Consider two binary equations of orders m and n respectively expressed' in non-homogeneous form, viz.
= 0, we find that, eliminating x, the resultant is a homogeneous function of y and z of degree mn; equating this to zero and solving for the ratio of y to z we obtain mn solutions; if values of y and z, given by any solution, be substituted in each of the two equations, they will possess a common factor which gives a value of x which, corn bined with the chosen values of y and z, yields a system of values which satisfies both equations.
The words " Asiatic " and " Oriental " are often used as if they denoted a definite and homogeneous type, but Russians resemble Asiatics in many ways, and Turks, Hindus, Chinese, &c., differ in so many important points that the common substratum is small.
In this he showed that a homogeneous fluid mass revolving uniformly round an axis under the action of gravity ought to assume the form of an ellipsoid of revolution.
Forms adapted to terrestrial life and to aerial respiration occur in various divisions of Gastropods, and do not constitute a single homogeneous group. Thus the Helicinidae, which are terrestrial, are now placed among the Aspidobranchia.
The people who finally abandoned the mainland and took their priests with them are the people who made the Venetian republic. But they were not as yet a homogeneous population.
Taking Syria as the strip limited by the sea, the edge of the Hamad, the Taurus and the Sinaitic desert, we have a remarkably homogeneous geographical area with very obvious natural boundaries; but these, for various reasons, have proved very Scale, 1:7.soo.000
Increase in size appears sometimes to be accompanied by the development of a new layer of fibres, whereas a difference in the method of preparation may give to a layer which appeared homogeneous in one specimen a decidedly fibrous aspect in another.
One hypothesis supposes that the shores of the Mediterranean were originally inhabited by a homogeneous race neither Aryan nor Semitic.
The conditions which are peculiar to the modern world are the large numbers we have to deal with, the vast and fairly homogeneous areas in which justice is administered and property secured, and the enormously increased facilities for transport and communication.
The origin of things, which is also their substance, is thus laid in the simplest and most homogeneous elements or principles.
The rail-failures mentioned above also drew renewed attention to the importance of the thermal treatment of the steel from the time of melting to the last passage through the rolling mill and to the necessity of the finishing temperature being sufficiently low if the product is to be fine grained, homogeneous and tough; and to permit of this requirement being met there was a tendency to increase the thickness of the metal in the web and flanges of the rails.
The vegetation of the Palaeozoic era, till towards its close, was apparently remarkably homogeneous all over the world.
Assuming that in its circumpolar origin the North Temperate flora was fairly homogeneous, it would meet in its centrifugal extension with a wide range of local conditions; these would favor the preservation of numerous species in some genera, their greater or less elimination in others.
It is clear, however, that the Chinese came from the west, and entered their present territory along the course of the Hwang-ho at an unknown period, possibly about 3000 B.C. In early historical times China consisted of a shifting confederacy of feudal states, but about 220 B.C. the state of Tsin or Chin (whence the name China) came into prominence, and succeeded in forming a homogeneous empire, which advanced considerably towards the south.
The coating of silver obtained by this process is coherent and homogeneous, while that deposited from a solution of silver nitrate, as the result of the primary action of the current, is crystalline and easily detached.
Assuming then 01 to have the coefficients B1, B2,...B,, and f l the coefficients A 1, A21...A,n, we may equate coefficients of like powers of x in the identity, and obtain m+n homogeneous linear equations satisfied by the m+n quantities B1, 2, ...B n, A 1, A 2, ...A m.
For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u 1 v 2 w3), and by Euler's theorem of homogeneous functions xu i +yu 2 +zu 3 = mu xv1 +yv2 +zv3 = /IV xw 1+y w 2+ zw 3 = pw; denoting now the reciprocal determinant by (U 1 V2 W3) we obtain Jx =muUi+nvVi+pwWi; Jy=ï¿½.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J.
The sum of the monomial functions of a given weight is called the homogeneous-product-sum or complete symmetric function of that weight; it is denoted by h.; it is connected with the elementary functions by the formula 1 7r1l7r2!7r3!
Schlafli 1 this equation may be directly formed and exhibited as the resultant of two given equations, and an arbitrary linear non-homogeneous equation in two variables.
THE Theory Of Binary Forms A binary form of order n is a homogeneous polynomial of the nth degree in two variables.
Invariantive forms will be found to be homogeneous functions alike of the coefficients and of the variables.
In the case of Italy, as in that of Germany, he frankly regretted the constitution of powerful homogeneous states upon the borders of France.
Spencer's " instability of the homogeneous " is perhaps an attempt to perform the impossible (First Principles, chap. xix.).
These may be called heterogeneous homoplasies, but it is necessary to recognize the existence of homogeneous homoplasies, here called multiradial apocentricities.
In some cases it shows, when submitted to a careful examination under the highest powers of the microscope, and especially when treated with reagents of various kinds, traces of a more or less definite structure in the form of a meshwork consisting of a clear homogeneous substance containing numerous minute bodies known as microsomes, the spaces being filled by a more fluid ground-substance.
