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An analytical proof of these conclusions may easily be given; Displacement of a thus we may write the cubic in the formbody, rigid or not, one point of

held fixed.

([Yy], [Yz]| + |[Z≈], [Zx] + [Xx], [Xy] which is [Zy], [Zz] [[X2], [Xx]_[Yx], [Yy]' +n2{[Xx] + [Yy] + [Zz]} − n3 = 0..................(3)

[Xx], [Xy], [X2] [Yx], [Yy], [Y≈] [Zx], [Zy], [Zz]

In the particular case of no strain, since [Xx], etc., are then equal, not merely proportional, to the direction cosines of three mutually perpendicular lines, we have by well-known geometrical theorems

|[Xx], [Xy], [X2]| = 1, and [Yy], [Y2] = [Xx], etc.
[Yx], [Yy], [Y]
[Zy], [Z]
[Zx], [Zy], [Zz]

Hence the cubic becomes

1 − (n − n3) {[Xx] + [Yy] + [Zz]} − n3 = 0,

of which one root is evidently 7 = 1. This leads to the above explained rotational solution, the line determined by the value 1 of n being the axis of rotation. Dividing out the factor 1-7, we get for the two remaining roots the equation

1 + (1 − [Xx] − [Yy] − [Zz]) n + 192 = 0,

whose roots are imaginary if the coefficient of ʼn lies between +2 and - 2. Now - 2 is evidently its least value, and for that case the roots are real, each being unity. Here there is no rotation. Also + 2 is its greatest value, and this gives us a pair of values each =-1, of which the interpretation is, that there is rotation through two right angles. In this case, as in general, one line (the axis of rotation) is determined by the equations (2) with the value + 1 for n; but with 7=-1 these equations are satisfied by any line perpendicular to the former.

The limiting case of two equal roots, when there is strain, is an interesting subject which may be left as an exercise. It separates the cases in which there is only one axis of directional identity from those in which there are three.

Let it next be proposed to find those lines of the body whose elongations are greatest or least. For this purpose we must find the equations expressing that x,y,+2, is a maximum, when x2 + y2 + z2 = r2, a constant. First, we have


x2 + y,* + z‚2 = Ax2 + By2 + Cz2 + 2 (ayz + bzx + cxy)....................... (4),



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a = [Xy] [Xz] + [Yy] [Yz] + [Zy][Z]
b = [Xz][Xx]+[Y2] [Yx] + [Zz ] [Zx]
c = [Xx][Xy] + [Yx] [Yy] + [Zx] [Zy]



Ax2 + By2+ Cz2 + 2 (ayz + bzx + cxy) = r, =r,2. (6), where r is any constant, represents clearly the ellipsoid which a spherical surface, radius r,, of the altered body, would become if the body were restored to its primitive condition. The problem of making r, a maximum when r is a given constant, leads to the following equations:


+ gay + d = 0,

} (8)

(Ax+cy + bz)dx + (cx + By + az)dy + (bx + ay + Cz)dz = 0. On the other hand, the problem of making r a maximum or minimum when r, is given, that is to say, the problem of finding maximum and minimum diameters, or principal axes, of the ellipsoid (6), leads to these same two differential equations (8), and only differs in having equation (6) instead of (7) to complete the determination of the absolute values of x, y, and z. Hence the ratios x y z will be the same in one problem as in the other; and therefore the directions determined are those of the principal axes of the ellipsoid (6). We know, therefore, by the properties of the ellipsoid, that there are three real solutions, and that the directions of the three radii so determined are mutually rectangular. The ordinary method (Lagrange's) for dealing with the differential equations, being to multiply one of them by an arbitrary multiplier, then add, and equate the coefficients of the separate differentials to zero, gives, if we take - as the arbitrary multiplier, and the first of the two equations the one multiplied by it,

x2 + y2 + x2 = p2

(A – n)x

+ cy

cx + (B − n)y

find what

+ bz = 0,

+az= = 0,


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+ ay + (C − y)z = 0.

n means if we multiply the first of these by x,

the second by y, and the third by z, and add; because we thus Displaceobtain

ment of a body, rigid or not, one point of which is held fixed.


Ax2 + By2+ Cz2 + 2 (ayz + bzx + cxy) − n (u3 + y2 + z3) = 0,


2 = 0,



which gives


Eliminating the ratios xy: z from (9), by the usual method, we
have the well-known determinant cubic



(A −n) (B — n)(C′ − n) — a2 (A — n) — b2 (B — n) — c2 (C′ − n) + 2abc = 0...(11),

of which the three roots are known to be all real. Any one of
the three roots if used for 7, in (9), harmonizes these three equa-
tions for the true ratios xy: z; and, making the coefficients of
x, y, z in them all known, allows us to determine the required.
ratios by any two of the equations, or symmetrically from the
three, by the proper algebraic processes. Thus we have only to
determine the absolute magnitudes of x, y, and z, which (7)
enables us to do when their ratios are known.

It is to be remarked, that when [Yz] = [Zy], [Zx] = [X2], and [Xy]=[Y], equation (3) becomes a cubic, the squares of whose roots are the roots of (11), and that the three lines determined by (2) in this case are identical with those determined by (9). The reader will find it a good analytical exercise to prove this directly from the equations. It is a necessary consequence of § 183, below.

