358.] To find the usual equations connecting, 0, & with the angular velocities about three rectangular axes fixed in the body. Having the value of q in last section in terms of the three angles, it may be useful to employ it, in conjunction with equation (3) of § 356, partly as a verification of that equation. Of course, this is an exceedingly roundabout process, and does not in the least resemble the simple one which is immediately suggested by qua ternions. We have whence or 2q = eq = {w1OA+w2OB+wz OC } q, -1 2 q ̄1 q = q ̄1 {w1 OA+w2 2 q = q(i w1 + jw2+kwz). This breaks up into the four (equivalent to three independent) Similarly, by eliminating between the same two equations, (1) sin+w2 cos. (1), (2), (3) are the forms in which the equations are usually given. 359.] To deduce expressions for the direction-cosines of a set of rectangular axes in any position in terms of rational functions of three quantities only. Let q Let a, B, y be unit-vectors in the directions of these axes. be, as in § 356, the requisite quaternion operator for turning the cöordinate axes into the position of this rectangular system. Then q = w + xi + yj +zk, = = (w2 + x2 — y2 — z2) i + 2 (w z + xy) j +2(xz—wy)k, where the coefficients of i, j, k are the direction-cosines of a as required. A similar process gives by inspection those of ẞ and y. As given by Cayley *, after Rodrigues, they have a slightly different and somewhat less simple form-to which, however, they are easily reduced by putting The geometrical interpretation of either set is obvious from the nature of quaternions. For (taking Cayley's notation) if be the angle of rotation: cosf, cosg, cos h, the direction-cosines of the axis, we have 0 q=w+xi+yj+zk = COS + sin(i cosf+j cos g + k cos h), 2 From these we pass at once to Rodrigues' subsidiary formulae, 360.] By the definition of Homogeneous Strain, it is evident that if we take any three (non-coplanar) unit-vectors a, ß, y in an unstrained mass, they become after the strain other vectors, not necessarily unit-vectors, a1, B1, 71 Hence any other given vector, which of course may be thus expressed, p = xa + yẞ + zy, and is therefore known if a1, B1, Y1 be given. More precisely becomes ρδ.αβγ = αδ. βγρ + βδ. γαρ + γδ. αβρ ριδ.αβγ = φρδ. αβγ = α δ.βγρ + β.δ. γαρ + γδ. αβρ. Thus the properties of 4, as in Chapter V, enable us to study with great simplicity strains or displacements in a solid or liquid. For instance, to find a vector whose direction is unchanged by the strain, is to solve the equation Урфр = 0, or фр = др, where g is a scalar unknown. [This vector equation is equivalent to three simple equations, and contains only three unknown quantities; viz. two for the direction of p (the tensor does not enter, or, rather, is a factor of each side), and the unknown g.] We have seen that every such equation leads to a cubic in g which may be written 2 where m2, m1, m are scalars depending in a known manner on the constant vectors involved in p. This must have one real root, and may have three. 361.] For simplicity let us assume that a, ß, y form a rectangular system, then we may operate by S.a, S.ß, and S.y; and thus at once obtain the equation for g, in the form Saa1 + J, δαβι, Sayı which, if the mass be rigid, becomes successively or —1—g (Saa1+Sßß1 + S¥¥1) +g2 (Saa1 + Sßß1 + Sy¥1)+g3 = 0, (g−1) (g2+g(1+Saa1 + Sẞ32+ Syy1) + 1) = 0. 362.] If we take Tp = C we consider a portion of the mass initially spherical. This becomes of course. an ellipsoid, in the strained state of the body. Or if we consider a portion which is spherical after the strain, i.e its initial form was another ellipsoid. The relation between these ellipsoids is obvious from their equations. (See § 311.) In either case the axes of the ellipsoid correspond to a rectangular set of three diameters of the sphere (§ 257). But we must carefully separate the cases in which these corresponding lines in the two surfaces are, and are not, coincident. For, in the former case there is pure strain, in the latter the strain is accompanied by rotation. Here we have at once the distinction pointed out by Stokes and Helmholtz + between the cases of fluid motion in which there is, or is not, a velocity-potential. In ordinary fluid motion the distortion is of the nature of a pure strain, i.e. is differentially non-rotational; while in vortex motion it is essentially accompanied by rotation. But the resultant of two pure strains is generally a strain accompanied by rotation. The question before us beautifully illustrates the properties of the linear and vector function. *Cambridge Phil Trans. 1845. + Crelle, vol. lv. 1857. See also Phil Mag. (Supplement) June 1867. 363.] To find the criterion of a pure strain. Take a, ẞ, y now as unit-vectors parallel to the axes of the strain-ellipsoid, they become after the strain aa, bẞ, cy. Hence P1 = opa a Sap-bẞSBp-cy Syp. φρ And we have, for the criterion of a pure strain, the property of the function 4, that it is self-conjugate, i. e. 364.] Two pure strains, in succession, generally give a strain accompanied by rotation. For if o, y represent the strains, since they by (1), and is not generally the same as py. (See Ex. 7 to Chapter V.) 365.] The simplicity of this view of the question leads us to suppose that we may easily separate the pure strain from the rotation in any case, and exhibit the corresponding functions. When the linear and vector function expressing a strain is selfconjugate the strain is pure. When not self-conjugate, it may be broken up into pure and rotational parts in various ways (analogous to the separation of a quaternion into the sum of a scalar and a vector part, or into the product of a tensor and a versor part), of which two are particularly noticeable. Denoting by a bar a selfconjugate function, we have thus either qã ( )q-1, or $=@1.q( )q1, where is a vector, and q a quaternion (which may obviously be regarded as a mere versor). That this is possible is seen from the fact that involves nine independent constants, while ↓ and each involve six, and e and each three. If o' be the function conjugate to p, we have which completely determine the first decomposition. This is, of |