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linear

the species of a genus can hardly ever be arranged in linear series.

5242is 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.

3731Hence it is impossible to form a satisfactory linear series.

2124in Crete, had given way entirely to a linear system by Period III.

1724Given certain linear and angular measurements, the area must be so and so.

1411Simspon concluded that for a given wind velocity dissipation is practically a linear function of ionization.

1217A theory of matrices has been constructed by Cayley in connexion particularly with the theory of linear transformation.

1115The 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.

1017There 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.

912The process is essentially a polar linear action, or differentiation from a common centre.

815Arithmetical 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.

77Schlafli 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.

69This 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.

36Organization 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.

36Peirce, "Mathematics is the science which draws necessary conclusions" (Linear Associative Algebra, ยง i.

23The metasternum is without the transverse linear impression that is found in most families of Adephaga.

24In 1895 he obtained a libation-table from the Dictaean cave with a linear dedication in the prehistoric writing (" Further Discoveries," &c., J.H.S.

24During 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.

24The 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.

24Foot narrow; dorsal papillae linear or fusiform, in several - series.

24" The symmetric function (m รฏยฟยฝ8 m' 2s m รฏยฟยฝ3s ...) 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."

25Assuming 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.

26The 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.

27Similarly, if a form in k variables be expressible as a quadratic function of k -1, linear functions X1, X2, ...

28The partial differential coefficient of T with respect to a component of velocity, linear or angular, will be the component of momentum, linear or angular, which corresponds.

10It 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.

13ALGEBRAIC 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.

13The present article is merely concerned with algebraical linear transformation.

13Linear Equations.-It is of importance to study the application of the theory of determinants to the solution of a system of linear equations.

13For further information concerning the compatibility and independence of a system of linear equations, see Gordon, Vorlesungen fiber Invariantentheorie, Bd.

13Resultant 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.

13By solving k linear equations we similarly express the latter functions as linear functions of the former, and this table will also be symmetrical.

13An 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.

13Naturally 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.

14As 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.

14The adjoint determinant will be seen subsequently to present itself in the theory of linear equations and in the theory of linear transformation.

14รฏยฟยฝ รฏยฟยฝ P1 v2 v3...) รฏยฟยฝ 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"`1mรฏยฟยฝ2m"`3...), and if X; 1 X 3 2X8 3 ' ..

14di 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.

14- 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.

15An 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.

16This arose from the study by Felix Klein and Sophus Lie of a new theory of groups of substitutions; it was shown that there exists an invariant theory connected with every group of linear substitutions.

00d d the last written relation having, in regard to each term on th right-hand side, to do with 17r successive linear operations.

00If the variables of the quantic f(x i, x 2) be subjected to the linear transformation x1 = a12Et2, x2 = a21E1+a2252, E1, being new variables replacing x1, x 2 and the coefficients an, all, a 21, a22, termed the coefficients of substitution (or of transformation), being constants, we arrive at a transformed quantic f% 1tn n n-1 n-2 52) = a S +(1)a11 E 2 + (2)a2E1 E 2 +รฏยฟยฝรฏยฟยฝรฏยฟยฝ in the new variables which is of the same order as the original quantic; the new coefficients a, a, a'...a are linear functions 0 1 2 n of the original coefficients, and also linear functions of products, of the coefficients of substitution, of the nth degree.

00F(a ' a ' a, ...a) =r A F(ao, a1, a2,รฏยฟยฝรฏยฟยฝรฏยฟยฝan), 0 1 2 n the function F(ao, al, a2,...an) is then said to be an invariant of the quantic gud linear transformation.

00To find the effect of linear transformation on the symbolic form of quantic we will disuse the coefficients a 111 a 12, a21, a22, and employ A1, Iรฏยฟยฝ1, A2, รฏยฟยฝ2.

00by the same scheme of linear substitution the two sets are said to be cogredient quantities."

00are said to be contragredient when the linear substitutions for the first set are x =A1X+u1Y-}-v1Z-?--..., y = A2X+,u2Y +v2Z รฏยฟยฝ..., Z = A 3 X +รฏยฟยฝ3Y -1v 3 Z - -..., and these are associated with the following formulae appertaining to the second set, .`"?.

00Since (ab) = a l b 2 -a 2 b l, that this may be the case each form must be linear; and if the forms be different (ab) is an invariant (simultaneous) of the two forms, its real expression being aob l -a l b 0.

00This will be recognized as the resultant of the two linear forms. If the two linear forms be identical, the umbral sets a l, a2; b l, b 2 are alternative, are ultimately put equal to one another and (ab) vanishes.

00A single linear form has, in fact, no invariant.

00If the forms be all linear and different, the function is an invariant, viz.

00the i t " power of that appertaining to a x and b x multiplied by the j t " power of that appertaining to a x and c x multiplied by &c. If any two of the linear forms, say p x, qx, be supposed identical, any symbolic expression involving the factor (pq) is zero.

00may be always viewed as a simultaneous invariant of a number of different linear forms a x, x, c x, ....

00For these only will the symbolic product be replaceable by a linear function of products of real coefficients.

00respectively, the linear forms a x, b., cg, ...

00It is always an invariant or covariant appertaining to a number of different linear forms, and as before it may vanish if two such linear forms be identical.

00All the forms obtained are invariants in regard to linear transformations, in accordance with the same scheme of substitutions, of the several sets of variables.

00- We have seen that (ab) is a simultaneous invariant of the two different linear forms a x, bx, and we observe that (ab) is equivalent to where f =a x, 4)=b.

00(f, 4)) k +(f, 4)) k +รฏยฟยฝ(f, 4/) k +aรฏยฟยฝ(1, 4)')k; and, moreover, if we require to find the kth transvectant of one linear system of forms over another we have merely to multiply the two systems, and take the k th transvectant of the separate products.

00+ X 2 - = D, X 1 -' j +X 2 =D 2 AA' ?2 / 2 1 3 - 5 -, =112 87,2 = ?1a a + ?2a a =Dรฏยฟยฝรฏยฟยฝ, 1 be linear operators.

00The invariants in question are invariants qud linear transformation of the forms themselves as well as qud linear transformation of the variables.

00Calling the discriminate D, the solution of the quadratic as =o is given by the formula a: = o (a0+a12_x2 (a0x+aix2 If the form a 2 be written as the product of its linear factors p.a., the discriminant takes the form -2(pq) 2.

00If the covariant (f,4) 1 vanishes f and 4 are clearly proportional, and if the second transvectant of (f, 4 5) 1 upon itself vanishes, f and 4) possess a common linear factor; and the condition is both necessary and sufficient.

00If (f,4) 1 be not a perfect square, and rx, s x be its linear factors, it is possible to express f and 4, in the canonical forms Xi(rx)2+X2(sx)2, 111(rx)2+1.2 (sx) 2 respectively.

00we find that Di must be equal to p x g x for then t x (p x) 3 +, u (g x) 3, Hence, if px, qx be the linear factors of the Hessian 64, the cubic can be put into the form A(p x) 3 +รฏยฟยฝ(g x) 3 and immediately solved.

00take and Uk = (af) k ai k the linear factor which occurs to the second power in f.

00the quartic to a quadratic. The new variables y1= 0 are the linear factors of 0.

00There are four invariants (i, i')2; (13, H)6; (f2, 151c.; (f t, 17)14 four linear forms (f, i 2) 4; (f, i 3) 5; (i 4, T) 8; (2 5, T)9 three quadratic forms i; (H, i 2)4; (H, 23)5 three cubic forms (f, i)2; (f, i 2) 3; (13, T)6 two quartic forms (H, i) 2; (H, 12)3.

00T = (j, j) 2 jxjx; 0 = (iT)i x r x; four other linear covariants, viz.

00Hermite expresses the quintic in a forme-type in which the constants are invariants and the variables linear covariants.

00If a, a be the linear forms, above defined, he raises the identity ax(0) =ax(aJ3) - (3x(aa) to the fifth power (and in general to the power n) obtaining (aa) 5 f = (a13) 5 az - 5 (a0) 4 (aa) ax?3 -F...

00On this principle the covariant j is expressible in the form R 2 j =5 3 + BS 2 a+4ACSa 2 + C(3AB -4C)a3 when S, a are the above defined linear forms.

