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modulus

It will be observed that in the first process the value of the modulus is in fact calculated from the formula.

10By solving the equations of transformation we obtain rE1 = a22x1 - a12x1, r = - a21x1 + allx2, aua12 where r = I = anon-anon; a21 a22 r is termed the determinant of substitution or modulus of transformation; we assure x 1, x 2 to be independents, so that r must differ from zero.

11In the theory of forms we seek functions of the coefficients and variables of the original quantic which, save as to a power of the modulus of transformation, are equal to the like functions of the coefficients and variables of the transformed quantic. We may have such a function which does not involve the variables, viz.

11Modulus.

00For the substitution rr xl =A 11 +1 2 12, 52=A21+ï¿½2E2, of modulus A1 ï¿½i = (Alï¿½.2-A2ï¿½1) = (AM), A 2 ï¿½2 the quadratic form a k xi -1-2a 1 x i x 2 +a 2 4 = x =f (x), becomes A41 +2A1E16 =At = OW, where Ao = aoA i +2a1AiA2 +a2Az, _ _ A 1 = ao A lï¿½l +ai(A1/.22+A2ï¿½1) +7,2X2/22, A2 = aoï¿½l +2a1ï¿½1/ï¿½2 +a 2ï¿½2 ï¿½ We pass to the symbolic forms a:= (aixi+a2x2) 2, A 2 = (A 151+ A 26) 2/ by writing for ao, al, a2 the symbols ai, a 1 a 2, a?

00= (A11+A22)n by the substitutions 51 = A l, E1+ï¿½1 2, 52 = A2E1+ï¿½2E2, the umbrae Al, A2 are expressed in terms of the umbrae al, a 2 by the formulae A l = Alai +A2a2, A2 = ï¿½la1 +ï¿½2a2ï¿½ We gather that A1, A2 are transformed to a l, a 2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, (A / .c).

00If u, a quantic in x, y, z, ..., be expressed in terms of new variables X, Y, Z ...; and if, n,, ..., be quantities contragredient to x, y, z, ...; there are found to exist functions of, n, ?, ..., and of the coefficients in u, which need, at most, be multiplied by powers of the modulus to be made equal to the same functions of E, H, Z, ...

00The transformation to the normal form, by the solution of a cubic and a quadratic, therefore, supplies a solution of the quartic. If (Xï¿½) is the modulus of the transformation by which a2 is reduced to 3 the normal form, i becomes (X /2) 4 i, and j, (Ap) 3 j; hence?

00Hesse's canonical form shows at once that there cannot be more than two independent invariants; for if there were three we could, by elimination of the modulus of transformation, obtain two functions of the coefficients equal to functions of m, and thus, by elimination of m, obtain a relation between the coefficients, showing them not to be independent, which is contrary to the hypothesis.

00Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution xl = X11 + J-s12, X 2 = 112 Again, in plane geometry, the most general equations of substitution which change from old axes inclined at w to new axes inclined at w' =13 - a, and inclined at angles a, l3 to the old axis of x, without change of origin, are x-sin(wa)X+sin(w -/3)Y sin w sin ' _sin ax y sin w a transformation of modulus sin w' sin w' The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry.

00We have cos w' = cos w = o and the substitution x 1 =cos OX, -sin 0(2 x 2 = sin OX i +cos 6X2, with modulus unity.

00If the senses of rotation be opposite we have the skew orthogonal substitution x1 =cos0Xi+sinOX2r x 2 = sin °Xicos OX2r of modulus -1.

00The whole of the rod will therefore be subject to a compressive longitudinal stress P, the associated contraction R, expressed as a fraction of the original length, being R = P/M = (B 2 -H2)/87-M, where M is Young's modulus.

002 The results are too numerous to discuss in detail; some of those to which special attention is directed are the following: In Swedish iron and tungsten-steel the change of elastic constants (Young's modulus and rigidity) is generally positive, but its amount is less than 0.5%; changes of Young's modulus and of rigidity are almost identical.

00In nickel the maximum change of the elastic constants is remarkably large, .amounting to about 15% for Young's modulus and 7% for rigidity; with increasing fields the elastic constants first decrease and then increase.

00Referring the reader to the article Elasticity for the theoretical and to the Strength Of Materials far the practical aspects of this subject, we give here a table of the "modulus of elasticity," E (column 2), for millimetre and kilogramme.

