) numbers, and real numbers generally, which include incommensurable numbers. - Bright colours (especially red) are frequent, and the white chemise is an
**integral**part of the dress. - Peacock powerfully aided the movement by publishing in 1820 A Collection of Examples of the Application of the Differential and
**Integral**Calculus. - The chronicler Saxo Grammaticus mentions in his Gesta Danorum the "rampart of Jutland" (Jutiae moenia) as having been once more extended by Valdemar the Great (1157-1182), which has been cited among the proofs that Schleswig (S4 nderjylland) forms an
**integral**part of Jutland (Manuel hist. - The French emperor, at the supposed request of the doge of Genoa, declared the Ligurian Republic to be an
**integral**part of the French empire. - By an instrument signed on the 24th of April 1815, the Austrian territories in north Italy were erected into the kingdom of Lombardo-Venetia, which, though an
**integral**part of the Austrian empire, was to enjoy a separate administration, the symbol of its separate individuality being the coronation of the emperors with the ancient iron crown of Lombardy (Proclamation de lempereur dAutriche, &c., April 7, 1815, State Papers, ii. - Any one introspectively apprehending the facts must grant, he thought, that benevolence was an
**integral**part of human nature and that conscience was rightfully supreme. - It remains to add that throughout we must carefully distinguish in theory, however hard this may be to do in practice, between legitimate ritual understood as such, whether
**integral**to prayer, such as its verbal forms, or accessory, such as gestures, postures, incense, oil or what not, and the formalism of religious decay, such as generally betrays itself by its meaninglessness, by its gibberish phrases, sing-song intonation and so forth. - In 1741 he received his first public distinction in being admitted a member of the Academy of Sciences, to which he had previously presented several papers, including a Memoire sur le calcul
**integral**(1739). - In 1746 and 1748 he published in the Memoirs of the Academy of Berlin "Recherches sur le calcul
**integral**," a branch of mathematical science which is greatly indebted to him. - This latter work included the differential and
**integral**calculus, the calculus of variations, the theory of attractions, and analytical mechanics. - As thus traced, the boundary in Central Asia includes the two khanates of Bokhara and Khiva, which, though nominally protected states, are to all intents and purposes
**integral**parts of the Russian empire. - South of this enclosed depression is another great hydrographic barrier which parts it from the low plains of the Amur, of China, Siam and India, bordered by the shallows of the Yellow Sea and the shoals which enclose the islands of Japan and Formosa, all of them once an
**integral**part of the continent. - Thanks to the exertions of Saliceti, one of the two deputies sent by the tiers etat of Corsica to the National Assembly of France, that body, on the 30th of November 1789, declared the island to be an
**integral**part of the kingdom with right to participate in all the reforms then being decreed. - The area of the complete curve is 2a 2, and the length of any arc may be expressed in the form f(1 - x 4) - i dx, an elliptic
**integral**sometimes termed the lemniscatic**integral**. - In the above example 2 R is an
**integral**real number, which is distinct from a rational integer, and from a cardinal number. - Although an
**integral**portion of the gut, it has ceased to assist in alimentation, its epithelium undergoes vacuolar differentiation and hypertrophy, and its lumen becomes more or less vestigial. - All symmetric functions are expressible in terms of the quantities ap g in a rational
**integral**form; from this property they are termed elementary functions; further they are said to be single-unitary since each part of the partition denoting ap q involves but a single unit. - - If, in the identity 1 (1 +anx = 1+aiox+aoly+a20x 2 +allxy+a02y 2 +..., we multiply each side by (I -ï¿½-P.x+vy), the right-hand side becomes 1 +(aio+1.1 ') x +(a ol+ v) y +...+(a p4+/ 1a P-1,4+ va Pr4-1) xPyq - - ...; hence any rational
**integral**function of the coefficients an, say f (al °, aol, ...) =f exp(ï¿½dlo+vdol)f d a P-1,4, dot = dapg The rule over exp will serve to denote that i udio+ vdo h is to be raised to the various powers symbolically as in Taylor's theorem. - Associated Forms.-A system of forms, such that every form appertaining to the binary form is expressible as a rational and
**integral**function of the members of the system, is difficult to obtain. - If, however, we specify that all forms are to be rational, but not necessarily
**integral**functions, a new system of forms arises which is easily obtainable. - To prove that this system is complete we have to consider (f, o) 2, 04') 1, (f,Q) 1, (f,Q) 2, (f,Q) 3, 0,Q) 1, (o,Q)2, and each of these can be shown either to be zero or to be a rational
**integral**function of f, 0 Q and R. - The discriminant is the resultant of ax and ax and of degree 8 in the coefficients; since it is a rational and
**integral**function of the fundamental invariants it is expressible as a linear function of A 2 and B; it is independent of C, and is therefore unaltered when C vanishes; we may therefore take f in the canonical form 6R 4 f = BS5+5BS4p-4A2p5. - In general any pencil of lines, connected with the line a x by descriptive or metrical properties, has for its equation a rational
**integral**function of the four forms equated to zero. - In spite of his doctrinal writings - which at the time made no little noise, so that his Compendium of Dogmatic (1760) was confiscated in Sweden, and the knighthood of the North Star was afterwards given him in reparation - it was the natural side of the Bible that really attracted him, and no man did more to introduce the modern method of studying Hebrew antiquity as an
**integral**part of ancient Eastern life. - The inroads made on the frontiers of Rio Grande and Sao Paulo decided the court of Rio to take possession of Montevideo; Brazil de- a force of 5000 troops was sent thither from Portugal, together with a Brazilian corps; and the irregulars
**integral**of Artigas, unable to withstand disciplined troops, were forced, after a total defeat, to take refuge beyond the river Uruguay. - Thus Siena was annexed to the Florentine state under the same ruler and became an
**integral**part of the grand-duchy of Tuscany.