page 207 note ^* This answer would be of the kind envisaged earlier, since the volume of the ellipsoid is expressible in terms of λ and the determinant of the quadratic form, which is the sole algebraic invariant of the form.

page 207 note ^† For brevity, I am using the word congruent to mean that the one body can be derived from the other by a translation only.

page 208 note ^* To avoid complications which are irrelevant to the argument, I have not defined the term “body” It will suffice we understand it as meaning any set of points which is open (i.e. every point is an interiror point), and possesses a volume, in any of the various ways in which “volume” can be defined.

page 209 note ^* See a later account by Blichfeldt in Math. Annalen, 101 (1929), pp. 605-8.

page 210 note ^* This proof was given by Hermite in 1853 (Oeuvres, I, p. 288). Of course he could not appeal to Minkowski’s theorem; in place of it he used his own result on the minimum of a positive definite quadratic form. Professor Forder has drawn my attention to a somewhat similar proof by Grace, J London Math. Soc., 2 (1927), pp. 3-8.

page 210 note ^† In connection with this step, I cannot refrain from referring to Professor Mordell’s note in this Gazette, 26 (1942), p. 52, on Lewis Carroll’s diary. Lewis Caroll found considerable difficulty in proving that if m is a sum of two squares, so is 2m.