00Let E be the bulk modulus of elasticity, defined as increase of pressure = decrease of volume per unit volume where the pressure increase is so small that this ratio is constant, w the small increase of pressure, and - (dy/dx) the volume decrease, then E=e/(- dy/dx) or w Ã†= - dy/dx (I) This gives the relation between pressure excess and displacement.

00But if y is the displacement at A, dy/dx is the extension at A, and the force acting is a pull across A equal to Y&uodyldx, where Y is Young's modulus of elasticity.

00Young's modulus may be obtained for the material of a rod by clamping it in the middle and obtaining the frequency of the fundamental when Y = 412n2p.

00Owing to the yielding of joints when a beam is first loaded a smaller modulus of elasticity should be taken than for a solid bar.

00It follows that the density of the aether must exceed io 18, and its elastic modulus must exceed Io 3, which is only about io s of the modulus of rigidity of glass.

00The second of these relations is an important one, as it shows that from a table of logarithms to base a, the corresponding table of logarithms to base b may be deduced by multiplying all the logarithms in the former by the constant multiplier i/logab, which is called the modulus of the system whose base is b with respect to the system whose base is a.

00The two systems of logarithms for which extensive tables have been calculated are the Napierian, or hyperbolic, or natural system, of which the base is e, and the Briggian, or decimal, or common system, of which the base is io; and we see that the logarithms in the latter system may be deduced from those in the former by multiplication by the constant multiplier /loge io, which is called the modulus of the common system of logarithms.

00The numerical value of this modulus is o 4342944 81 9 03251 82765 11289..

00It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x and cos x, for such a series could only represent a uniform function, and in fact the equation log(I +x) =x -",, x2 +3x 3 -4x 4 + is true only when the analytical modulus of x is less than unity.

00to 276 places of decimals, and deduced the value of log e lo and its reciprocal M, the modulus of the Briggian system of logarithms. The value of the modulus found by Adams is Mo = 0-43429 44 81 9 03251 82765 11289 18916 60508 22 943 97 00 5 80366 65661 14453 78316 58646 4920-8870 77 47292 2 4949 33 8 43 17483 18706 106 74 47 6630-3733 64167 92871 58963 90656 92210 64662 81226 58521 27086 56867 03295 9337 0 86965 88266 88331 16360 773849 0514 28443 48666 76864 65860 85135 56148 212 34 87653 43543 43573 25 which is true certainly to 272, and probably to 273, places (Proc. Roy.

00If the logarithms are to be Briggian all the series in the preceding formulae must be multiplied by M, the modulus; thus, log i o(I +x) = M(x-2x2+3x3-4x4+&c.), and so on.

00The length of 100 divisions of the scale, or the length of the uniform stem the volume of which would be equal to that of the hydrometer up to the zero graduation, Francceur called the "modulus" of the hydrometer.

00lo g i op =o 6640+8.585t/e-4.70(log109/Bo-Mt/6), where t=9 -273, and M =0.4343, the modulus of common logarithms. These formulae are practically accurate for a range of 20° or 30° C. on either side of the melting-point, as the pressure is so small that the vapour may be treated as an ideal gas.

00Modulus of logarithms.

00The constant -r is called the modulus of decay of the oscillations; if it is large compared with 2irfa the effect of friction on the period is of the second order of small quantities and may in general be ignored.

00whenever increases by 4K/cr, when K is the complete elliptic integral of the first kind with respect to the modulus k.

00That least distance is called by Moseley the modulus of stability.

00The following are some of the ratios of the modulus of stability to the depth of the joint which occur in practice :

00Modulus of a Machine.In the investigation of the properties of a machine, the useful resistances to be overcome and the useful work to be performed are usually given.

00The result of this investigation, expressed in the form of an equation between this energy and the useful work, is called by Moseley the modulus of the machine.

00The general form of the modulus may be expressed thus E~U+cb(U, A)+~(A), (54)

00The twist or surface-shear being proportional to the torque, the horse-power can be calculated if the modulus of rigidity of the steel employed is known or if the amount of twist corresponding to a given power has previously been ascertained by direct experiment on the shaft before it has been put in place.

00modulus, a measure, or standard.

00bulk modulus, toxicity, effect on environment, and biodegradability.

00Is this because cast gold has a higher, or lower, modulus of elasticity compared to cobalt chromium?