00Now, evidently, the third transvectant of f, expressed in this form, with the cubic pxgxrx is zero, and hence from a property of the covariant j we must have j = pxgxrx; showing that the linear forms involved are the linear factors of j.

00This canonical form depends upon j having three unequal linear factors.

00The 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.

00The system of four forms, of which two are linear and two quadratic, has been investigated by Perrin (S.

00If ai, bx, cx be different forms we obtain, after development of the squared determinant and conversion to the real form (employing single and double dashes to distinguish the real coefficients of bx and cz), a(b'c"+b"c'-2 f'f") +b(c'a"+c"a'-2g'g") +c(a' +a"b'-2h'h")+2f(g'h"+g"h'-a' + 2g (h ' f"+h"f'-b'g"-b"g')+2h(f'g"+f"g'-c'h"-c"h'); a simultaneous invariant of the three forms, and now suppressing the dashes we obtain 6 (abc+2fgh -af t - bg 2 -ch2), the expression in brackets being the S well-known invariant of az, the vanishing of which expresses the condition that the form may break up into two linear factors, or, geometrically, that the conic may represent two right lines.

00Hesse showed independently that the general ternary cubic can be reduced, by linear transformation, to the form x3+y3+z3+ 6mxyz, a form which involves 9 independent constants, as should be the case; it must, however, be remarked that the counting of constants is not a sure guide to the existence of a conjectured canonical form.

00He proves, by means of the six linear partial differential equations satisfied by the concomitants, that, if any concomitant be expanded in powers of xi, x 2, x 3, the point variables-and of u 8, u 2, u3, the contragredient line variables-it is completely determinate if its leading coefficient be known.

00When R =0, and neither of the expressions AC - B 2, 2AB -3C vanishes, the covariant a x is a linear factor of f; but, when R =AC - B 2 = 2AB -3C =0, a x also vanishes, and then f is the product of the form jx and of the Hessian of jx.

00When a z and the invariants B and C all vanish, either A or j must vanish; in the former case j is a perfect cube, its Hessian vanishing, and further f contains j as a factor; in the latter case, if p x, ax be the linear factors of i, f can be expressed as (pa) 5 f =cip2+c2ay; if both A and j vanish i also vanishes identically, and so also does f.

00If, however, the condition be the vanishing of i, f contains a linear factor to the fourth power.

00In order to obtain the seminvari ants we would write down the (w; 0, n) terms each associated with a literal coefficient; if we now operate with 52 we obtain a linear function of (w - I; 8, n) products, for the vanishing of which the literal coefficients must satisfy (w-I; 0, n) linear equations; hence (w; 8, n)-(w-I; 0, n) of these coefficients may be assumed arbitrarily, and the number of linearly independent solutions of 52=o, of the given degree and weight, is precisely (w; 8, n) - (w - I; 0, n).

00symbolically, to be the fundamental form of seminvariant of degree 0 and weight w; he observes that every form of this degree and weight is a linear lic expressions.

00a irl aรฏยฟยฝ 2 a a3 ...Ev 1 02 2 ?3 3 ...; and, if we express Ea l v2 2 0-3 3 in terms of A2, A3 i ..., and arrange the whole as a linear function of products of A2, A3,..., each coefficient will be a seminvariant, and the aggregate of the coefficients will give us the complete asyzygetic system of the given degree and weight.

00Solving the equation by the Ordinary Theory Of Linear Partial Differential Equations, We Obtain P Q 1 Independent Solutions, Of Which P Appertain To S2Au = 0, Q To 12 B U =0; The Remaining One Is Ab =Aobl A 1 Bo, The Leading Coefficient Of The Jacobian Of The Two Forms. This Constitutes An Algebraically Complete System, And, In Terms Of Its Members, All Seminvariants Can Be Rationally Expressed.

00Thus, For Two Linear Forms, P =Q = I, We Find 1 Z 2 1 Az.

001 Ab' Establishing The Ground Forms Of Degrees Order (I, O; I), (O, I; I), (I, I; O), Viz: The Linear Forms Themselves And Their Jacobian J Ab.

00Similarly, For A Linear And A Quadratic, P= I, Q= 2, And The Reduced Form Is Found To Be 1 A2B2Z2 1 Az.

00Taking the variables to be x, y and effecting the linear transformation x = X1X+1.11Y, y = X2X+It2Y, X 2 +Y2X Y Xl - X2 y = _ x X I + AI R X 122 so that - รฏยฟยฝl b it is seen that the two lines, on which lie (x, y), (X, Y), have a definite projective correspondence.

00The linear transformation replaces points on lines through the origin by corresponding points on projectively corresponding lines through the origin; it therefore replaces a pencil of lines by another pencil, which corresponds projectively, and harmonic and other properties of pencils which are unaltered by linear transformation we may expect to find indicated in the invariant system.

00Or, instead of looking upon a linear substitution as replacing a pencil of lines by a projectively corresponding pencil retaining the same axes of co-ordinates, we may look upon the substitution as changing the axes of co-ordinates retaining the same pencil.

00The substitution for x, y in terms of X, Y is the most general linear substitution in virtue of the four degrees of arbitrariness introduced, viz.

00Besides the invariants and covariants, hitherto studied, there are others which appertain to particular cases of the general linear substitution.

00then of course (AB) = (ab) the fundamental fact which appertains to the theory of the general linear substitution; now here we have additional and equally fundamental facts; for since A i = Xa i +,ia2, A2= - รฏยฟยฝay + X a2, AA =A?-}-A2= (X2 +M 2)(a i+ a z) =aa; A B =AjBi+A2B2= (X2 +, U2)(albi+a2b2) =ab; (XA) = X i A2 - X2 Ai = (Ax i + /-Lx2) (- /-jai + Xa2) - (- / J.x i '+' Axe) (X a i +%Ga^2) = (X2 +, u 2) (x a - = showing that, in the present theory, a a, a b, and (xa) possess the invariant property.

00For the linear forms aoxi+aix2=ax = b x there are four fundamental forms ax=a:,x i --+a i x 2 of degree-order (1, 1), x7-1--4_ รฏยฟยฝ (0, 2), a i x, 1), a b =a2+ai รฏยฟยฝ (2, 0), (iii.) and (iv.) being the linear covariant and the quadrinvariant respectively.

00Every other concomitant is a rational integral function of these four forms. The linear covariant, obviously the Jacobian of a x and x x is the line perpendicular to x and the vanishing of the quadrinvariant a x is the condition that a x passes through one of the circular points at infinity.

00There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines ax, or, as we may say, the common harmonic conjugates of the lines and the lines x x .

00The linear invariant a s is such that, when equated to zero, it determines the lines ax as harmonically conjugate to the lines xx; or, in other words, it is the condition that may denote lines at right angles.

00- The so-called " ballistic " method of measuring induction is based upon the fact that a change of the induction through a closed linear conductor sets up in the conductor an electromotive force which is proportional to the rate of change.

00five times linear.

00Magnified five times linear.

00- Ventral view of Limulus polyphemus, one-fourth the natural size, linear.

00Dorsal view of male, enlarged five times linear.

00Enlarged fifteen times linear.

00The slender filaments of the stamens vary widely, often in the same flower; the anthers are linear to ovate in shape, attached at the back to the filament, and open lengthwise.

00The style is simple or branched, and the stigma is linear, capitate or globose in form; these variations afford means for distinguishing the different genera.

00He also showed that every equation of an even degree must have at least one real quadratic factor, reduced the solution of linear differential equations to definite integrals, and furnished an elegant method by which the linear partial differential equation of the second order might be solved.

00An n-tuple linear algebra (also called a complex number system) deals with quantities of the type A=/aiei derived from n special units e l, e 2 ...

00The types of linear associative algebras, not assumed to be commutative, have been enumerated (with some omissions) up to sextuple algebras inclusive by B.

00Quaternions afford an example of a quadruple algebra of this kind; ordinary algebra is a special case of a duplex linear algebra.

00If, in the extensive calculus of the nth category, all the units (including i and the derived units E) are taken to be homologous instead of being distributed into species, we may regard it as a (2'-I)-tuple linear algebra, which, however, is not wholly associative.

00It should be observed that while the use of special units, or extraordinaries, in a linear algebra is convenient, especially in applications, it is not indispensable.

00Any linear quantity may be denoted by a symbol (a 1, a2, ...

