Page:Proceedings of the Royal Society of London Vol 60.djvu/534

Rh The diagrams show how a table of frequency of the various combinations of two independent and normal variables may be changed into one of A/C, B/C, where 0 is also an independent and normal variable in respect to its intrinsic qualities, but subjected to the condition that the same value of C is to be used as the divisor of both members of the same couplet of A and B. In short, th at the couplets shall always be of the form A/C„, B/C«, and never th at of A/C„ B/C*.

For the sake of clearness, the simplest possible suppositions, that are at the same time serviceable, will be made in regard to the particular case illustrated by the diagrams, namely, that A, B, and C, severally, are sharply divided into three, and only into three, equal grades of magnitude, distinguished as AI, A ll, A III; BI, BIT, B ill; and Cl, CII, C I I I ; also that the frequency with which these three grades occur is expressed by the three terms of the binomial ( l + l ) 2. Consequently there is one occurrence of I to two occurrences of II and to one occurrence of III. Roman and italic figures are here used to keep the distinction clear between magnitudes and frequencies. It will be easily gathered as we proceed, without the need of special explanation, that the smallness of the value of the binomial index has no influence either on the general character of the operation or on its general result.

The large figures in the outlined square, occupying the lower right hand portion of fig. 1, show the distribution of frequency of the various combinations of A and B. The scales running along the top and down the left side of the figure, which are there assigned to the “values of A/G, B/C, &pply to these entries also. The latter run in the same way as those in Table I below, or when quadrupled, as they will be for purposes immediately to be explained, as in Table II.

Table I. Table II. 1 2 1 2 4 2 1 2 1 4 8 4 8 16 8 4 8 4

Let us now follow the fortunes of one of the large figures in fig. 1, say that which refers to A = I, B = III, of which the frequency is only 1. When the latter is expanded into the three possible values of the form A/C, B/C, caused by the three varieties of C, it yields i case of frequency to (I/I, III/I), f case to (I/ll, III/II), and i case to ( Hil,III/III), for entry at the intersections of the lines (I, III), (I/ll, III/II), and (I/III, I) respectively.

But, in order to avoid the inconvenience of quarter values, it is better to suppose the original figures in the fig. and in Table I above to have been replaced by those in Table I I; then the original entry