00Residual measurements carried out on glass/polyester rods showed that the longitudinal tensile modulus remains constant over the entire lifetime of the unidirectional composite.

00elastic modulus, a new local memory can be induced.

00enciphering modulus.

00impact excitation is used to determine the Young's modulus of the coating independent of Poisson's ratio.

00Modulus and strain to failure are measured using an extensometer (mechanical or non-contact ).

00Around t g there is a large fall in the value of the modulus.

00Around T g there is a large fall in the value of the modulus.

00The difference in the shear modulus has been suggested as the main factor in determining the hardness of the multilayer system.

00To obtain a reliable estimate of indentation modulus, each indentation modulus, each indentation experiment was repeated 20 times at 50 µm intervals.

00The distance modulus = the apparent magnitude - absolute magnitude.

00modulus of elasticity of metals in crystal in different directions.

00modulus of the coating independent of Poisson's ratio.

00At least two tests were carried out at each temperature interval and the mean frequency value used to calculate the elastic modulus.

00PPX 7125 has a flexural modulus of 1,340 MPa and an instrumented impact strength of 37 Nm at -30 degC.

00A - Yes, by overcoming the elastic modulus, a new local memory can be induced.

00tensile modulus was reduced by 15% in the same timescale.

00We have managed to explain, and quantify, the origin of the surprising negative gradient of the bulk modulus in this model.

00The transfer function is the complex viscoelastic modulus of the interface.

00modulus silicone.

00modulus blanks.

00modulus carbon fiber was used to strengthen the struts, totalling 1.5 tons in weight.

00modulus checking system in place.

00modulus measurements at low strain levels, uncertainties of better than ~ ± 1% should be readily achievable.

00modulus m and which is the residue r.

00The elastic modulus is defined as a secant modulus at service load condition.

00In reality the values of shear modulus G are much lower.

00To obtain a reliable estimate of indentation modulus, each indentation experiment was repeated 20 times at 50 µm intervals.

00For a gas, the adiabatic bulk modulus B = P, where P is the ambient pressure.

00The ratio of loss and storage modulus, tan d, is also reported.

00The distance modulus = the apparent magnitude - absolute magnitude.

00rhenium alloy that maintains a high modulus of elasticity even at high temperatures.

00Seat them down on a bead of clear low modulus silicone sealer before the screws are inserted.

00secant modulus at service load condition.

00shear modulus G are much lower.

00Fix it to the base unit by bedding the sink onto beads of low modulus silicone.

00The reduction factor for elastic modulus of light gage steel is assumed to be identical to that of carbon steel.

00This sections starts by defining modulus, tensile strength, elongation at break and yield strength on a typical stress-strain graph.

00tangent modulus is given.

00tensile modulus was reduced by 15% in the same timescale.

00Recommendations This method is inherently unsuitable for low modulus polymeric material systems.

00535 British oak '99 The resistance to breaking or rupture of Australian timber is very high; grey iron-bark with a specific gravity of 1.18 has a modulus of rupture of 17,900 lb per sq.

00No Australian timber in the foregoing list has a less modulus than 13,100 lb per sq.

00By solving the equations of transformation we obtain rE1 = a22x1 - a12x1, r = - a21x1 + allx2, aua12 where r = I = anon-anon; a21 a22 r is termed the determinant of substitution or modulus of transformation; we assure x 1, x 2 to be independents, so that r must differ from zero.

00In the theory of forms we seek functions of the coefficients and variables of the original quantic which, save as to a power of the modulus of transformation, are equal to the like functions of the coefficients and variables of the transformed quantic. We may have such a function which does not involve the variables, viz.

00For the substitution rr xl =A 11 +1 2 12, 52=A21+Ã¯¿½2E2, of modulus A1 Ã¯¿½i = (AlÃ¯¿½.2-A2Ã¯¿½1) = (AM), A 2 Ã¯¿½2 the quadratic form a k xi -1-2a 1 x i x 2 +a 2 4 = x =f (x), becomes A41 +2A1E16 =At = OW, where Ao = aoA i +2a1AiA2 +a2Az, _ _ A 1 = ao A lÃ¯¿½l +ai(A1/.22+A2Ã¯¿½1) +7,2X2/22, A2 = aoÃ¯¿½l +2a1Ã¯¿½1/Ã¯¿½2 +a 2Ã¯¿½2 Ã¯¿½ We pass to the symbolic forms a:= (aixi+a2x2) 2, A 2 = (A 151+ A 26) 2/ by writing for ao, al, a2 the symbols ai, a 1 a 2, a?