00This applies also to quaternions, but not to extensive quantities, nor is it true for linear algebras in general.

00One of the most important questions in investigating a linear algebra is to decide the necessary relations between a and b in order that this product may be zero.

00Peirce, " Linear Associative Algebra," Amer.

00Smith, on Linear Indeterminate Equations, Phil.

00If 2R be the diameter of the objectglass and D the distance of the object, the angle subtended by AP is E/D, and the angular resolving power is given by X/2 D sin a = X/2 R (3) This method of derivation (substantially due to Helmholtz) makes it obvious that there is no essential difference of principle between the two cases, although the results are conveniently stated in different forms. In the case of the telescope we have to deal with a linear measure of aperture and an angular limit of resolution, whereas in the case of the microscope the limit of resolution is linear, and it is expressed in terms of angular aperture.

00In a somewhat similar way a dark linear interruption in a bright ground may be visible, although its actual width is much inferior to the half wave-length.

00and, as we have just seen, the term in x 2 corresponds to a linear error in the spacing.

00A linear error in the spacing, and a general curvature of the lines, are eliminated in the ordinary use of a grating.

00The linear width of the band (e) is the increment of which alters p by 27r, so that e =27r /tr.

00Its coefficient of linear expansion between 0ยฐ and 100ยฐ is 0.002717; its specific heat 0.0562; its thermal and electrical conductivities are 145 to 152 and '14.5 to 140.

00The following table gives the linear expansions from oยฐ to 100ยฐ C. according to Fizeau (Comptes rendus, lxviii.

00Beginning with a single body in liquid extending to infinity, and denoting by U, V, W, P, Q, R the components of linear and angular velocity with respect to axes fixed in the body, the velocity function takes the form = Ucb1+V42+W43+ P xi+Qx2+Rx3, (I) where the 0's and x's are functions of x, y, z depending on the shape of the body; interpreted dynamically, C -p0 represents the impulsive pressure required to stop the motion, or C +p4) to start it again from rest.

00These equations are proved by taking a line fixed in space, whose direction cosines are 1, then dt=mR-nQ,' d'-t = nP =lQ-mP. (5) If P denotes the resultant linear impulse or momentum in this direction P =lxl+mx2+nx3, ' dP dt xl+, d y t x2' x3 +1 dtl dt 2 +n dt3, =1 ('+m (dt2-x3P+x1R) ' +n ('-x1Q-{-x2P) ' '= IX +mY+nZ, / (7) for all values of 1, Next, taking a fixed origin and axes parallel to Ox, Oy, Oz through 0, and denoting by x, y, z the coordinates of 0, and by G the component angular momentum about 1"2 in the direction (1, G =1(yi-x2z+x3y) m 2-+xlz) n(y(y 3x 1 x3x y + x 2 x) (8) Differentiating with respect to t, and afterwards moving the fixed.

00(9) c 2 Ci If the shot is moving as if fired from a gun of calibre d inches, in which the rifling makes one turn in a pitch of n calibres or nd inches, so that the angle S of the rifling is given by tan S = ird/nd = 2 d p/u, (10) 'ยฐIf a denotes the density of the metal, and if the shell has a cavity homothetic with the external ellipsoidal shape, a fraction f of the linear scale; then the volume of a round shot being sird 3, and sird 3 x of a shot x calibres long W =*ird 3 x(I -f 3)v, (20) 2 Wki 2= 61rd 3 xo(I-f 5)Q, (21) Wk22=67rd3x 2 2+0 2(I - f5)Q.

00which is the ratio of the linear velocity of rotation 2dp to u, the velocity of advance, -T2 d2 C 22 tans = n 2 = 4 = (c 2 - Ct) cg C12 2 W!

00The body is held fixed, and the reactio of the mechanism and the resultant of the impulsive pressure on th surface are a measure of the impulse, linear,, , and angula A, ยต, v, required to start the circulation.

00The effect of an external circulation of vortex motion on the motion of a cylinder has been investigated in ยง 29; a similar procedure will show the influence of circulation through a hole in a solid, taking as the simplest illustration a ring-shaped figure, with uniplanar motion, and denoting by the resultant axial linear momentum of the circulation.

00As the ring is moved from 0 to 0' in time t, with velocity Q, and angular velocity R, the components of liquid momentum change from aM'U +E and SM'V along Ox and Oy to aM'U'+ and /3M'V' along O'x' and O'y', (I) the axis of the ring changing from Ox to O'x'; and U = Q cos 0, V = Q sin 0, U' =Q cos (o - Rt), V' =Q sin (0 - Rt), (2) so that the increase of the components of momentum, X 1, Y 1, and N1, linear and angular, are X 1 = (aM'U'+ 0 cos Rt - aM'U - - 1 3M'V' sin Rt =(a - (3)M'Q sin_(0 - Rt) sin Rt - ver Rt (3) Y 1 = (aM'U'+) sin Rt-[-13M'V' cos Rt - (3M'V = (a - (3) M'Q cos (0 - Rt) sin Rt +t sin RT, N1=[ - (aM'U'+E) sin (0 - Rt)+ 1 3M'V' cos (o - Rt)]OO' = [- (a - 1 3) M'Q cos (o - Rt) sin (o - Rt) - sin (o - Rt) ]Qt.

00"ounce," and see As), the twelfth part of a linear foot.

00The coefficient of linear expansion is 0.002,905 for 100ยฐ from oยฐ upwards (Fizeau).

00The vapour tension may approximate to a linear function of the composition, and the curve will then be practically a straight line.

00cyclamineus is a pretty dwarf subspecies, native of Portugal, with narrow linear leaves and drooping flowers with reflexed lemon-yellow segments and an orange-yellow corona N.

00Somewhat more successful has been an attemptinaugurated by Hashimoto GahO and Kawabata Gyokushoto combine the art of the West with that of Japan by adding to the latter the chiaroscuro and the linear perspective of the former.

00This variation, however, is not always linear.

00If we assume that s is a linear function of 0, s= so(I +aO), the adiabatic equation takes the form, s 0 log e OW +aso(0 - Oo) +R loge(v/vo) =o

00Other favourite types' of equation for approximate work are (I) p=RO/vยฑf(v), which makes p a linear function of 0 at constant volume, as in van der Waal's equation; (2) v=RO/p+f(p), which makes v a linear function of 0 at constant pressure.

00Reynolds, in his investigation, introducing no new form of law of distribution of velocities, uses a linear quantity, proportional to the mean free path of the gaseous molecules, which he takes to represent (somewhat roughly) the average distance from which molecules directly affect, by their convection, the state of the medium; the gas not being uniform on account of the gradient of temperature, the change going on at each point is calculated from the elements contributed by the parts at this particular distance in all directions.

00The word is also used as a unit of linear measurement of the magnifying power of a lens or microscope.

00According to Quincke, the surface tension of pure water in contact with air at 20ยฐ C. is 81 dynes per linear centimetre, while that of alcohol is only 25.5 dynes; and a small percentage of alcohol produces much more than a proportional decrease in the surface tension when added to pure water.

00In this instrument a considerable linear relative movement of the divided lens corresponds with a comparatively small separation of the double image, so that simple verniers reading to 6 1 0 in.

00'SNOWDROP,' Galanthus nivalis, the best known representative of a small genus of the order Amaryllidaceae, all the species of which have bulbs, linear leaves and erect flower-stalks, destitute of leaves but bearing at the top a solitary pendulous bell-shaped flower.

00Apart from speculation, the first definite evidence for the molecular structure of matter occurs when it is found that certain physical phenomena change their whole nature as soon as we deal with matter of which the linear dimensions are less than a certain amount.

00It follows that the average distance apart of the molecules in the gaseous state is roughly ten times as great as in the solid or liquid state, and hence that in the gaseous state the molecules are at distances apart which are large compared with their linear dimensions.

00The genus is represented in Europe, north Asia, North America and Australia, and is characterized by oblong or linear stem-leaves, flowers in.

00The failure seems (ยง 2) to be due to difficulty in realizing the numerical expression of an area or a solid in terms of a specified unit, while the same difficulty does not arise in the case of linear measure or liquid measure, where the number of units can be ascertained by direct counting.

