^{page 16 note *} Spearman, , The Abilities of Man, Appendix, pp. iiGoogle Scholar, iii.

^{page 17 note *} Spearman, loc. cit.; Maxwell Garnett, Proc. Roy. Soc., A, xcvi, pp. 91 et seq.

^{page 17 note †} Proc. Boy. Soc., A, 1919, xcv, p. 400.

^{page 17 note ‡} Thomson, , “A worked-out Example of the Possible Linkages of Four Correlated Variables on the Sampling Theory,” British Journ. Psychol., 1927, 18, 68Google Scholar; Mackie, , “The Probable Value of the Tetrad-difference on the Sampling Theory,” British Journ. Psychol., 1928, 19, 65.Google Scholar

^{page 17 note §} See Garnett's article referred to.

^{page 18 note *} For a discussion of this geometrical representation of qualities and of the correlation between them see Garnett, , “On Certain Independent Factors in Mental Measurements,” Proc. Roy. Soc., A, 1919, xcvi.Google Scholar

^{page 20 note *} It has been pointed out to me that the same result may be obtained thus. The volume of an N-dimensional sphere is V = ∫ ∫ … ∫dx ₁dx ₂ … dx _N. Let N new variables be defined by equations (7), where the l's are replaced by x's, along with r ² = x ₁² + x ₂² + x ₃² + … + x _N². The integral is transformed into V = ∫ ∫ … ∫ J(x ₁, …, x _N)drdθ₁ … dθ_N–1, where J is the Jacobian The surface area is = ∫ ∫ … ∫J dθ₁ … dθ_N–1, so that the surface element is dθ₁dθ₂ … dθ_N–1. When this is evaluated, and r put equal to 1, the result (8) is obtained.

^{page 22 note *} In essence this is a geometrical theorem. E.g., if there are only two variables x and y, and the lines OQ₁, OQ₂, OQ₃, OQ₄ make the angles α, β, γ, δ with OX, we have cos (γ – α) cos (δ – β) – cos (γ – β) cos (δ – α) = (cos α sin β – cos β sin α)(cos γ sin δ - cos δ sin γ), a result easily proved by elementary trigonometry.

^{page 27 note *} The above analysis supposes that n is an even number. If n is odd, there will be throughout slight modifications in the various integrations, but the final result as expressed in equation (19) will be the same.

^{page 28 note *} This is because the distribution of every l is the same. If we take the (n + l)th axis as our example, we have from equations (12) ₁l ₂l = sin θ_n sin ф_n. Multiplying by the frequency and integrating, we have ∫ ∫, … ∫ cos θ2 cos₂ θ₃ … cos^n–2 θ_n–1 cos^n–1 θ_n sin θ_ndθ₁dθ₂ … dθ_n, × a similar expression in ф as before.

^{page 31 note *} See Brown, and Thomson, , The Essentials of Mental Measurement (1925), chap. x.Google Scholar

^{page 31 note †} Mackie, loc. cit.

^{page 32 note *} Mackie, loc. cit., equation (21).

^{page 34 note *} Mackie, loc. cit.

^{page 34 note †} Brit. Journ. Psychol., 1924, xv, 18, 19.