and we see by §§ 435, 436 that this is the vector-force exerted by a small plane current at the origin (its plane being perpendicular to a) upon a magnetic particle, or pole of a solenoid, at p. This expression, being a pure vector, denotes an elementary rotation caused by the distortion of the solid, and it is evident that the above value of o satisfies the equations (2), (3), and the distortion is therefore producible by external forces. Thus the effect of an element of a current on a magnetic particle is expressed directly by the displacement, while that of a small closed current or magnet is represented by the vector-axis of the rotation caused by the displacement. σ It is evident that o satisfies (2), and that the right-hand side of the above equation may be written Hence the corresponding displacement is producible by external forces, and Vo is the rotation axis of the element at p, and is seen as before to represent the vector-force exerted on a particle of magnetism at p by an element a of a current at the origin. 457.] It is interesting to observe that a particular value of o in this case is α σ = − √ √ SaUp — Tp' as may easily be proved by substitution. Τρ Sap Sap Now, as is the potential of a small magnet a, at the origin, Tp3 on a particle of free magnetism at p, o is the resultant magnetic force, and represents also a possible distortion of the elastic solid by external forces, since Vo = V2σ = 0, and thus (2) and (3) are both satisfied. 458.] We conclude with some examples of quaternion integration of the kinds specially required for many important physical problems. It may perhaps be useful to commence with a different form of definition of the operator V, as we shall thus, if we desire it, entirely avoid the use of ordinary Cartesian coordinates. For this purpose we write S.av=-da, where a is any unit-vector, the meaning of the right-hand operator (neglecting its sign) being the rate of change of the function to which it is applied per unit of length in the direction of the unit-vector a. If a be not a unit-vector we may treat it as a vector-velocity, and then the right-hand operator means the rate of change per unit of time due to the change of position. Let a, ẞ, y be any rectangular system of unit-vectors, then by a fundamental quaternion transformation ▼=—aSa▼ — BSßV —уSyv = ada+ẞdß+ydy, which is identical with Hamilton's form so often given above. (Lectures, § 620.) 459.] This mode of viewing the subject enables us to see at once that the effect of applying V to any scalar function of the position of a point is to give its vector of most rapid increase. Hence, when it is applied to a potential u, we have Vu vector-force at ρ. If u be a velocity-potential, we obtain the velocity of the fluid element at p; and if u be the temperature of a conducting solid we obtain the flux of heat. Finally, whatever series of surfaces is represented by u = = C, the vector vu is the normal at the point p, and its length is inversely as the normal distance at that point between two consecutive surfaces of the series. the left-hand member therefore expresses total differentiation in virtue of any arbitrary, but small, displacement dp. 460.] To interpret the operator V.aV let us apply it to a potential function u. Then we easily see that u may be taken under the vector sign, and the expression √(av) u = V. avu denotes the vector-couple due to the force at p about a point whose relative vector is a. Again, if σ be any vector function of p, we have by ordinary quaternion operations that V(av).σ = S.al Vo+aSVo-V Sao. The meaning of the third term (in which it is of course understood operates on σ alone) is obvious from what precedes. It remains that we explain the other terms. 461.] These involve the very important quantities (not operators such as the expressions we have been hitherto considering), S.Vo and V.Vo, which form the basis of our investigations. Let us look upon o as the displacement, or as the velocity, of a point situated at P, and consider the group of points situated near to that at p, as the quantities to be interpreted have reference to the deformation of the group. T 462.] Let be the vector of one of the group relative to that situated at p. Then after a small interval of time t, the actual coordinates become and P+ to by the definition of V in § 458. Hence, if o be the linear and vector function representing the deformation of the group, we have The farther solution is rendered very simple by the fact that we may assume to be so small that its square and higher powers may be neglected. If ' be the function conjugate to 4, we have The first three terms form a self-conjugate linear and vector function of 7, which we may denote for a moment by @τ. Hence Hence the deformation may be decomposed into-(1) the pure strain , (2) the rotation t 2 Thus the vector-axis of rotation of the group is If we were content to avail ourselves of the ordinary results of Cartesian investigations, we might at once have reached this conclusion by noticing that and remembering as in (§ 362) the formulae of Stokes and Helmholtz. 463.] In the same way, as SVo dε dn αξ we recognise the cubical compression of the group of points considered. It would be easy to give this a more strictly quaternionic form by employing the definition of § 458. But, working with quaternions, we ought to obtain all our results by their help alone; so that we proceed to prove the above result by finding the volume of the ellipsoid into which an originally spherical group of points has been distorted in time t. For this purpose, we refer again to the equation of deformation and form the cubic in & according to Hamilton's exquisite process. We easily obtain, remembering that t2 is to be neglected*, or 0 = 43 — (3 — tS▼o) p2 + (3 — 2 tSV o) p—(1 — tSVo), 0 = (-1)2 (—1+tSVo). The roots of this equation are the ratios of the diameters of the ellipsoid whose directions are unchanged to that of the sphere. Hence the volume is increased by the factor 1-1SVo, from which the truth of the preceding statement is manifest. * Thus, in Hamilton's notation, A, μ, v being any three non-coplanar vectors, and m, my, m2 the coefficients of the cubic, -ης λμν = 5.φ'λφ' μόν =S.(^—t▼Sλo) (μ—t▼ Sμo) (v—t▼Svo) =S.(^—t▼Sλo) (Vμv−tVμ▼Svo + t VvVSμo) =S.λuv-t[S.μv▼ Sλo+S. vλ▼ Sμo+S.λμ▼ Svo] mδ.λμν=S.λό'μόν + 5. μόνφ'λ + δ νφ'λφ'μ =S.λμv-tS.λμ▼Svo―tS.vλ▼Sμo+&c. -ηδ.λμν=S.λ μφ'ν + δ. μνφ' λ + δνλφ' μ =S.λμv-tS.λμ▼Svo +&c. =38.λμν-ισ .λμν 464.] As the process in last section depends essentially on the use of a non-conjugate vector function, with which the reader is less likely to be acquainted than with the more usually employed forms, I add another investigation. Let Then ==&T=T-1S (TV) σ. + = $-1w = ☎ +tS (wV) o. Hence since if, before distortion, the group formed a sphere of radius 1, we have TT = 1, where x is now self-conjugate. Hamilton has shewn that the reciprocal of the product of the squares of the semiaxes is -S.xixjxk, whatever rectangular system of unit-vectors is denoted by i, j, k. Substituting the value of x, we have — S.(i+t▼ Sio+tS (i▼) o) (j+&c.) (k+&c.) =S.(i+tv Sio+tS (iv) o) (i + 2 tiSVo-tS (iv) o-tv Sio) The ratio of volumes of the ellipsoid and sphere is therefore, as before, 1 √1+2tSVo = 1-tSvo. 465.] In what follows we have constantly to deal with integrals extended over a closed surface, compared with others taken through the space enclosed by such a surface; or with integrals over a limited surface, compared with others taken round its bounding curve. The notation employed is as follows. If Q per unit of length, of surface, or of volume, at the point p, Q being any quaternion, be the quantity to be summed, these sums will be denoted by SQds and JSS Qds, when comparing integrals over a closed surface with others through the enclosed space; and by Qds and SQTdp, when comparing integrals over an unclosed surface with others round its boundary. No ambiguity is likely to arise from the double use of SS Q ds, |