# Linear Sentence Examples

- The species of a genus can hardly ever be arranged in
**linear**series. - Is a large tree remarkable for the variability in the shape of its leaves, which are
**linear**in young trees and vigorous shoots, and broad and ovate on older branches. - In Crete, had given way entirely to a
**linear**system by Period III. - Given certain
**linear**and angular measurements, the area must be so and so. - 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. - Simspon concluded that for a given wind velocity dissipation is practically a
**linear**function of ionization. - Serratifolia, having
**linear-lanceolate**sharply serrulate leaves, and B. - This is equally true both of the pictographic and the
**linear**Aegean systems. Its nearest affinities are with the "Asianic" scripts, preserved to us by Hittite, Cypriote and south-west Anatolian (Pamphylian, Lycian and Carian) inscriptions. - A theory of matrices has been constructed by Cayley in connexion particularly with the theory of
**linear**transformation. - Hence it is impossible to form a satisfactory
**linear**series. - There is evidence that the use in Crete of both
**linear**and pictorial signs existed in the Early Minoan period, contemporary with the first Egyptian dynasties. - Foot narrow; dorsal papillae
**linear**or fusiform, in several - series. - Peirce, "Mathematics is the science which draws necessary conclusions" (
**Linear**Associative Algebra, § i. - Arithmetical groups, connected with the theory of quadratic forms and other branches of the theory of numbers, which are termed "discontinuous," and infinite groups connected with differential forms and equations, came into existence, and also particular
**linear**and higher transformations connected with analysis and geometry. - The present article is merely concerned with algebraical
**linear**transformation. **Linear**Equations.-It is of importance to study the application of the theory of determinants to the solution of a system of**linear**equations.- 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. - He first divides by the factor x -x', reducing it to the degree m - I in both x and x' where m>n; he then forms m equations by equating to zero the coefficients of the various powers of x'; these equations involve the m powers xo, x, - of x, and regarding these as the unknowns of a system of
**linear**equations the resultant is reached in the form of a determinant of order m. - The law of reciprocity shows that p(s) = zti (m 1te2tmtL3t) t=1 st It 2t 3t viz.: a
**linear**function of symmetric functions symbolized by the k specifications; and that () St =ti ts. - By solving k
**linear**equations we similarly express the latter functions as**linear**functions of the former, and this table will also be symmetrical. - Naturally very many other factors have to be considered, but this alone is a sufficient reason to restrain attempts to place existing forms in
**linear**phylogenetic series. - As in all poplars, the catkins expand in early spring, long before the leaves unfold; the ovaries bear four
**linear**stigma lobes; the capsules ripen in May. - The metasternum is without the transverse
**linear**impression that is found in most families of Adephaga. - In 1895 he obtained a libation-table from the Dictaean cave with a
**linear**dedication in the prehistoric writing (" Further Discoveries," &c., J.H.S. - During the Third Middle Minoan period, the lower limits of which approach 1600 B.C., this pictographic script finally gives way to a still more developed
**linear**system - which is itself divided into an earlier and a later class. - The later class (B) of the
**linear**script is that used on the great bulk of the clay tablets of the Cnossian palace, amounting in number to nearly 2000. - Clay tablets were here found belonging to the earlier type of the
**linear**script (Class A), together with a great number of clay sealings with religious and other devices and incised countermarks. - But the bulk of the work consists of problems leading to indeterminate equations of the second degree, and these universally take the form that one or two (and never more)
**linear**or quadratic functions of one variable x are to be made rational square numbers by finding a suitable value for x. - Their language may prove to be that of the
**Linear**tablets. - The regions of this cuticle have a markedly segmental arrangement, and the definite hardened pieces (sclerites) of the exoskeleton are in close contact with one another along
**linear**sutures, or are united by regions of the cuticle which are less chitinous and more membranous, so as to permit freedom of movement. - - We have al 17 ready seen that in numerous lower insects the abdomen 18 is formed from twelve divi 19 sions placed in
**linear**fashion. - It is hard to arrange the Exopterygota in a
**linear**series, for some of the orders that are remarkably primitive in some respects