which, if the mass be rigid, becomes successively

= 0,

or

-1-g(Saa1+Sßß1 + S¥¥1) +g2 (Saa1 + Sßß1 + Sy1)+g3

(g−1) (g2 + g (1 + Saa1 + Sßß1 + 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

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. Iv. 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 = ppa a Sap-b3SBp-cy Syp.

And we have, for the criterion of a pure strain, the property of the function 4, that it is self-conjugate, i. e.

δροσ = Οσφρ.

364.] Two pure strains, in succession, generally give a strain accompanied by rotation. For if p, y represent the strains, since they

by (1), and is not generally the same as y. (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

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 € and each three. If ' be the function conjugate to p, we have

which completely determine the first decomposition. This is, of

course, perfectly well known in quaternions, but it does not seem to have been noticed as a theorem in the kinematics of strains that there is always one, and but one, mode of resolving a strain into the geometrical composition of the separate effects of (1) a pure strain, and (2) a rotation accompanied by uniform dilatation perpendicular to its axis, the dilatation being measured by (sec. 0—1) where 0 is the angle of rotation.

In the second form (whose solution does not appear to have been attempted), we have

where the pure strain precedes the rotation, and from this

or in the conjugate strain the rotation (reversed) is followed by the pure strain. From these

and is to be found by the solution of a biquadratic equation *. It is evident, indeed, from the identical equation

which shew the relations between po', 'p, and q.

write a for p'p in the given equation, and by its help this may be written as

Eliminating ☎, we have

(@+g) w+g1 @ + J2 = 0 = ☎ (∞ + g1) +gw+92•

a2+(2g,-g3) a2+(g12-2992)∞-g2=0.

This must agree with the (known) cubic in w,

w3 + mw2 + m, w + m2 = 0,

whatever be a and ẞ; and the rest of the solution follows at once. A similar process gives us the solution when the rotation precedes pure strain.

the

366.] In general, if

P1 = ppa1 Sap-B1S3p-11 SYP,
φρ =-

the angle between any two lines, say p and σ, becomes in the altered state of the body

cos-1(-S.UppUpo).

The plane Sep = 0 becomes (with the notation of § 144)

SSP1 =0= S5pp = Spp's.

Hence the angle between the planes Sp=0, and Snp = 0, which is cos-1-S.UÇUn), becomes

cos-1-S.Up'Up'n).

The locus of lines equally elongated is, of course,

367.] In the case of a Simple Shear, we have, obviously,

The vectors which are unaltered in length are given by

The intersection of this plane with the plane of a, ẞ is perpendicular to 23+ B2a. Let it be a +xß, then

ß2a.

• For the axes of the strain, one is of course aß, and the others are found by making ToUp a maximum and minimum.

S.(a+x1ẞ) (a+xß) = − 1 + ẞ2 x1 X2,

and should be 0. It is so, since, by the equation,

Again

=

S{a+(x1−1)ẞ} {a+(x2−1)ẞ} = −1+ß2 {x1x ̧−(x1+x) + 1 }, which ought also to be zero. And, in fact, 1 += 1 by the equation; so that this also is verified.

368.] We regret that our limits do not allow us to enter farther upon this very beautiful application.

But it may be interesting here, especially for the consideration of any continuous displacements of the particles of a mass, to introduce another of the extraordinary instruments of analysis which Hamilton has invented. Part of what is now to be given has been anticipated in last Chapter, but for continuity we commence afresh. C.....

If

Fp

(1)

be the equation of one of a system of surfaces, and if the differential of (1) be

Svdp= 0,.......

(2)

v is a vector perpendicular to the surface, and its length is inversely proportional to the normal distance between two consecutive surfaces. In fact (2) shews that v is perpendicular to dp, which is any tangent vector, thus proving the first assertion. Also, since in passing to a proximate surface we may write

It is evident from the above that if (1) be an equipotential, or an isothermal, surface, v represents in direction and magnitude the force at any point or the flux of heat. And we have seen (§ 317) that if