00(iii) Association of solid measure with liquid measure, presenting numerical measurement in a different aspect, should be retained by testing volumes as found from linear dimensions with the volumes of the same bodies as found by the use of measures of capacity.

00The ideas of moment and of centroid are extended to geometrical figures, whether solid, superficial or linear.

00The simplest case is that in which u is constant or is a linear function of x, i.e.

00Generally, if the area of a trapezette for which u is an algebraical function of x of degree 2n is given correctly by an expression which is a linear function of values of u representing ordinates placed symmetrically about the mid-ordinate of the trapezette (with or without this mid-ordinate), the same expression will give the area of a trapezette for which u is an algebraical function of x of degree 2n + 1.

00The velocity of propagation of a torsional disturbance along a wire of circular section may be found by the transfer of momentum method, remembering that we must now replace linear momentum by angular momentum.

00The unit of linear measure is the wall, which is subdivided into wah or sauk, a wah or kup, and into 9 1 6 wah or niew.

00Let E be the effective elasticity of the aether; then E = pc t, where p is its density, and c the velocity of light which is 3 X 10 10 cm./sec. If = A cos" (t - x/c) is the linear vibration, the stress is E dE/dx; and the total energy, which is twice the kinetic energy Zp(d/dt) 2 dx, is 2pn2A2 per cm., which is thus equal to 1.8 ergs as above.

00It has in fact been found, with the very great precision of which optical experiment is capable, that all terrestrial optical phenomenareflexion, refraction, polarization linear and circular, diffraction - are entirely unaffected by the direction of the earth's motion, while the same result has recently been extended to electrostatic forces; and this is our main experimental clue.

00Another powerful reason for taking the aether to be stationary is afforded by the character of the equations of electrodynamics; they are all of linear type, and superposition of effects is possible.

00Now the kinetics of a medium in which the parts can have finite relative motions will lead to equations which are not linear - as, for example, those of hydrodynamics - and the phenomena will be far more complexly involved.

00The polarization itself is determined from the electric force (P,Q,R) by the usual statical formula of linear type which becomes tor an isotropic medium (.f',g',h') = c2(P,Q,R), because any change of the dielectric constant K arising from the convection of the material through the aether must be independent of the sign of v and therefore be of the second order.

00In these the furcal branches are linear or rudimentary, the shell is without rostral sinus, and, besides distinguishing characters of the second 2ntennae, they have always a branchial plate well developed on the first maxillae, which is inconstant in the other tribe.

00(b) In determinations of volume which do not admit of a high degree of accuracy the cubic decimetre can be taken as equivalent to the litre; and in these determinations expressions of volumes based on the cube of the unit of linear measure can be substituted for expressions based on the litre as defined above.

00(76 millimetres), which denotes a variation of 103 grammes per square centimetre in the pressure of the atmosphere, the change caused thereby in the length of a standard of linear measurement would appear to be as follows: --

00Its coefficient of linear expansion is only 0.0000008 for 1ยฐ C. See: Rapport du Yard, Dr Benoit (1896).

00Bรยถckh, to the study of water-volumes and weights, even deriving linear measures therefrom; V.

00For engineering and manufacturing purposes the more important linear gauges are, however, now used, adjusted to some fundamental unit of measure as the inch; although in certain trades, as for wires and flat metals, gauges continue to be used of arbitrary scales and of merely numerical sizes, having no reference to a legal unit of measure; and such are rarely accurate.

00Both as regards structure and habits, the leopard may be reckoned as one of the more typical representatives of the genus Felis, belonging to that section in which the hyoid bone is loosely connected with the skull, owing to imperfect ossification of its anterior arch, and the pupil of the eye when contracted under the influence of light is circular, not linear as in the smaller cats.

00In 1857 he published his best known work, the System of Analytical Mechanics, which was, however, surpassed in brilliant originality by his Linear Associative Algebra (lithographed privately in a few copies, 1870; reprinted in the Amer.

00The twigs are densely clothed with flat spreading linear leaves of a fine glossy green above and glaucous beneath; in the old trees they become shorter and more rigid and partly lose their distichous habit.

00This frame or tube is so constructed of iron and brass (one-third iron and two-thirds brass) that its temperature coefficient of linear expansion is the same as that of the platinumsilver alloy.

00As the frame has the same linear expansion as the wire, external changes of the temperature will not affect their relative length, but if the fine wire is heated by the passage of an electric current, its expansion will move the indicating needle over the scale, the motion being multiplied by the gear.

00The linear ridges of this mIddle section are often called the Alleghany Mountains.

00They may be seen on the western border, in the Hoosac range along the boundary of Massachusetts and New York; in the linear series of the Green Mountain summits (Mt Mansfield.

00the east, where the disturbances of the Appalachian system have developed ridges and valleys of linear trends, which are wanting or but faintly seen elsewhere.

00" The other dimensions may be had from the drawing, which is one-sixth of the linear magnitude of the instrument itself.

00europaea) is, when grown in perfection, a stately tree with tall erect trunk, gradually tapering from root to summit, and horizontal branches springing at irregular intervals from the stem, and in old trees often becoming more or less drooping, but rising again towards the extremities; the branchlets or side shoots, very slender and pendulous, are pretty thickly studded with the spurs each bearing a fascicle of thirty or more narrow linear leaves, of a peculiar bright light green when they first appear in the spring, but becoming of a deeper hue when mature.

00They agree with primitive nephridia in being of ectodermic origin, in consisting of perforated cells in linear series, and in having no communication with the coelom.

00The rice plant is an annual grass with long linear glabrous leaves, each provided with a long sharply pointed ligule.

00The slit of the collimator confines the light to a nearly linear source, the beam diverging from each point of the source being subsequently made parallel by means of a lens.

00In order to record line spectra it is by no means necessary that the receiving instrument (bolometer or radiometer) should be linear in shape, for the separation of adjacent lines may be obtained if the linear receiver be replaced by a narrow slit in a screen placed at the focus of the condensing lens.

00From the known coefficients of compressibility and thermal expansion we find that V may be represented by the linear equation V=1.000+0.0008 A, where A is the lowering of the freezing point below oยฐ.

00Its coefficient of linear expansion by heat is 0.0000222 (Richards) or 0.0000231 (RobertsAusten) per 1ยฐ C. Its mean specific heat between oยฐ and iooยฐ is 0.227, and its latent heat of fusion loo calories (Richards).

00cephalotes, ft., is a larger plant, with tufts of linear lance-shaped leaves, and abundant globular heads of deep rose flowers, in June and July.

00The Carboniferous forerunners of the tiny club-moss were then great trees with dichotomously branching stems and crowded linear leaves, such as Lepidodendron (with its fruit cone called Lepidostrobus), Halonia, Lepidophloios and Sigillaria, the largest plants of the period, with trunks sometimes 5 ft.

00If the linear velocity of the cups in feet a second is V 1, and the linear velocity of the jet is V2, then the velocity of the jet relative to the cup is V2 - V1 feet a second, and if the whole energy of the water is to be given up to the cups, the water must leave the cup with zero absolute velocity.

00This will become zero if V 1 is 2V 2, that is, if the linear velocity of the cupcentres is one-half that of the jet of water impinging upon them.

00The form of the leaf shows a very great variety ranging from the narrow linear form with parallel sides, as in grasses or the needle-like leaves of pines and firs to more or less rounded or orbicular - descriptions of these will be found in works on descriptive botany - FIG.

00The first leaves produced by the young plant are linear,the second are sagittate and hastate, the third are rounded-cordate and the next are orbicular.

00The flow of heat may still be linear if the horizontal layers of the soil are of uniform composition, but the quantity flowing through each layer is no longer the same.

00This gives an average value of the conductivity over the range, but it is better to observe the temperatures at three distances, and to assume k to be a linear function of the temperature, in which case the solution of the equation is still very simple, namely, 0+Ze6 2 =a log r+b, (3) where e is the temperature-coefficient of the conductivity.

00The chief uncertainty of this method is the area from which the heat is collected, which probably exceeds that of the central column, owing to the disturbance of the linear flow by the projecting bulb of the calorimeter.

00As in the case of steady-flow methods, by far the simplest example to consider is that of the linear flow of heat in an infinite solid, which is most nearly realized in nature in the propagation of temperature waves in the surface of the soil.

00The equation of the method is the same as that for the linear flow with the addition of a small term representing the radiation loss.

