Page:Popular Science Monthly Volume 29.djvu/61

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51
THE PROBLEM OF CRYSTALLIZATION.

we observe that every ball of the pyramid bears the weight only of those balls that are arranged in three lines parallel to upper edges of the pyramid respectively, and meeting in the center of the ball. Thus, in Figs. 15 a and 15 b are represented in plan the four layers of a pyramid of twenty balls. The ball a, of the lowest layer, can only receive the weights of the balls b1 b1 b1 of the second layer, transmitted in direction-lines parallel respectively to the three upper edges of the pyramid (Fig. 14), namely, D A, D A0 , and D A3.

PSM V29 D061 Stress line comparisons between pyramid and crystal structures 1.jpg
Fig. 14.

The ball a can not receive the weight of any other ball of the pyramid; it can not receive the weight of the topmost ball, d, inasmuch as the weight of this ball is transmitted only in the lines DA, DA, and DA3 , the three upper edges of the pyramid; nor can it receive the weight of the ball C of the third layer, for that is only transmitted in three lines, of which two, C A, and C A3, can be seen in the figure. By a simple application of the physical principle known as the parallelogram of forces, we arrive at the deduction that all balls equidistant from the vertex of the pyramid are solicited by the same force; or, in other words, that every ball of the pyramid is repulsed from the vertex with a force proportional to its distance from the vertex, as a direct con-sequence of this stress distribution. At the vertex itself the repulsion is zero. The weight of the pyramid is uniformly distributed over its base; a result which can readily be verified by experiment, and is also a verification of the stated force law. Now, an exactly analogous action occurs among the invisibly small particles of a crystal. In the pyramid of balls, it is the pull of the earth upon each ball which is active; in the crystal it is the mutual attraction of the particles.

PSM V29 D061 Stress line comparisons between pyramid and crystal structures 2.jpg

Fig. 15.

In the pyramid of balls, there are only three stress direction-lines respectively parallel to the upper edges of the pyramid, inasmuch as the pull of the earth acts only vertically downward, hence there is no weight transmission in the three horizontal direction-lines parallel to the basal edges respectively; in the crystal, however, there are