* However, at most [min (N, Q) − 1] and at least one possibility is of practical use.

* If λ₁ ≠ 0 and λ₂ ≠ 0, then in addition to equation (4) a corresponding equation (4′), obtained by interchanging in (4) x _ν − m _x and y _q − m _ν, N and Q, and Greek and Latin indices, is found from (2) by eliminating the x _ν − m _x instead of the y _q − m _ν.

† It will be obvious by an argument parallel to the following one, that equations (1 b) and (4′) [see footnote^* above] would be sufficient conditions for (1) and (2) as well. Thus we may assume from now on that N ≥ Q.

‡ Thus, if ρ² = 0, then all y _q = 0 = m _ν. But we shall show later that we can always find at least one solution of (4) and (1 a) with ρ² > 0 except in the case of absolute non-correlation, i.e. if p _νq = P _ντ_q. In this case, however, the problem is trivial.

* See p. 521, note †. If N > Q, then at least N − Q of the (ρ⁽ⁱ⁾)² must vanish. For if (ρⁱ)² > 0, then by (ii) λ₂ = 1, λ₁ = (ρ⁽ⁱ⁾)² > 0, but there are at most Q (linearly independent) characteristic sets of (4′) (see p. 521, note ^*). Furthermore it may occur that for some i = 2, …, N two coordinates and are equal. However, this property is of no statistical relevance, since it depends on the exact values of the p _νq. The same applies if two characteristic-values are equal.

* Provided that the variance (7) is 1, which is true for the

† Usually Pearson's Mean Square Contingency is only defined for N = Q. But if N > Q (say) there are at least two possibilities of defining a measure having its properties: one is given by (18), another would have Q − 1 instead of N − 1 (as the corresponding argument for the y _q would show).