By 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.
In 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.
For 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?
= (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).
If 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, ...
The 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?
Hesse'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.
Thus 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.
We 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.
If the senses of rotation be opposite we have the skew orthogonal substitution x1 =cos0Xi+sinOX2r x 2 = sin °Xicos OX2r of modulus -1.
2 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.
In 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.
Referring 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.
Let 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.
But 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.
Young'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.
Owing to the yielding of joints when a beam is first loaded a smaller modulus of elasticity should be taken than for a solid bar.
It 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.
The 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.
The numerical value of this modulus is o 4342944 81 9 03251 82765 11289..
It 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.
It will be observed that in the first process the value of the modulus is in fact calculated from the formula.
To 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.
If 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.
Lo 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.
Modulus of logarithms.
The 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.
Whenever increases by 4K/cr, when K is the complete elliptic integral of the first kind with respect to the modulus k.
That least distance is called by Moseley the modulus of stability.
The following are some of the ratios of the modulus of stability to the depth of the joint which occur in practice :
Modulus 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.
The 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.
The general form of the modulus may be expressed thus E~U+cb(U, A)+~(A), (54)
The 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.
Modulus, a measure, or standard.