00Although this correction should be made if the definition were strictly followed, it is more convenient in practice to include the small effect of linear expansion in the temperature-coefficient in the case of solid bodies.

00Smoothing them, Carlheim-Gyllenskold gives f = too' -7.3c as the most probable linear relation between c, the amount of cloud, and f, the frequency, assuming the latter to be loo when there is no cloud.

00Those of the earliest period, the lower limit of which is put about 150o B.C., are aeneolithic, metal being, however, rare and only found in the form of small ornaments; pottery with linear decoration is abundant.

00The earlier Choripetalae embraces the Polypetalae and Monochlamydae of the French systems. It includes 21 series, and is an attempt to arrange as far as possible in a linear series those orders which are characterized by absence or freedom of petals.

00They range from subjects of the homeliest and most mirthful realism to others serious and devout, and from literal or almost literal transcripts of natural form to the most whimsically abstract combinations of linear pattern and tendril .and flourish.

00The two carried out some of the earliest thermochemical investigations, devised apparatus for measuring linear and cubical expansions, and employed a modification of Joseph Black's ice calorimeter in a series of determinations of specific heats.

00Hottonia (water violet) is a floating water plant with submerged leaves cut into fine linear segments.

00In some cases these transitions may be unsatisfactory and forced; it is apparent that the linear development from " being " to the " idea " is got by transforming into a logical order the sequence that has roughly prevailed in philosophy from the Eleatics; cases might be quoted where the reasoning seems a play upon words; and it may often be doubted whether certain ideas do not involve extra-logical considerations.

00Other bees, the species of Osmia for example, choose the hollow stem of a bramble or other shrub, the female forming a linear series of cells in each of which an egg is laid and a supply of food stored up. J.

00focal length a discrepancy of I unit in the 3rd decimal place in the above proportional figures would cause a linear error in the focus for that colour of only about .025 in., so that the largest deviation implied by the tables would be a focus for the extreme violet H ray about 037 longer than the normal.

00But if it be possible to mount a fixed telescope by which a solar or stellar image can be formed within a laboratory we give the following advantages: - (1) There is no mechanical limit to the length of the telescope; (2) the clockwork and other appliances to move the mirror, which reflects the starlight along the axis, are much lighter and smaller than those required to move a large telescope; (3) the observer remains in a fixed position, and spectroscopes of any weight can be used on piers within the laboratory; and (4) the angular value of any linear distance on a photographic plate can be determined by direct measurement of the distance of the photographic plate from the optical centre of the object-glass.

00long including the tail, and prettily marked with dark longitudinal stripes, and spots which have a distinctly linear arrangement.

00This Formula Has Since Been Very Generally Applied Over The Whole Range Oยฐ To 200ยฐ C., But The Experiments Could Not In Reality Give Any Information With Regard To The Specific Heat At Temperatures Below 100ยฐ C. The Linear Formula Proposed By J.

00When the parallax of a star is known, we are able to infer from its proper motion its actual linear speed in miles per hour, in so far as the motion is transverse to the line of sight.

00Regarded as a linear velocity, the parallactic motion is the same for all stars, being exactly equal and opposite to the solar motion; but its amount, as measured by the corresponding angular displacement of the star, is inversely proportional to the distance of the star from the earth, and foreshortening causes it to vary as the sine of the angular distance from the apex.

00To arrive at some estimate of the speed of the solar motion, we may consider the motions of those stars whose parallaxes have been measured, and whose actual linear speed is accordingly known (disregarding motion in the line of sight).

00Somewhere, apparently, in the 4th millennium B.C., we begin to find inscriptions written on clay, in an almost linear script, in the Sumerian tongue.

00The step he took is really nothing more than the kinematical principle of the composition of linear velocities, but expressed in terms of the algebraic imaginary.

00In its essential nature a set is a linear function of any number of " distinct " units of the same species.

00Even in Hamilton's earlier work it was shown that all such questions were reducible to the solution of linear equations in quaternions; and he proved that this, in turn, depended on the determination of a certain operator, which could be represented for purposes of calculation by a single symbol.

00This has a reciprocal Q -1= p-r = qq-1 - wp1 rq1, and a conjugate KQ (such that K[QQ'] = KQ'KQ, K[KQ] = Q) given by KQ = Kq-}-rlKp+wKr; the product QQ' of Q and Q' is app'+nqq'+w(pr'+rq'); the quasi-vector RI - K) Q is Combebiac's linear element and may be regarded as a point on a line; the quasi-scalar (in a different sense from the rest of this article) 2(1+K)Q is Combebiac's scalar (Sp+Sq)+Combebiac's plane.

00Again a unit point p. may be regarded as by multiplication changing (a) from octonion to point-plane-scalar, (b) from point-plane-scalar to octonion, (c) from plane-scalar to linear element, (d) from linear element to plane-scalar.

00(II) This relation gives a linear formula for the variation of the total heat, a result which agrees in form with that found by Regnault for steam, and implies that the coefficient of t in his formula should be equal to the specific heat S of steam.

00Perry (Steam Engine, p. 580), assuming a characteristic equation similar to Zeuner's (which makes v a linear function of the temperature at constant pressure, and S independent of the pressure), calculates S as a function of the temperature to satisfy Regnault's formula (10) for the total heat.

00(20) from which we deduce that the ratio 0'/0" of the temperatures at which the vapour-pressures are the same is a linear function of the temperature 0' of one of the substances.

00Under these conditions both S and s may be regarded as approximately constant, so that L is a linear function of the temperature.

00A formula of the same type was given by Athenase Dupre (Theorie de chaleur, p. 96, Paris, 1869), on the assumption that the latent heat was a linear function of the temperature, taking the instance of Regnault's formula (io) for steam.

00Evans for a photograph of the Cretan linear script, and to Professors A.

00Another proof will be indicated later (~ 8) in connection with the properties of the linear complex.

00For the conditions of equilibrium of the forces on each pin furnish vi equations, viz, two for each point, which are linear in respect of the stresses and the extraneous forces.

00The translation parallel to this axis is lox + mly + nhz (Xf + uv + vi) Ic. (8) The linear magnitude which measures the ratio of translation to rotation in a screw is called the pitch.

00if a point of the body be restricted to lie on a given surface, the mathematical expression of this fact leads to a homogeneous linear equation between the infinitesimais f, 77, i, X,, s, v, say A~+Bi7+Ci~+FX+GfL+Hv=O.

00The complex is therefore of the type called linear (in relation to the degree of this locus).

00This is the equation of 1 linear complex (cf.

00It is assumed that the form can be sufficiently represented by a plane curve, that the stress (tension) at any point P of the curve, between the two portions which meet there, is in the direction of the tangent at P, and that the forces on any linear element s must satisfy the conditions of equilibrium laid down in I.

00If we construct the quadric Axi+By2+Czi 2Fyz2Gzx 2HXy = M~4, (3c~) where e is an arbitrary linear magnitude, the intercept r which it makes on a radius drawn in the direction X, u, v is found by putting x, y, z=Ar, ur, Pr.

003I) that the linear moment of each particle about the line may be found by means of a funicular polygon.

00Since the equations (3~) are linear, these two solutions are to be superposed.

00Thus taking nny point 0 as base, we have first a linear momentum whose components referred to rectangular axes through 0 are ~(m~), Z(m~), ~(mb); - (I)

00The linear momentum is the same as if the whole mass wer(concentrated at the centre of mass G, and endowed with thi velocity of this point.

00At the instant t+t5t the momenta of the system are equivalent to a linear momentum represented by a localized vector ~(m).(U+U) in a line through G tangential to the path of G, together with a certain angular momentum.

00The first of these may be called the Principle of Linear Momentum.

00If there are no extraneous forces, the resultant linear momentum is constant in every respect.

00If extraneous forces act, it is seen in like manner that the resultant linear momentum of the system is in any given time modified by the geometric additiofi of the total impulse of the extraneous forces.

00On this basis we can predicate the principles of linear and angular momentum, as in 15.

00and so express that the rate of change of the linear momentum in any fixed direction (e.g.

00If we integrate with respect to I between fixed limits, we obtain the principles of linear and angular momentum in the form previously given, Hence, whichever form of postulate we adopt, we are led to the principles of linear and angular momentum, which form in fact the basis of all our subsequent work.

