Our characteristic or regression-equation will now be of the form xx = bx2x2 + bx3x3................................. (9),
bl2 and &13 being the unknowns to be determined from the observations by the method of least squares. I have omitted a constant term on the right-hand side, since its least-square value would be zero as before. The two normal equations are now—
S{aq&2) = bX2S(x22) S(»i®s) = &i2S(a?2%) + &13S(«32),
or replacing the sums by the symbols defined above, and simplifying— whence
r12o-i = bX2a2-\- bx3r23a1 — &12^23°'2 ^13^3 / .......... (10), - 7 ^12—rl3r23 W l - r 232 7^ ^13 ^12^23 0
- .
°"2 r | ......... (11)* 1 w eo 1 rH <*3J
That is, the characteristic relation between xL and is— Xi rl2— ^13^23 ^1 , ^13 — ^12^23^1 —----- ~ “ ^2+ ------ ^-----#3 1—^23 °2 1 ^23" ( 12).
Now Bravais showed that if the correlation were , and we selected a group or array of X^s with regard to special values h2 and li3 of x 2and x3, then hi being the deviation of the mean of the selected Xj’s from the Xr mean of the whole material, K — liJii + lifh,
where bn and &13 have the values given in (11). But evidently the relation is of much greater generality; it holds good so long as hx is a linear function of h2 and h3, whatever be the law of frequency. Further, the values of bX2 and &13 above determined, are, under any circumstances, such that Sr2 = S [xx — (bx2x2 + ]2,
is a minimum. If we insert in this expression the values of bx2 and bx3 from (11), we have, after some reduction,
SO2) = No-!21 1- rjj = No-!2{l —Rx2} (13),
