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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 ◊ is the angle of rotation.

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

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where the pure strain precedes the rotation, and from this

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

and is to be found

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by the solution of a biquadratic equation *.

It is evident, indeed, from the identical equation

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or

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-1

q ̄1(pp′p) q = ☎2 (g ̄1pq) = p′p (q ̄1pg),
4-1

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

To determine q we have

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

(☎ +g) w + g1☎ +J2 = 0 = (w +91) +gw+92.

Eliminating, we have

2

w3 + (2g1 −g2) ∞2 + (g12 — 2992)w—g22 = 0.

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

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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 the pure strain.

366.] In general, if

P1 =ppa1 Sap-B1Sẞp-Y1Syp,

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)

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

cos-1(— S.Up'¿Up ́n).

The locus of lines equally elongated is, of course,

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= 0, which

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

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The vectors which are unaltered in length are given by

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S. ap=0,
S.ap

Sp (2B+B2a) = 0.

The intersection of this plane with the plane of a, ẞ is perpen

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

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S.(a+x1B) (a+x ̧ß3) = − 1 + ẞ2 x1 X2,

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

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S{a+(x,−1)ẞ} {a+(x,−1)ß} = −1 + ß2 { x1X ̧1⁄2− (X1 + x12) + 1 }, which ought also to be zero. And, in fact, 1+2 = 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.

If

Fp

=

C

(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

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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

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This is due to Hamilton (Lectures on Quaternions, p. 611).

369. From this it follows that the effect of the vector operation V, upon any scalar function of the vector of a point, is to produce the vector which represents in magnitude and direction the most rapid change in the value of the function.

Let us next consider the effect of V upon a vector function as

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and in this semi-Cartesian form it is easy to see that :

If σ represent a small vector displacement of a point situated at the extremity of the vector p (drawn from the origin)

SV represents the consequent cubical compression of the group of points in the vicinity of that considered, and

Vo represents twice the vector axis of rotation of the same group of points.

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or is equivalent to total differentiation in virtue of our having passed from one end to the other of the vector σ.

370.] Suppose we fix our attention upon a group of points which originally filled a small sphere about the extremity of p as centre, whose equation referred to that point is

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After displacement p becomes p+o, and, by last section, p+w becomes p+w+o-(SwV)o. Hence the vector of the new surface which encloses the group of points (drawn from the extremity of p+o) is

w1 = w―(SwV)σ.

Hence is a homogeneous linear and vector function of w1; or

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(2)

the equation of the new surface, which is evidently a central surface of the second order, and therefore, of course, an ellipsoid.

We may solve (2) with great ease by approximation, if we remember that Ts is very small, and therefore that in the small term we may put o1 for w; i.e. omit squares of small quantities; thus w = w1+ (Sw1V)σ.

371.] If the small displacement of each point of a medium is in the direction of, and proportional to, the attraction exerted at that point by any system of material masses, the displacement is effected without

rotation,

For if Fp C be the potential surface, we have Sodp a complete differential; i. e. in Cartesian cöordinates.

§ dx + ŋ dy + Ċ dz

is a differential of three independent variables. Hence the vector axis of rotation

dn

αζ

i

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dy

dz

vanishes by the vanishing of each of its constituents, or

V.Vo = 0.

Conversely, if there be no rotation, the displacements are in the direction of, and proportional to, the normal vectors to a series of surfaces. For

0 = V.dpV.Vo

=

(Sdpv)o-Sodp, where, in the last term, V acts on σ alone.

Now, of the two terms on the right, the first is a complete differential, since it may be written -Dapσ, and therefore the remaining term must be so.

Thus, in a distorted system, there is no compression if

and no rotation if

and evidently merely

which is one case of

SVσ = 0,

V.Vo = 0;

transference if σ = a = a constant vector,

In the important case of

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there is evidently no rotation, since

Vσ = eV2Fp

is evidently a scalar. In this case, then, there are only translation and compression, and the latter is at each point proportional to the density of a distribution of matter, which would give the potential Fp. For if r be such density, we have at once

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372.] The Moment of Inertia of a body about a unit vector a as axis is evidently Mk2m(Vap)2,

where ρ is the vector of the portion m of the mass, and the origin of p is in the axis.

*Proc. R. S. E., 1862-3.

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