00The components of linear momentum are then M~, Mi, and the angular momentum relative to G as base is JO, where M is the mass and I the moment of inertia about G.

00It may also be deduced from the principles of linear and angular momentum as embodied in the equations (9).

00The components ~, il, ~ of linear momentum are a~ aT

00j To prove these, we may take fixed axes Ox, Oy, Oz coincident with the moving axes at time t, and compare the linear and angular momenta E+E, ~ ~ ?~+~X, u+u, v+~v relative to the new position of the axes, Ox, Oy, Oz at time t+t with the original momenta ~, ~ ~, A, j~i, v relative to Ox, Oy, Oz at time t.

00This solution, taken by itself, represents a motion in which each particle of the system (since its displacements parallel to Cartesian co-ordinate axes are linear functions of the qs) executes a simple vibration of period 21r/u.

00The series of equations of the type (3) is then replaced by a single linear partial differential equation, or by a set of two or three such equations, according to the number of dependent variables.

00To determine the free oscillations we assume a time factor e~1 the equations then become linear differential equations between the dependent variables of the problem and the independent variables x, or x, y, or x, y, 1 as the case may be.

00Let the angular velocity of the rotation be denoted by a=dO/dt, then the linear velocity of any point A at the distance r from the axis is or; and the path of that point is a circle of the radius r described about the axis.

00Let v~ denote the linear velocity of the point C. Then vc=a.CF=-y.CG -

00Let yr be the linear velocity of the point E fixed in the plane of axes AOB.

00Now, as the line of contact OT is for the instant at rest on the rolling cone as well as on the fixed cone, the linear velocity of the point E fixed to the plane AOB relatively to the rolling cone is the same with its velocity relatively to the fixed cone.

00Application to a Shifting Piece and a Turning Piece.Let a shifting piece be connected with a turning piece, and at a given instant let ai be the angular velocity of the turning piece, r~ the common perpendicular of its axis of rotation and the line of connection, Oi the angle made by the line of connection with the plane of rotation, Oi the angle made by the line of connection with the direction of motion 01 the shifting piece, v2 the linear velocity -of that piece.

00That the linear velocity of a shifting piece in rolling contact with a turning piece is equal to the product of the angular velocity of the turning piece by the perpendicular distance from its axis to a pair of points of contact.

00Let a small body of the weight w undergo translation in a circulai path of the radius p, with the angular velocity of deflexion a, so that the common linear velocity of all its particles is v=ap. Then the actual energy of that body is WV2/2g = Waip2/2g.

00The principles according to which variations of angular velocity store and restore energy are the same as those of 117, only substituting moment of inertia for mass, and angular for linear velocity.

00These investigations led him to the announcement of the fundamental law of action between elements of current, or currents in infinitely short lengths of linear conductors, upon one another at a distance; summed up in compact expression this law states that the action is proportional to the product of the current strengths of the two elements, and the lengths of the two elements, and inversely proportional to the square of the distance between the two elements, and also directly proportional to a function of the angles which the line joining the elements makes with the directions of the two elements respectively.

00evidence of continuity comes from the peculiar Cypriote script, a syllabary related to the linear scripts of Crete and the south Aegean, and traceable in Cyprus to the Mycenaean age.'

00Its specific heat is o 0899 at 0ยฐ C. and 0.0942 at 10o; the coefficient of linear expansion per 1ยฐ C. is o o01869.

00For infinitely distant objects the radius of the chromatic disk of confusion is proportional to the linear aperture, and independent of the focal length (vide supra," Monochromatic Aberration of the Axis Point "); and since this disk becomes the less harmful with an increasing image of a given object, or with increasing focal length, it follows that the deterioration of the image is proportional to the ratio of the aperture to the focal length, i.e.

00If r be the number of quotients in the recurring cycle, we can by writing down the relations connectin g the successive p's and q's obtain a linear relation connecting p nr +m, t'(n-1)r +m, +m in which the coefficients are all constants.

00We Will Assume That When, As In Most Cases, Viscosity Maybe Neglected, The Mass (M) Of A Drop Depends Only Upon The Density (V), The Capillary Tension (T), The Acceleration Of Gravity (G), And The Linear Dimension Of The Tube (A).

00Its specific heat is 0.05701 (Regnault) or 0.0559 (Bunsen); its coefficient of linear expansion is 0.0000-1921.

00The form of the body is very various: some are rounded or oval, others elongated, almost linear; some are covered with warty protuberances, whilst others are smooth and shining, often with a metallic lustre.

00It is usually the case that the secondary caustic is easier to determine than the caustic, and hence, when determined, it affords a ready means for deducing, the primary caustic. It may be shown by geometrical considerations that the secondary caustic is a curve similar to the first positive pedal of the reflecting curve, of twice the linear dimensions, with respect to the luminous point.

00In this work he introduced the use of linear functions in place of the ordinary co-ordinates; he also made the fullest use of the principles of collineation and reciprocity.

00Lynxes are found in the northern and temperate regions of both the Old and New World; they are smaller than leopards, and larger than true wild cats, with long limbs, short stumpy tail, ears tufted at the tip, and pupil of the eye linear when contracted.

00They are hypogynous, and have long and very delicate filaments, and large, linear or oblong two-celled anthers, dorsifixed and ultimately very versatile, deeply indented at each end, and commonly exserted and pendulous.

00On the other, posterior, side of the grain is a more or less evident, sometimes punctiform, sometimes elongated or linear mark, the hilum, the place where the ovule was fastened to the wall of the ovary.

00He has used all his acquired science of linear and aerial perspective to create an almost complete illusion to the eye, but an illusion that has in it nothing trivial, and in heightening our sense of the material reality of the scene only heightens its profound spiritual impressiveness and gravity.

00Leonardo sought to achieve that conquest and at the same time to carry the old Florentine excellences of linear drawing and psychological expression to a perfection of which other men had not dreamed.

00As a stonemason's measure, a "perch" =1 linear perch in length by 12 ft.

00I) on the branched or unbranched stem; (b) the growth of the main stem through the female flower; (c) the presence of a prominent single vein in the linear pinnae; (d) the structure of the female flower, which is peculiar in not having the form of a cone, but consists of numerous independent carpels, each of which bears two or more lateral ovules.

00Leaves simple, small, linear or short and scale-like, usually persisting for more than one year.

00the needle-form represented by Pinus, Cedrus, Larix, &c., the linear flat or angular leaves, appressed to the branches, of Thuja, Cupressus, Libocedrus, &c. The flat and comparatively broad leaves of Araucaria imbricata, A.

00The long linear leaves of some species of Podocarpus, in which the lamina is traversed by a single vein, recall the pinnae of Cycas; the branches of some Dacrydiums and other forms closely resemble those of lycopods; these superficial resemblances, both between different genera of conifers and between conifers and other plants, coupled with the usual occurrence of fossil coniferous twigs without cones attached to them, render the determination of extinct types a very unsatisfactory and frequently an impossible task.

00Linear, 109 Xearth's equatorial diameter =864,000 M.

00the linear equivalents of the above equations in the following manner.

00From this first theca originates a second, opening in the same direction, and from the second a third, and so on, in a continuous linear series until the polypary is complete.

00(iii) Finally, there are the cases of linear measurement, where it is theoretically possible to find, by geometrical methods, an exact submultiple of a given unit, but both the unit and the submultiple are not really concrete objects, but are spatial relations embodied in objects.

00a curve is of the first order, second order, third order, &c., according as it is represented by an equation of the first order, ax+by+c = o, or say (1 x, y, 1) = o; or by an equation of the second order, ax 2 +2hxy+by e +2fy+2gx+c=o, say (*I x, y, 1) 2 =o; or by an equation of the third order, &c.; or what is the same thing, according as the equation is linear, quadric, cubic, &c.

00A linear relation a+bn+c'=o between these co-ordinate determines a point, viz.

00In particular if D =o, that is, if the given curve be unicursal, the transformed curve is a line, 4 is a mere linear function of 0, and the theorem is that the co-ordinates x, y, z of a point of the unicursal curve can be expressed as proportional to rational and integral functions of 0; it is easy to see that for a given curve of the order m, these functions of 0 must be of the same order m.

00The long, sparingly branched stem bore at the somewhat swollen nodes whorls of six to eighteen wedge-shaped or linear leaves, which did not alternate in successive whorls.

