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# XII.—The Concomitants (including Differential Invariants) of Quadratic Differential Forms in Four Variables

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The characteristic properties of quadratic differential forms, involving two or more independent variables, have been investigated from the days of Gauss onwards. Initially, the discussion arose for the case of two variables; and, in its most general trend, it was concerned with a form

associated with surfaces, E, F, G being integral functions of p and q. But the relation does not, by itself, define a surface completely. When a surface is deformed in any manner, without stretching and without tearing, the quantity ds2 preserves its measure unchanged; the measure is of fundamental importance. Consequently, the measure must remain unchanged whatever changes of the variables are admitted. Further, changes of the variables, of any kind, allow the existence of covariant concomitants which therefore persist through these changes. In particular, there is one function, of E, F, G and of their derivatives up to the second order inclusive, which persists unaltered; it is the Gauss measure of the curvature of the surface.

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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. 164; 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. 441528; (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 A12, = A21; 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. 725791, 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 H1 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. 281345; Gram, , Math. Ann., t. vii (1874), pp. 230240.

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 Bμv 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.

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