Page:Proceedings of the Royal Society of London Vol 60.djvu/536

Rh from which we start will have become four, to he expanded into three derivative entries, having respectively the frequencies 1, 2, and 1; these latter figures are entered in fig. 1 at the intersections of the lines just named. Under this arrangem ent the large figure from which we started, which had been changed from 1 to 4, again assumes its original value of 1. It will easily be understood, that the positions of the three derivative entries necessarily lie in the same straight line, and th at this line necessarily runs towards the (0, 0 ) corner of the figure. The same is true for every other set of derivative entries, w ith the result that whereas the original set of large figures, referring to the combinations of A and B, are symmetrically disposed on either side of the horizontal, of the vertical, and of the diagonal lines passing through their common centre at (II, II), the derivative values of A/C, B/C are disposed symmetrically only in respect to the diagonal line th at runs from the (0, 0 ) corner. Their symmetry, in this sense, is well shown by the dotted connections between the corresponding figures on either side of the diagonal. Also, it will be seen that the diagonal passes through the regions of greatest frequency. It follows th at the diagonal in question represents the locus of average frequency. Now, along th at diagonal, each value of A/C is associated with identically the same value of B/C; in other words, a correlation is found to have become established between them, which is solely due to the fact th at each member in every couplet of A/C, B/C values is divided by the same value of the variable C. We will now submit the above process to the te3t of extreme cases. First, let the variability of A be so small that it may be treated as a constant, and take it = 1. Then the values of A/C and B/C, that are sevei*ally associated with the three values of C, are as follows:—

Table III. c. A/C. B/C. Corresponding j frequencies. j I I I II IIL - 1 2 1 II I/II I/II I m in 1 2 1 III I/III I/III II/III i 1 2 1

These frequencies are laid down at their proper places in fig. 2, where the three entries, corresponding to each successive value of A/C, run in vertical lines, but, on connecting the entries of maximum