00= (A11+A22)n by the substitutions 51 = A l, E1+Ã¯¿½1 2, 52 = A2E1+Ã¯¿½2E2, the umbrae Al, A2 are expressed in terms of the umbrae al, a 2 by the formulae A l = Alai +A2a2, A2 = Ã¯¿½la1 +Ã¯¿½2a2Ã¯¿½ We gather that A1, A2 are transformed to a l, a 2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, (A / .c).

00If u, a quantic in x, y, z, ..., be expressed in terms of new variables X, Y, Z ...; and if, n,, ..., be quantities contragredient to x, y, z, ...; there are found to exist functions of, n, ?, ..., and of the coefficients in u, which need, at most, be multiplied by powers of the modulus to be made equal to the same functions of E, H, Z, ...

00The transformation to the normal form, by the solution of a cubic and a quadratic, therefore, supplies a solution of the quartic. If (XÃ¯¿½) is the modulus of the transformation by which a2 is reduced to 3 the normal form, i becomes (X /2) 4 i, and j, (Ap) 3 j; hence?

00Hesse's canonical form shows at once that there cannot be more than two independent invariants; for if there were three we could, by elimination of the modulus of transformation, obtain two functions of the coefficients equal to functions of m, and thus, by elimination of m, obtain a relation between the coefficients, showing them not to be independent, which is contrary to the hypothesis.

00Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution xl = X11 + J-s12, X 2 = 112 Again, in plane geometry, the most general equations of substitution which change from old axes inclined at w to new axes inclined at w' =13 - a, and inclined at angles a, l3 to the old axis of x, without change of origin, are x-sin(wa)X+sin(w -/3)Y sin w sin ' _sin ax y sin w a transformation of modulus sin w' sin w' The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry.

00We have cos w' = cos w = o and the substitution x 1 =cos OX, -sin 0(2 x 2 = sin OX i +cos 6X2, with modulus unity.

00If the senses of rotation be opposite we have the skew orthogonal substitution x1 =cos0Xi+sinOX2r x 2 = sin Ã‚°Xicos OX2r of modulus -1.

00The whole of the rod will therefore be subject to a compressive longitudinal stress P, the associated contraction R, expressed as a fraction of the original length, being R = P/M = (B 2 -H2)/87-M, where M is Young's modulus.

002 The results are too numerous to discuss in detail; some of those to which special attention is directed are the following: In Swedish iron and tungsten-steel the change of elastic constants (Young's modulus and rigidity) is generally positive, but its amount is less than 0.5%; changes of Young's modulus and of rigidity are almost identical.

00In nickel the maximum change of the elastic constants is remarkably large, .amounting to about 15% for Young's modulus and 7% for rigidity; with increasing fields the elastic constants first decrease and then increase.

00Referring the reader to the article Elasticity for the theoretical and to the Strength Of Materials far the practical aspects of this subject, we give here a table of the "modulus of elasticity," E (column 2), for millimetre and kilogramme.

00Let E be the bulk modulus of elasticity, defined as increase of pressure = decrease of volume per unit volume where the pressure increase is so small that this ratio is constant, w the small increase of pressure, and - (dy/dx) the volume decrease, then E=e/(- dy/dx) or w Ãƒâ€ = - dy/dx (I) This gives the relation between pressure excess and displacement.

00But if y is the displacement at A, dy/dx is the extension at A, and the force acting is a pull across A equal to Y&uodyldx, where Y is Young's modulus of elasticity.

00There are shears of the order dry/dx and the simple Young's modulus system can no longer be taken to represent the actual condition (see Rayleigh, Sound, i.

00Young's modulus may be obtained for the material of a rod by clamping it in the middle and obtaining the frequency of the fundamental when Y = 412n2p.

00Owing to the yielding of joints when a beam is first loaded a smaller modulus of elasticity should be taken than for a solid bar.

00It follows that the density of the aether must exceed io 18, and its elastic modulus must exceed Io 3, which is only about io s of the modulus of rigidity of glass.

00The second of these relations is an important one, as it shows that from a table of logarithms to base a, the corresponding table of logarithms to base b may be deduced by multiplying all the logarithms in the former by the constant multiplier i/logab, which is called the modulus of the system whose base is b with respect to the system whose base is a.