We have precisely the same problem to solve when the question proposed is, to find what radii of a sphere remain perpendicular to the surface of the altered figure. This is obvious when viewed geometrically. The tangent plane is perpendicular to the radius when the radius is a maximum or minimum. Therefore, every plane of the body parallel to such tangent plane is perpendicular to the radius in the altered, as it was in the initial condition.

The analytical investigation of the problem, presented in the second way, is as follows:

1 x + m1y, +n, z1 = 0 0 ...

be the equation of any plane of the altered substance, through
the origin of co-ordinates, the axes of co-ordinates being the
same fixed axes, OX, OY, OZ, which we have used of late. The
direction cosines of a perpendicular to it are, of course, propor-
tional to l, m,, n ̧.
If, now, for x, y, z,, we substitute their

Displacement of a body, rigid or not, one point of which is held fixed.

Pure strain.

values, as in (1), in terms of the co-ordinates which the same point of the substance had initially, we find the equation of the same plane of the body in its initial position, which, when the terms are grouped properly, is this—


{l,[Xx] + m ̧[Yx] +n,[Zx]}x+ {l,[Xy] +m,[Yy] +n,[Zy]}y + {l[X2] + m,[Yz] + n ̧[Zz]} z = ...(13). The direction cosines of the perpendicular to the plane are proportional to the co-efficients of x, y, z. Now these are to be the direction cosines of the same line of the substance as was altered into the line l, m, n,. Hence, if 1: m : n are quantities proportional to the direction cosines of this line in its initial position, we must have


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where n is arbitrary. Suppose, to fix the ideas, that l, m, n, are the co-ordinates of a certain point of the substance in its altered state, and that l, m, n are proportional to the initial coordinates of the same point of the substance. Then we shall have, by the fundamental equations, the expressions for l, m, n, in terms of l, m, n. Using these in the first members of (14), and taking advantage of the abbreviated notation (5), we have precisely the same equations for l, m, n as (9) for x, y, z above.

Analysis of a 182. From the preceding analysis it follows that any homo

strain into distortion and rotation

geneous strain whatever applied to a body generally changes a sphere of the body into an ellipsoid, and causes the latter to rotate about a definite axis through a definite angle. In particular cases the sphere may remain a sphere. Also there may be no rotation. In the general case, when there is no rotation, there are three directions in the body (the axes of the ellipsoid) which remain fixed; when there is rotation, there are generally three such directions, but not rectangular. Sometimes, however, there is but one.

183. When the axes of the ellipsoid are lines of the body whose directions do not change, the strain is said to be pure, or unaccompanied by rotation. The strains we have already considered were more general than this, being pure strains

accompanied by rotation. We proceed to find the analytical Pure strain. conditions of the existence of a pure strain.

Let OE, OF', OE" be the three principal axes of the strain,
l, m, n, l', m', n', l", m", n",

and let

be their direction cosines. Let a, a', a" be the principal elonga-
tions. Then, if έ, έ', έ" be the position of a point of the un-
altered body, with reference to OE, OE', OE", its position in
the body when altered will be aέ, a'§', a′′§”. But if x, y, z be
its initial, and x,, y,,, its final, positions with reference to
OX, OY, OZ, we have


§ = lx+my+nz, §' = etc., έ"= etc.
x1 = la§ + l'a'§' + l"a"", y, etc., z1 = etc.



For έ, '," substitute their values (15), and we have x,, y,, 2, in
terms of x, y, z, expressed by the following equations :-

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[Xx] = al2 + a'l'2 + a′′l"3, etc. ;


[Zy] = [Y2] = amn + a'm'n' + u"m"n", etc.

x1 = (al2 + a'l'2 + a' l'2) x+ (alm +al′m′ +a′′l′′m") y+(aln + a′l'n' +a′′l′′n") z y1 = (aml + a'm'l' + a'm'l'')x+(am2 +a'm'2 + a′′m”2) y + (amn + a'm'n' + a′′m′′n') z §.(16). 2 = (anl +a'n'l' +a'n'l') x+(anm + a'n'm' + a′′n'm”)y + (an2 + a'n'2 + a′′n"3) z. Hence, comparing with (1) of § 181, we have




In these equations, l, l', l'', m, m', m", n, n', n", are deducible
from three independent elements, the three angular co-ordinates
(§ 100, above) of a rigid body, of which one point is held fixed;
and therefore, along with a, a', a", constituting in all six in-
dependent elements, may be determined so as to make the six
members of these equations have any six prescribed values.
Hence the conditions necessary and sufficient to insure no rotation


[Zy]=[Y], [X2] = [Zx], [Xy] = [Y]............. (18).

tion of pure

184. If a body experience a succession of strains, each un- Composiaccompanied by rotation, its resulting condition will generally strains. be producible by a strain and a rotation. From this follows the remarkable corollary that three pure strains produced one after another, in any piece of matter, each without rotation, may be so adjusted as to leave the body unstrained, but rotated through some angle about some axis. We shall have, later, most important and interesting applications to fluid motion,

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