00What is known at present, while it does not indicate the phylogeny of the Lycopodiales, at least shows that such living orders as Lycopodiaceae and Selaginellaceae cannot be regarded as forming a linear series.

00The leaves are simple and linear in Pilularia, but in Marsilia bear a pinnate four-lobed lamina.

00This concept is extended to algebra: since a line, surface and solid are represented by linear, quadratic and cubic equations, and are of one, two and three dimensions; a biquadratic equation has its highest terms of four dimensions, and, in general, an equation in any number of variables which has the greatest sum of the indices of any term equal to n is said to have n dimensions.

00It further appears that a determinant is a linear function' of the elements of each column thereof, and also a linear function of the elements of each line thereof; moreover, that the determinant retains the same value, only its sign being altered, when any two columns are interchanged, or when any two lines are interchanged; more generally, when the columns are permuted in any manner, or when the lines are permuted in any manner, the determinant retains its original value, with the sign + or - according as the new arrangement (considered as derived from the primitive arrangement) is positive or negative according to the foregoing rule of signs.

00By what precedes it appears that there exists a function of the n 2 elements, linear as regards the terms of each column (or say, for shortness, linear as to each column), and such that only the sign is altered when any two columns are interchanged; these properties completely determine the function, except as to a common factor which may multiply all the terms. If, to get rid of this arbitrary common factor, we assume that the product of the elements in the dexter diagonal has the coefficient + 1, we have a complete definition of the determinant, and it is interesting to show how from these properties, assumed for the definition of the determinant, it at once appears that the determinant is a function serving for the solution of a system of linear equations.

00Reverting to the system of linear equations written down at the beginning of this article, consider the determinant ax+by+cz - d,b,c a' x+b' y+c'z - d', b', c" a"x+b"y+c"z - d", b", c" it appears that this is viz.

00the second and third terms each vanishing, it is a, b, c a', c', , a", bn c",, But if the linear equations hold good, then the first column of the 1 The expression, a linear function, is here used in its narrowest sense, a linear function without constant term; what is meant is that the determinant is in regard to the elements a, a', a", ..

00=x d,b,c d ', c', , d "b" c",, original determinant is = o, and therefore the determinant itself is = o; that is, the linear equations give x'a,b,c - d,b,c =o; a', b', c' d', b', c' a", b', c" d", b", c" which is the result obtained above.

00Join to the original equations the new equation ax+(33'+yz=8; a like process shows that, the equations being satisfied, we have a,a,'Y,S a,b,c,d a' b ' c ' d'a",, b", c", or, as this may be written, a,13,y - 8a,b,c =07 a,b,c,d a',b'c' a' b r c r d'a", b N c" a", b", c", d",, , which, considering b as standing herein for its value ax+0y+yz, is a consequence of the original equations only: we have thus an expression for ax+/3y+yz, an arbitrary linear function of the unknown quantities x, y, z; and by comparing the coefficients of a, /3, y on the two sides respectively, we have the values of x, y, z; in fact, these quantities, each multiplied by a,b,c a' b ' c',, a N b r/ c", b", are in the first instance obtained in the forms a,b,c,d a 'b' c' d'a", b N' c N' dN,, , respectively.

00To indicate the method of proof, observe that the determinant on the left-hand side, qua linear function of its columns, may be I The reason is the connexion with the corresponding theorem for the multiplication of two matrices.

00The germ of the theory of determinants is to be found in the writings of Gottfried Wilhelm Leibnitz (1693), who incidentally discovered certain properties when reducing the eliminant of a system of linear equations.

00Its specific gravity is given variously from 5.395 to 5'959; its specific heat is 0.083, and its coefficient of linear expansion 0.0000-0559 (at 40ยฐ C.).

00Similar phenomena occur in the case of many other couples, and it is found that the thermoelectric power p is not in general a constant, and that the simple linear formula (I) is applicable only for small differences of temperature.

00was not a linear function of the temperature difference.

00If we suppose that s is not quite constant, but increases or diminishes slightly with change of temperature according to a linear formula s=50-1-2cT (in which so represents the constant part of s, and c may have either sign), we obtain a more general formula which is evidently the sum of the two previous solutions and may be made to cover a greater variety of cases.

00According to this formula, the Peltier effect is a linear function of the temperature.

00This might be appropriately represented, as already suggested, by a linear formula s=so - cT.

00They are generally of a more or less oval or elliptical form, or they may be globular, as in Mercurialis annua; at other times linear or clavate, curved, flexuose, or sinuose, as in bryony and gourd.

00The free lamina, however, was always considerably more developed than in the recent family; in form it was usually linear or narrowly lanceolate.

00In Asterophyllites., the generic distinction of which from Annularia is not always clear, the narrow linear leaves are in crowded whorls, and the ultimate branches distichously arranged; in the Calamocladus of Grand' Eury - characteristic of the Upper Coal Measures - the whorls are more remote, and the twigs polystichous in arrangement.

00As examples of these doubtful forms may be mentioned Thinnfeldia, characteristic of Rhaetic and Lower Jurassic rocks; Dichopteris, represented by some exceptionally fine Jurassic specimens, described by Zigno, from Italy; and Ctenis, a genus chiefly from Jurassic beds, founded on pinnate fronds like those of Zamia and other Cycads, with linear pinnae characterized by anastomosing veins.

007), which is characterized by tripinnate fronds with short linear ultimate segments, bearing a single row of A, Summit of petiole.

0015, 4 and 7), as investigated in English, French, Italian, and American specimens, may be briefly described as a short lateral shoot or peduncle, arising in a leaf-axil and terminating in a bluntly rounded apex, bearing numerous linear bracts enclosing a central group of appendages, some of which consist of slender pedicels traversed by a vascular strand and bearing a single terminal ovule enclosed in an integument, which forms a distal canal or micropyle.

00A number of hairy linear bracts enclose the whole; internal to these occur 12 to 20 crowded pinnate leaves (sporophylls), with their apical portions bent over towards the axis of the flower, the bases of the petioles being fused laterally into a disk surrounding the base of the conical receptacle.

00I simply did not want to engage with a linear narrative.

00In 2006 a further three linear accelerators will be bought to bring the total at the new West of Scotland Cancer Center to 11.

00Figure 4. [below] Linear regression of plasma ACTH following fasting stress on daily fatigue severity rating.

00Study the mode of CO 2 adsorption on the hydrotalcite surface, to determine whether linear or bridged adsorption on the hydrotalcite surface, to determine whether linear or bridged adsorption occurs.

00Outline content: The module starts with review of linear algebra.

00While linear methods have proven invaluable over many years in generating first-order approximations, they are severely limited in both qualitative and quantitative predictions.

00A simple artifice is used to represent any spanwise twist distribution by a combination of linear segments.

00assemblyting valves have solid stainless steel equal percentage or linear characteristic control plug and spindle assemblies with stainless steel screw in body seats.

00Sharaf al-Din al-Tusi (born 1201) invented the linear astrolabe.

00Users of industrial automation are more likely to specify linear motor systems rather than separate linear components.

00Parametric analyzes showed a significant linear relationship between increasing proportions of horizontally averted gaze and increased rCBF in the MPF cortex.

00They grow in the upper axils of the linear or lance shaped, deep green arching leaves.

00Designs include linear phase filters, a wideband phase shifter, and chirp filters with fractional bandwidths exceeding 100% .

00It never falls into a linear plot, which remains the bane of the script writer.

00Whether the force is fighting in a linear or a nonlinear battlefield, the factors of METTT will determine location of the FARP.

00The corrugated cloth surround is coated with a damping material to make diaphragm movement more linear and discourage cone breakup.

00See also categorical predictor variables, design matrix; or General Linear Models.

00Thus, linear causality is the bedrock of technology and social contracts.

00Following a section on normed linear spaces, we will consider periodic approximation using classical Fourier series.

00We therefore suggest that the linear attenuation coefficient be treated as a separate unit.

00In the longitudinal cohorts, the rate of change was determined by linear regression.

00There are two main design proposals for the future linear collider.

00A major component of modern linear accelerators which assists this technique is the multi-leaf collimator (MLC ).

00Discrete Random Variables: mean and variance; linear combinations; covariance and correlation.

00Mechanical coolers used in space are generally based upon a linear compressor with clearance seals.

