Tangential displacement in a

solid, in terms of

which, according to the formula for projection of areas, is equal components

double the area of the orthogonal projection of the curve on that plane; and similarly for the other integrals.

(b) From this and § 190, it follows that if the body is rigid, and therefore only rotationally displaced, if at all, [Zy]-[Y] is equal to twice the sine of the angle of rotation multiplied by the cosine of the inclination of the axis of rotation to the line of reference OX.

(c) And in general [Zy] - [Y2] measures the entire tangential displacement, divided by the area on ZOY, of any closed curve given, if a plane curve, in the plane YOZ, or, if a tortuous curve, given so as to have zero area projections on ZOX and XOY. The entire tangential displacement of any closed curve given in a plane, A, perpendicular to a line whose direction cosines are proportional to w, p, σ, is equal to twice its area multiplied by √(w2+p3+o2). And the entire tangential displacement of any closed curve whatever is equal to twice the area of its projection on A, multiplied by √(w2 + p2 + o3).

In the transformation of co-ordinates, w, p, o transform by the elementary cosine law, and of course w2 + p2 +σ2 is an invariant; that is to say, its value is unchanged by transformation from one set of rectangular axes to another.

(d) In non-rotational homogeneous strain, the entire tangential displacement along any curve from the fixed point to (x, y, ≈), reckoned along the undisplaced curve, is equal to

(4-1)x+(B-1) y + (C-1)2 + 2 (ayz + bzx + cxy)}. Reckoned along displaced curve, it is, from this and § 187,

{(A − 1) x2 + (B − 1) y3 + (C' − 1 ) ≈2 + 2 (ayz + bzx + cxy)}

+ } {[(4 − 1) x + cy + bz]2 + [cx + (B −1) y + az]3

+[bx+ay + (C-1)]}.

And the entire tangential displacement from one point along any curve to another point, is independent of the curve, i.e., is the same along any number of conterminous curves, this of

of strain.

Hetero

geneous strain.

course whether reckoned in each case along the undisplaced or along the displaced curve.

(e) Given the absolute displacement of every point, to find the strain. Let a, ẞ, y, be the components, relative to fixed axes, OX, OY, OZ, of the displacement of a particle, P, initially in the position x, y, z. That is to say, let x + a, y + ß, ≈+y be the γ co-ordinates, in the strained body, of the point of it which was initially at x, y, z.

Consider the matter all round this point in its first and second positions. Taking this point P as moveable origin, let §, n, Š be the initial co-ordinates of any other point near it, and §,, 71, 5, the final co-ordinates of the same.

The initial and final co-ordinates of the last-mentioned point, with reference to the fixed axes OX, OY, OZ, will be

are the components of the displacement of the point which had
initially the co-ordinates x + έ, y +ŋ, z+%, or, which is the same
thing, are the values of a, ß, y, when x, y, z are changed into
x + έ, y + n, z + 6.

the higher powers and products of έ, 7, ( being neglected. Comparing these expressions with (1) of § 181, we see that they express the changes in the co-ordinates of any displaced point of a body relatively to three rectangular axes in fixed directions through one point of it, when all other points of it are displaced relatively to this one, in any manner subject only to the condition of giving a homogeneous strain. Hence we perceive that at distances all round any point, so small that the first terms only of the expressions by Taylor's theorem for the differences of displacement are sensible, the strain is sensibly homogeneous,

geneous

and we conclude that the directions of the principal axes of the Heterostrain at any point (x, y, z), and the amounts of the elongations strain. of the matter along them, and the tangential displacements in closed curves, are to be found according to the general methods described above, by taking

[xx] =da +1, [xy]-da, [X]=da,

dx

=

dz

If each of these nine quantities is constant (i.e., the same for all Homo

values of x, y, z), the strain is homogeneous: not unless.

geneous strain.

Infinitely

(ƒ) The condition that the strain may be infinitely small is that small strain.

general mo

matter.

(g) These formulæ apply to the most general possible motion Most of any substance, and they may be considered as the fundamental tion of equations of kinematics. If we introduce time as independent variable, we have for component velocities u, v, w, parallel to the fixed axes OX, OY, OZ, the following expressions; x, y, z, t being independent variables, and a, B, y functions of them :—

(h) If we introduce the condition that no line of the body experiences any elongation, we have the general equations for the kinematics of a rigid body, of which, however, we have had Change of enough already. The equations of condition to express this rigid body.

position of a

will be six in number, among the nine quantities

da
"
dx

etc.,

which

(g) are, in this case, each constant relatively to x, y, z. There are left three independent arbitrary elements to express any angular motion of a rigid body.

Non-rotational strain.

(i) If the disturbed condition is so related to the initial condition that every portion of the body can pass from its initial to its disturbed position and strain, by a translation and a strain without rotation; i.e., if the three principal axes of the strain at any point are lines of the substance which retain their parallelism, we must have, § 183 (18),

and if these equations are fulfilled, the strain is non-rotational, as
specified. But these three equations express neither more nor
less than that
adx + ẞdy + ydz

is the differential of a function of three independent variables.
Hence we have the remarkable proposition, and its converse, that
if F (x, y, z) denote any function of the co-ordinates of any point
of a body, and if every such point be displaced from its given
position (x, y, z) to the point whose co-ordinates are

the principal axes of the strain at every point are lines of the substance which have retained their parallelism. The displacement back from (x,, y1, z,) to (x, y, z) fulfils the same condition, and therefore we must have

partial differential coefficients with reference to this system of variables. The relation between F and F, is clearly

This, of course, may be proved by ordinary analytical methods, applied to find x, y, z in terms of x, y, z,, when the latter are given by (1) in terms of the former.

(j) Let α, β, γ be any three functions of x, y, z. Let ds be any element of a surface; l, m, n the direction cosines of its normal.

Then ffds (dy-da)+m(da-dy) + n (de da)}

dz

dx

= [(adx+ßdy+ydz)..........................

dy

(5),

the former integral being over any curvilinear area bounded by a
closed curve; and the latter, which may be written

being round the periphery of this curve line*. To demonstrate
this, begin with the part of the first member of (5) depending on
a; that is

da);

dy

and to evaluate it divide S into bands by planes parallel to ZOY,
and each of these bands into rectangles.

1

The breadth at x, y, z,

of the band between the planes x- dx and x +

dx is

Ꭵf Ꮎ

2

sin 0

2

denote the inclination of the tangent plane of S to the plane x.
Hence if ds denote an element of the curve in which the plane
x cuts the surface S, we may take

Hetero

geneous strain.

The limits of the s integration being properly attended to we see
that the remaining integration, fada, must be performed round
the periphery of the curve bounding S. By this, and correspond-
ing evaluations of the parts of the first member of (5) depending
on ẞ and y, the equation is proved.

This theorem was given by Stokes in his Smith's Prize paper for 1854 (Cambridge University Calendar, 1854). The demonstration in the text is an expansion of that indicated in our first edition. A more synthetical proof is given in § 69 (q) of Sir W. Thomson's paper on "Vortex Motion," Trans. R. S. E. 1869. A thoroughly analytical proof is given by Prof. Clerk Maxwell in his Electricity and Magnetism (§ 24).