00The two systems of logarithms for which extensive tables have been calculated are the Napierian, or hyperbolic, or natural system, of which the base is e, and the Briggian, or decimal, or common system, of which the base is io; and we see that the logarithms in the latter system may be deduced from those in the former by multiplication by the constant multiplier /loge io, which is called the modulus of the common system of logarithms.

00The numerical value of this modulus is o 4342944 81 9 03251 82765 11289..

00It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x and cos x, for such a series could only represent a uniform function, and in fact the equation log(I +x) =x -",, x2 +3x 3 -4x 4 + is true only when the analytical modulus of x is less than unity.

00It will be observed that in the first process the value of the modulus is in fact calculated from the formula.

00to 276 places of decimals, and deduced the value of log e lo and its reciprocal M, the modulus of the Briggian system of logarithms. The value of the modulus found by Adams is Mo = 0-43429 44 81 9 03251 82765 11289 18916 60508 22 943 97 00 5 80366 65661 14453 78316 58646 4920-8870 77 47292 2 4949 33 8 43 17483 18706 106 74 47 6630-3733 64167 92871 58963 90656 92210 64662 81226 58521 27086 56867 03295 9337 0 86965 88266 88331 16360 773849 0514 28443 48666 76864 65860 85135 56148 212 34 87653 43543 43573 25 which is true certainly to 272, and probably to 273, places (Proc. Roy.

00If the logarithms are to be Briggian all the series in the preceding formulae must be multiplied by M, the modulus; thus, log i o(I +x) = M(x-2x2+3x3-4x4+&c.), and so on.

00The length of 100 divisions of the scale, or the length of the uniform stem the volume of which would be equal to that of the hydrometer up to the zero graduation, Francceur called the "modulus" of the hydrometer.

00lo g i op =o 6640+8.585t/e-4.70(log109/Bo-Mt/6), where t=9 -273, and M =0.4343, the modulus of common logarithms. These formulae are practically accurate for a range of 20Ã‚° or 30Ã‚° C. on either side of the melting-point, as the pressure is so small that the vapour may be treated as an ideal gas.

00Modulus of logarithms.

00The constant -r is called the modulus of decay of the oscillations; if it is large compared with 2irfa the effect of friction on the period is of the second order of small quantities and may in general be ignored.

00whenever increases by 4K/cr, when K is the complete elliptic integral of the first kind with respect to the modulus k.

00That least distance is called by Moseley the modulus of stability.

00The following are some of the ratios of the modulus of stability to the depth of the joint which occur in practice :

00Modulus of a Machine.In the investigation of the properties of a machine, the useful resistances to be overcome and the useful work to be performed are usually given.

00The result of this investigation, expressed in the form of an equation between this energy and the useful work, is called by Moseley the modulus of the machine.

00The general form of the modulus may be expressed thus E~U+cb(U, A)+~(A), (54)

00The twist or surface-shear being proportional to the torque, the horse-power can be calculated if the modulus of rigidity of the steel employed is known or if the amount of twist corresponding to a given power has previously been ascertained by direct experiment on the shaft before it has been put in place.

00modulus, a measure, or standard.

00In portions of the structure liable to important changes of pressure from the rise, and fall of the water and subject to the additional stresses which expansion and contraction by changes of temperature and of moisture induce, and in view of the great difficulty of securing that the average modulus of elasticity in all parts of the structure shall be approximately the same, it is probably desirable to limit the calculated load upon any external work, even of the best kind, to 15 or 20 tons per sq.

00The posts are made of a molybdenum- rhenium alloy that maintains a high modulus of elasticity even at high temperatures.

00Seat them down on a bead of clear low modulus silicone sealer before the screws are inserted.

00The reduction factor for elastic modulus of light gage steel is assumed to be identical to that of carbon steel.

00This sections starts by defining modulus, tensile strength, elongation at break and yield strength on a typical stress-strain graph.

00An example of estimating the elasto-plastic buckling stress through use of the material tangent modulus is given.

00Recommendations This method is inherently unsuitable for low modulus polymeric material systems.

00The bulk modulus K ' relates the change in stress to the volumetric strain.

00The Luhn formula is sometimes referred to as the "modulus 10" or "mod 10" algorithm.

00

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