Cells treated by reagents, has been interpreted by many observers as a network of threads embedded in a homogeneous ground-substance.
Evidence is not wanting, however, that the cytoplasm must be regarded as, fundamentally, a semifluid, homogeneous substance in which by its own activity, granules, vacuoles, fibrils, &c., can be formed as secondary structures.
The outermost, newly formed layer is composed of a more homogeneous, denser substance than the inner one, and can be distinguished in all starch-grains that are in process of development.
Centrosome.The centrosome is a minute homogeneous granule found in the cytoplasm of some cells in the neighborhood of the nucleus.
The chromatin substance increases in amount; the thread stains moie deeply, and in most cases presents a homogeneous appearance.
Amongst nomads the tribe is the unit of government, the political bond is personal, and there is no definite territorial association of the people, who may be loyal but cannot be patriotic. The idea of a country arises only when a nation, either homogeneous or composed of several races, establishes itself in a region the boundaries of which may be defined and defended against aggression from without.
Not indeed altogether so homogeneous as the Nearctic area, it presents, however, even at its extreme points, no very striking difference between the bulk of its birds.
He professed the most open materialism, denied immortality in all forms and taught that the soul of man is homogeneous with the soul of animals and plants, material in origin and incapable of separate existence.
If all the industries belong to one economic area over which, so far as we can tell from general statistics of wages and prices, and other information, fairly homogeneous conditions prevailed, we may be able to reach some useful conclusions as to the operation of the act.
In so large and so homogeneous a group as that of the true Passerines, a constant 1 An abstract is contained in the Minute-book of the Scientific Meetings of the Zoological Society, 26th June and 10th July 1838.
The difference in the appearance of the basement membrane - sometimes wholly homogeneous, sometimes eminently fibrillar = can more especially be observed in differently preserved specimens of the genus Polia.
From this it would appear that the documents were merely materials collected from various sources and possibly with a view to the compilation of a homogeneous law.
(1789-1807), who initiated them, to break down the barriers to the formation of a homogeneous nation.
Resultants.-When we are given k homogeneous equations in k variables or k non-homogeneous equations in k - i variables, the equations being independent, it is always possible to derive from them a single equation R = o, where in R the variables do not appear.
Resultant Expressible as a Determinant.-From the theory of linear equations it can be gathered that the condition that p linear equations in p variables (homogeneous and independent) may be simultaneously satisfied is expressible as a determinant, viz.
The general theory of the resultant of k homogeneous equations in k variables presents no further difficulties when viewed in this manner.
But even with homogeneous light it would be unreasonable to expect an oblique direction of perfect polarization.
They are composed of a homogeneous proteid substance, and often contain albuminoid or proteid crystals of the same kind as those which form the pyrenoid.
The Structure of the Nucleus.In the living condition the resting nucleus appears to consist of a homogeneous ground substance containing a large number of small chromatin granules and one or more large spherical granulesnucleolithe whole being surrounded by a limiting membrane which separates it from the cytoplasm.
It is often vacuolar, sometimes granular, and in other cases it is a homogeneous body with no visible structure or differentiation.
The external menace to their independence had welded together the place and the people; the same pressure had brought about the fusion of the conflicting parties in the lagoon townships into one homogeneous whole.
9 illustrates the first case: the ordinates represent specific volumes, and the abscissae denote the composition of isomorphous mixtures of ammonium and potassium dihydrogen phosphates, which mutually take one another up to the extent of 20% to form homogeneous crystals.
In such cases the molten rock cannot have been homogeneous, and as it flowed along the ground the different portions of it were drawn out into long parallel streaks.
If al, a2, ...a, n be the roots of f=o, (1, R2, -Ai the roots of 0=o, the condition that some root of 0 =o may qq cause f to vanish is clearly R s, 5 =f (01)f (N2) ï¿½ ï¿½;f (Nn) = 0; so that Rf,q5 is the resultant of f and and expressed as a function of the roots, it is of degree m in each root 13, and of degree n in each root a, and also a symmetric function alike of the roots a and of the roots 1 3; hence, expressed in terms of the coefficients, it is homogeneous and of degree n in the coefficients of f, and homogeneous and of degree m in the coefficients of 4..
It forms shiny, homogeneous masses, quite insoluble in cold water and in salt solutions, but soluble in alkalies.
Excellent political training such a government unquestionably offered; but it became unworkable as disparities of social condition increased, as the number of legal voters (above 7000 in 1822) became greater, and as the population ceased to be homogeneous in blood.
Case of Three Variables.-In the next place we consider the resultants of three homogeneous polynomials in three variables.
Discriminants.-The discriminant of a homogeneous polynomial in k variables is the resultant of the k polynomials formed by differentiations in regard to each of the variables.
The population was not homogeneous, as a considerable part of the immigration came by way of Pennsylvania and included Germans, the Protestant Scotch-Irish and settlers from the states farther.