^{page 151 note *} For convenience, I refer specially to the general account given by him in his memoir “Die Grundlage der allgemeinen Relativitätstheorie,” *Annalen der Physik*, Bd. xlix (1916), pp. 769–822.

^{page 151 note †} Particular mention should be made of three important and comprehensive accounts of various investigations connected with the general theory of relativity which have recently appeared, viz.: Donder, De, La gravifique einsteinienne (Gauthier-Villars, Paris, 1921); Pauli, W. Jr., “Relativitätstheorie,” Encycl. d. math. Wiss., Bd. v, 2, Heft 4 (1921); Marcolongo, , Relatività (Principato, Messina, 1921). The amplest references are given.

^{page 152 note *} *Crelle*, t. lxx (1869), pp. 46–70, 241–5. An account is also given in Bianchi's *Lezioni di geometria differenziale*, vol. i, chaps, ii, xi.

^{page 152 note †} *Ges. Werke*, pp. 384 *et seq.* Riemann died in 1866.

^{page 152 note ‡} *Math. Ann.*, t. xxiv (1884), pp. 537–578.

^{page 152 note §} In particular, reference may be made to a paper by Žorawski, , Acta Math., vol. xvi (1892–3), pp. 1–64; and to two papers by myself, *Phil. Trans.*, vol. 201 (1903), pp. 329–402, *ib.* vol. 202 (1903), pp. 277–333. In the present paper, the detailed calculations are systematised in a fashion distinct from the processes in these papers.

^{page 152 note ‖} For the establishment of the various propositions, reference may be made to the following:—

(1) Lie, , (a) Math. Ann., Bd. xvi (1880), pp. 441–528; (b) the paper quoted in last note but one; (c) *Theorie der Transformationsgruppen*, Bd. i, Kap. 25.

(2) Campbell, *Theory of Continuous Groups*, chaps, iii, iv, v.

(3) Wright (J. E.), *Invariants of Quadratic Differential Forms*.

^{page 153 note *} For the nature and properties of a complete Jacobian system, see my *Theory of Differential Equations*, vol. v, ch. iii.

^{page 153 note †} For the purpose of estimating the number, a coefficient of the quadratic form and all its derivatives are reckoned as independent.

^{page 160 note *} These ten sets of symbols can be grouped into a single set, giving

for all values of *p, q, r, s*; but the forms are not so easy to manipulate as are the more diffuse expressions in the text.

^{page 165 note *} For the characteristics and the properties of such a system, see my *Theory of Differential Equations*, vol. v, §§ 37–46.

^{page 165 note †} *Loc. cit.*, p. 86.

^{page 176 note *} The reason for the apparently arbitrary notation A, …, V for the integrals will appear later (§ 26).

^{page 176 note †} In the double summation for *s, t* = 1, 2, 3, 4, all the combinations are to be taken. Thus we must retain

while A_{12}, = A_{21}; and so for other combinations in A and in the succeeding integrals.

^{page 181 note *} See a memoir of my own, “Systems of Quaternariants that are algebraically complete,” *Camb. Phil. Trans.*, vol. xiv (1889), pp. 409–466, specially pp. 430 *et seq.*, 460 *et seq.* The subsequent analysis is modified from the analysis in that memoir.

^{page 182 note *} The quadratic line-complex was first considered by Plücker, , “New Geometry of Space,” Phil. Trans. (1865), pp. 725–791, and subsequently in his *Neue Geometrie des Raumes* (1868). I have preserved his notation so far as regards the coefficients of the complex, because it has been used by other writers, and variations of notation tend to be confusing; but a notation which runs

would codify the expression of the operators. Thus the operator H_{1} becomes

and so for the others: the relation N + O + V = 0 becomes *q* _{16} + *q* _{25} + *q* _{34} = 0, and similarly for other invariants: the notation immediately suggests (or is suggested by) the umbral notation used in my memoir which has just been quoted. As the umbral notation is not used here, I have adhered to the Plücker coefficients.

^{page 184 note *} See the memoir quoted in § 26 (footnote), at p. 431.

^{page 189 note *} *Math. Ann.*, vol. vii, pp. 145–207.

^{page 195 note *} This selection of the independent, invariants, so as to make up the aggregate, was effected by using the canonical forms of § 32.

^{page 202 note *} Aronhold, , Crelle, t. lxii (1863), pp. 281–345; Gram, , Math. Ann., t. vii (1874), pp. 230–240.

^{page 203 note *} The notion of amplitudes with a constant (or variable) measure of curvature originated with Riemann. The literature dealing with amplitudes having a constant measure of curvature is copious, and the developments really belong to the domain of differential geometry. Some account is given by Bianchi, *Lezioni di geometria differenziale*, t. i, cap. xi.

^{page 203 note †} He obtained it for a space of *n* dimensions.

^{page 203 note ‡} Bianchi, *Lezioni di geometria differenziale*, t. i, p. 72; Einstein, *Ann. d. Phys.*, Bd. xlix, p. 800.

^{page 204 note *} “Deshalbs liegt es nahe, für das materiefreie Gravitationsfeld das Verschwinden des Tensors B_{μv} zu verlangen,” *l.c.*, p. 803.

^{page 212 note *} *Ann. d. Physik*, Bd. xlix, pp. 789, 801.

^{page 212 note †} *Sitzungsb. Berlin* (1915), p. 801.

^{page 212 note ‡} *Ib.* (1916), p. 192.