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Now as dx, dy, dz are in the most general case arbitrary functions of x, y, and 3*,
dě = dx + 0,04 dx + 0,04 dy + d.medz,
But as ôx,ồy, ôz are variations of the x — , Y-, and z— coordinates respectively, it is evident in reference to ox that
dy ane dz
d.ox are infinitesimals of a higher order than ; and analogous results are true of the differentials of dy and òz; so that taking infinitesimals of the lowest order in (17), we have
the order of the symbols d and having been changed by reason of (7). And this is the same result as (15).
Now, as 8.ds is the change in length which the arc-element undergoes by reason of the displacement of ds, it has been called the linear dilatation of the arc-element; and similarly, de dr dn ds. Ždr, dy, dz are respectively the linear dilatations of dx, dy, dz. We shall have important applications of this and of similar theorems in the higher mechanics.
* Many of the brackets which are indicative of partial differentiation are omitted in this and the following Articles, that the heaviness of the formulæ may be relieved.
298.] Next let dy dx be the infinitesimal whose variation is to be determined, which is the area-element of a plane surface. Let &, n, as in the last article be the varied values of x and y; so that & = x+8X, n = y + 8y; then
and consequently, by the method of Art. 211,
where infinitesimals of the higher orders are omitted. Thus, since 8.dx dy = the excess of the area of the displaced element over that of the original element = df dn-dx dy,
I d.ox d.dy . 8.dx dy =>
(23) I dx dy s As 8.dx dy is the variation which the infinitesimal area-element undergoes by reason of its displacement, it is called the superficial dilatation of the area-element; and this is of course expressed by the right-hand member of (23).
Let us however consider this subject by the light of first principles, for we shall thereby be able to trace not only the area of the displaced element, but also its form; and this is important in a problem wherein great clearness and precision are required. Let A, B, C, be the four angular points of the original area-element, dx dy, where a is (x,y); B is (x + dx,y); c is (x, y+dy); D is(x + dx,y+dy); also let A', B, C, D', be the places of A, B, C, D after the displacement; so that a' is (X +8x, y + 8y);
and thus the figure a'B'c'd' is a parallelogram. If infinitesimals of the higher orders are neglected, these results are the same as (18).
Also let w be the angle between the two sides A'B' and a'c'; then since
A'B”? + A'd2-B'c? ... cos w =
2 A'B' X A'C' d.dx d.dy
. (24) - dy I dx if infinitesimals of the higher orders are neglected. As those however involved in the right-hand member are of a higher order than those retained in (23), it appears that approximately cos w = 0; consequently w= 90°, and the displaced area-element is a rectangle as it was in its original state.
Again, let us investigate the variation of that general surfaceelement, which is given in (30), Art. 236; viz. da? = dy dz2 + dz2 dx2 +. de dya.
(25) Let da' be the area of the displaced surface-element, so that if (€, n, 5) is the varied place of (x, y, z), da“ = dn2 d62+d62 d&2 + de 2 dna.
(26) And substituting for d$, dn, ds the values given in (18), we have
all infinitesimals of the higher orders having been omitted. Now since 8.da is the excess of the area of the surface-element in its displaced state over that of it in its original state, 0.da =da'-da; and consequently
d.8x) dz2 d.x2 tdz 1
. dx I da (d.oxd.dy) dxdy: (28)
Tdx dy da which is the required result; and expresses the areal dilatation of the surface-element. I may observe that (28) may be deduced from (25) by taking the variation of (25) according to the rules of differentiation.
If a, b, y are the direction-cosines of the normal to the surface at the point (x, y, z), then employing the values given in (26), (27), (28), Art. 236, (28) becomes
which shews that the variation of the general surface-element is equal to the sum of the projections on its plane of the several variations of its projections on the coordinate planes.
299.] Again, let the infinitesimal whose variation is to be determined be dx dy dz; that is, be the volume-element in reference to a system of rectangular coordinates in space.
Let (&, n, 8) be as heretofore the varied place of (x, y, z); then by reason of (17),
dx dy dz; (30)
And as the variation of the volume-element is equal to the excess of the volume in its displaced position over that of it in its original position,
This is called the cubical dilatation of the volume-element.
As this subject is of peculiar importance in reference to subsequent investigations, let us examine it more closely, as in the preceding article, by means of first principles, and retain infinitesimals of a higher order ; so that hereby we may clear up all the obscurity which surrounds it, depending as it does on the preceding partial differentials.
Let A, B, C, D be the four angular points of that face of the elemental parallelepipedon which is parallel to and nearest to the plane of (x,y); where a is (x, y, z); B is (x + dx, y, z); c is (x, y + dy, z); dis (x + dx, y+dy, z). Let a', B', d', D' be the places of A, B, C, D when the variation has taken place; so that a' is (x + 8x, y + 8y, z+öz),
so that the figure a'B'c'D' is also a parallelogram. If infinitesimals of a higher order are neglected, these results are the same as the first two of (18).
Similarly if the variations of the other faces of the new volumeelement are calculated, it will be found that they are parallelograms; so that the volume-element is a parallelepipedon, its volume being given in (31), so far as infinitesimals of the first order are involved. Also let a, ß, y be the angles between the edges dn and ds, ds and dệ, df and dn respectively; then, as y is the angle between the two sides a'B' and a'c', and as
b'c'? = A'B'2 – 2 A'B' x A'c'cos y + A'da; and PRICE, VOL. II.