Vectorial Mechanics

Cover Vectorial Mechanics
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Heaviside, loc. Cit.
io8 VECTORIAL MECHANICS The most general vector may be expressed by a > A 7 being freely variable scalars. Now, if we take al instead of 1, both the numerator and the denominator will be simply multiplied by a, so that the quotient x will not be changed; again, adding /3m, we get in the numerator the complementary term ft . WmVcomcon, which vanishes identically, by (vn. ), and similarly also in the denominator (3 . MVmn = o ; hence x will still be unaltered. And it will rem
...ain invariable if the term yn be added. Hence, if 1 be changed in the most general way, the value of x will not change ; and, similarly, if instead of m, n any other vectors be taken, x will still keep its value. Q. E. D.
Applying the theorem to ft (as a special kind of w), we see by (57) that the ratio dr'jdr is independent of the dimensions and of . The shape of the volume-element dr. , and depends only on the properties of ft, i. E. On the properties of the given strain, at the place where dr is taken.


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