are rather highly specialized in others. - Equally exaggerated are the statements as to the
**linear**and lateral extent of the catacombs, and their intercommunication with one another. - Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "
**Linear**Substitutions," with the topics determinants, &c.,**linear**substitutions, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers. - ALGEBRAIC FORMS. The subject-matter of algebraic forms is to a large extent connected with the
**linear**transformation of algebraical polynomials which involve two or more variables. - The adjoint determinant will be seen subsequently to present itself in the theory of
**linear**equations and in the theory of**linear**transformation. **Linear**Equations.-It is of importance to study the application of the theory of determinants to the solution of a system of**linear**equations.- For further information concerning the compatibility and independence of a system of
**linear**equations, see Gordon, Vorlesungen fiber Invariantentheorie, Bd. - ï¿½ ï¿½ P1 v2 v3...) ï¿½ For, writing as before, Xm 'Xm 2 Xm '= zzo(SQls:2s73...) xi'x12x13..., 1 2 3" 1231 2 3 = EPxi l x A2 x A3, P is a
**linear**function of separations of(/ 1 / 2 A2 / 4 3 3 ...) of specification (m"`1mï¿½2m"`3...), and if X; 1 X 3 2X8 3 ' .. - A table may be formed expressing the k expressions Pa l), P(2),...P(1) as
**linear**functions of the k expressions (m"`'smï¿½2smï¿½3s...), s =1, 2, ...k, and the numbers BSc occurring therein is 2s 3s possess row and column symmetry. - " The symmetric function (m ï¿½8 m' 2s m ï¿½3s ...) whose is 2s 3s partition is a specification of a separation of the function symbolized by (li'l2 2 l3 3 ...) is expressible as a
**linear**function of symmetric functions symbolized by separations of (li 1 12 2 13 3 ...) and a symmetrical table may be thus formed." - Di denotes, in fact, an operator of order s, but we may transform the right-hand side so that we are only concerned with the successive performance of
**linear**operations. - An important notion in the theory of
**linear**operators in general is that of MacMahon's multilinear operator (" Theory of a Multilinear partial Differential Operator with Applications to the Theories of Invariants and Reciprocants," Proc. Lond. - From the above D p4 is an operator of order pq, but it is convenient for some purposes to obtain its expression in the form of a number of terms, each of which denotes pq successive
**linear**operations: to accomplish this write d ars and note the general result exp (mlodlo+moldol +... - Two cases then arise: (I) the properties may be expressed as
**linear**functions of the composition, the terminal values being identical with those obtained for the individual components, and there being a break in the curve corresponding to the absence of mixed crystals; or (2) similar to (I) except that different values must be assigned to the terminal values in order to preserve collinearity. - An important distinction separates true mixed crystals and crystallized double salts, for in the latter the properties are not
**linear**functions of the properties of the components; generally there is a contraction in /10.591 volume, while the re fractive indices and other physical properties do not, in general, obey the additive law. - Formerly map makers contented themselves with placing upon their maps a
**linear**scale of miles, deduced from the central meridian or the equator. - The
**linear**scale of maps can obviously be used only in the case of maps covering a small area, for in the case of maps of greater extension measurements would be vitiated owing to the distortion or exaggeration inherent in all projections, not to mention the expansion or shrinking of the paper in the process of printing. - Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "
**Linear**Substitutions," with the topics determinants, &c.,**linear**substitutions, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers. - Organization and tactics did not affect the issue directly, for the conduct of the men and their junior officers gave abundant proof that in the hands of a competent leader the "
**linear**" principle of delivering one shattering blow would have proved superior to that of a gradual attrition of the enemy here, as on the battlefields of the Peninsula and at Waterloo, and this in spite of other defects in the training of the Prussian infantry which simultaneously caused its defeat on the neighbouring field of Auerstadt. - 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. - The first error is to assert that history unfolds in a basically
**linear**fashion, that there is a fundamental continuity between the past, present, and future. - The process is essentially a polar
**linear**action, or differentiation from a common centre. - Similarly, if a form in k variables be expressible as a quadratic function of k -1,
**linear**functions X1, X2, ...