EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES.
209
Or, if we set
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(437)
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we shall have
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(438)
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It will be observed that
represents the sum of the squares of the nine minors which can be formed from the determinant in (437), and that
represents the sum of the squares of the nine constituents of the same determinant.
Now we know by the theory of equations that equation (431) will be satisfied in general by three different values of
, which we may denote by
, and which must represent the squares of the ratios of elongation for the three principal axes of strain; also that
are symmetrical functions of
, viz.,
![{\displaystyle E=r_{1}^{2}+r_{2}^{2}+r_{3}^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd08691ba0d9d1feccfdbbe94521065365a08e5a) |
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![{\displaystyle F=r_{1}^{2}r_{2}^{2}+r_{2}^{2}r_{3}^{2}+r_{3}^{2}r_{1}^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2687fc286852cee8f67dfcc559778ad4c7b9eecf) |
(439)
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Hence, although it is possible to solve equation (431) by the use of trigonometrical functions, it will be more simple to regard
as a function of
and the quantities
(or
), which we have expressed in terms of
Since
, is a single-valued function of
and
(with respect to all the changes of which the body is capable), and a symmetrical function with respect to
, and since
are collectively determined without ambiguity by the values of
and
, the quantity
must be a single-valued function of
and
. The determination of the fundamental equation for isotropic bodies is therefore reduced to the determination of this function, or (as appears from similar considerations) the determination of
, as a function of
and
.
It appears from equations (439) that E represents the sum of the squares of the ratios of elongation for the principal axes of strain, that
represents the sum of the squares of the ratios of enlargement for the three surfaces determined by these axes, and that
represents the square of the ratio of enlargement of volume. Again, equation (432) shows that
represents the sum of the squares of the ratios of elongation for lines parallel to
and
; equation (434) shows that
represents the sum of the squares of the ratios of enlargement for surfaces parallel to the planes
; and equation (438), like (439), shows that
represents the square of the ratio of