00cone spectral sensitivities are defined as linear combinations of the Stiles and Burch (1959) 10-deg CMFs,, and.

00This is done by coding DTDs and integrity constraints with linear constraints on the integers.

00Should we think in terms of a linear expansion of western consumerism ending in global convergence?

00It's a gameplay contrivance that keeps you on a linear path through each area.

00Modern power recorders tend to have linear convertors but with a high bit number.

00convoluted plot has been replaced by a far more linear slasher storyline.

00mechanical coolers used in space are generally based upon a linear compressor with clearance seals.

00Two voxel-based similarity measures, the linear correlation coefficient and the entropy correlation coefficient, are used.

00Medium to large linear blocks with large rectangular blocks in the lower stage of the tower; irregular coursing.

00This approach has been proven to be highly resistant against many attacks such as differential and linear cryptanalysis.

00History is linear and not cyclical and going on for ever.

00defensive linear earthwork.

00discrepancy vector is B, a linear combination of the elements of B.

00The series of linear earthworks, just north of the fort, also merit further investigation.

00Most likely, Wansdyke was a defensive linear earthwork.

00efficient parallel algorithms for solving Linear Algebra Problems.

00The program tests for linear dependence by calculating the eigenvalues of the overlap matrix.

00Course Description Solving linear systems Ax = b or finding the eigenvalues of a matrix may appear to be trivial tasks.

00The poled microstructure determines the linear elastic, dielectric and piezoelectric moduli.

00The current work assumes linear elasticity which is a weak model for real soft tissue.

00Von Koch's first results were on infinitely many linear equations in infinitely many unknowns.

00Identifying strategies by some kind of behavioral equivalence we expect to receive a linear category.

00Wednesday 2pm, 3rd May, 2006 Title: parameter estimation of Non Linear Pathway Models.

00Simple methods define an arbitrary law, such as an inverse square relationship, or a linear falloff.

00The problem being that Flash downloads in a linear fashion from frame 1 through to the last frame.

00Whence the concept of linear karma dominates the foreground of ethics.

00frustration factors, with some bland level maps, linear routes and samey gunfights.

00generalized linear modeling.

00Biographical data, treatment patterns, and clinical impressions were analyzed by the generalized linear model and generalized estimating equations method.

00globules shown on Figure (6) have linear dimensions which vary from 2 microns to about 13 microns.

00The most obvious effect is that it stimulates linear growth of the skeleton.

00Solution method: A molecular Hamiltonian describing carbon dioxide, which is a linear triatomic, is diagonalized.

00In addition there are numerous stone circles and several possible henges and linear stone alignments.

00Also focuses on the decipherment and decoding of cuneiform, Egyptian hieroglyphs, Linear B, and Mayan Glyphs.

00homogeneous, linear partial differential equations.

00The free layer is now being swept in a hard axis, and shows reduced hysteresis, and a more linear response.

00intaglio printmaking, engraving and etching, were purely linear in approach.

00integrity constraints with linear constraints on the integers.

00In turn point counting gave lower values than linear intercepts.

00Such metabolic pathways are not linear but often intersect to form a complex network.

00Invariant Theory of Finite Groups This introductory lecture will be concerned with polynomial invariant Theory of Finite Groups This introductory lecture will be concerned with polynomial invariants of finite groups which come from a linear group action.

00The set of linear equations is then solved by means of a matrix inversion.

00The later mechanics chapters again logically separate content into linear and angular kinematics and kinetics.

00It also shows how causation distinguishes time from space, makes time linear, gives it its direction and enables us to perceive it.

00linear problems usually involve linear algebra in their solution.

00linear equations in infinitely many unknowns.

00linear regression.

00linear accelerator can treat a tumor with around 1mm precision.

00linear interpolation.

00linear earthwork.

00As individual elements we choose piecewise linear or smooth maps, and compare results obtained in both cases.

00Agents of the Enlightenment thought history was strictly linear, he says.

00First, they have tended to see the global in terms of an essentially linear process, or linear processes, of globalization.

00Memory usage is approximately linear in RE size, and largely insensitive to RE complexity, except for bounded repetitions.

00This fall in heart rate is fairly linear, decreasing by approximately 1 beat per minute per year.

00Therefore, these algorithms rely on a locally linear internal model of the black box.

00To rectify this I relocated both handles to the bottom edge in which the two points remain linear.

00Have you established that the relationship is linear or can be transformed to become linear?

00If you place the dots with even spacing from frame to frame, the motion will look linear.

00linear in the size of the graph ).

00linear in the dose range tested.

00linear with respect to the size of set of samples.

00In order to deliver the radiation to the patient we use devices called medical linear accelerators, or " linacs " .

00The Radiotherapy Department has eight linear accelerators and a Pantak superficial x-ray therapy unit.

00Rail Gun: This highly accurate weapon uses a linear accelerator to throw a depleted uranium slug at an incredible rate of speed.

00Szilard's ideas included the linear accelerator, cyclotron, electron microscope, and nuclear chain reaction.

00Oxford Radcliffe Hospitals NHS Trust is to get two new linear accelerators to cut delays and improve treatment for cancer patients in Oxfordshire.

00linear accelerators at the same time before.

00Even where new linear accelerators have been purchased, often with Lottery money, there are not always the staff to operate them.

00This implied that there should be 48 additional linear accelerators in the UK, of which 22 would replace old cobalt machines.

00Core modules include mathematical ecology, mathematical biology and medicine and applied linear algebra.

00Areas covered include linear algebra, optimization, quadrature, differential equations, regression analysis, and time series analysis.

00Knowledge of Maple as a tool for doing linear algebra.

00Here's an example describing *one* type of process associated with matrices, that I've used in teaching linear algebra.

00In numerical linear algebra he developed backward error analysis methods.

00Such a differential structure, expressing the local versus global nature of brain structure, is lacking in adaptive linear algebra.

00We shall concentrate on a particularly nice class of codes called linear codes, a beautiful application of elementary linear algebra.

00linear equationribed by linear dynamic equations subject to linear inequalities involving real and integer variables.

00linear equationms are normally modeled using linear integro-differential equations.

00The Babylonians studied problems which lead to simultaneous linear equations and some of these are preserved in clay tablets which survive.

00linear equationtudies Fourier Series and their applications to the solution of boundary value problems associated with certain linear partial differential equations.

00linear (induction) motortes an innovative three-phase linear servo motor with a precision guide assembly and a displacement encoder.

00Topics will include a review of multiple linear regression and will then focus on logistic regression and multinomial logistic regression.

00These include complete sets of curriculum material for various college and university level courses, including calculus, linear algebra and engineering mathematics.

00metreking on behalf of Defense Estates, Colas installed sealant to over 150,000 linear meters of concrete joints.

00mimic system allows an author to define a linear route through a set of documents.

00Analysis involved the use of a longitudinal linear model ANCOVA.

00As a general principle, off-line moorings are always preferable to creating new linear moorings.

00multiplication of matrices is defined so as to agree with what happens when we combine, or compose linear transformations.

00nightingale wards are being removed and today we have had the wonderful announcement that Clatterbridge will be part of the linear accelerator program.

00We work mostly with nucleic acids here, incorporating modified nucleosides into linear as well as small circular DNA/RNA chains.

00linear odes and systems of first order linear ODEs form the first topic to be treated in depth.

00This powerful method is applicable to linear odes or systems of linear ODEs with constant coefficients.

00oesophagusory Weiss tear is a linear mucosal tear near the gastro-oesophageal junction either in the distal esophagus or proximal stomach.

00overload protection for circuit components or limit the force or torque exerted by a linear actuator or rotary motor.

00parsimonious representations, but generally are unable to reproduce features missed by linear models.

00Topiramate exhibits linear pharmacokinetics with low levels of protein binding.

00This means that the action will be linear in beta, delta phi, and alpha.

00A new system has also been implemented for supporting the moving platen on the machine base, using linear bearings.

00The marked linear polarization of the emission from the condensate is also measured.

00Target materials are low and linear low density polyethylene, but the new masterbatches can also be used with HDPE and PP.

00polynomial invariants of finite groups which come from a linear group action.

00Part 2: Duality theory The theory of duality for linear programming.

00promote learning with respect to linear programming.

00purine absorbed is linear.

00