^{page 327 note *} In the case of each expression under a root-sign a certain amount of simplification is also possible, e.g., we find

^{page 331 note *} Instead of the biquadratic of this section another might readily have been obtained from the single equation numbered (9) in Professor Nanson's paper, viz.,

where

μ = A+B+C+D-½Δ−1−BC−CA−AB.

Observe also that this equation gives a much simpler expression for Δ, viz.:—

^{page 334 note *} In effect the substitutions are the same as the circular substitution if we consider cos(β+γ) cos(γ+α), cos(α+β) as invariant.

^{page 335 note *} It is interesting to note the mode in which the more general relation connecting cos (α+β+γ), cos α, cos β, cos γ, passes over into this on putting γ = 0 in the former. The result of the substitution is

where the elements of the 4th column are easily transformed into zeros with the exception of the last element which becomes

1 + 2 cos α cos β cos (α + β) − cos² (α + β) cos² β − cos² α,

so that the value of the determinant is seen to be

With this mode of degeneration may be compared that seen on p. 377 of Proc. Roy. Soc. Edin., xx

^{page 336 note *} See Quart. Journ. of Math, xviii. pp. 170, 171.

^{page 339 note *} Proc. Roy. Soc. Edin., xxi. p. 333.