152 Proceedings of the Royal Society of Edinburgh. [Sess. 
The mathematical process adopted was begun by Christoffel ; * * * § and it leads 
to the quantities known as the Christoffel-Riemann symbols, the second 
name being added because these quantities occur in a posthumous fragment 
of Riemann’s.-]* It is proved that, under any transformations of the variables, 
these quantities are subject to linear transformations ; but they are not in- 
variants, they do not belong to any of the three types of covariants, and 
indeed (except for their connection with the equivalence of the forms) they 
are not brought into relation with the body of invariantive functions. 
8. As my aim is the construction of a complete body of invariantive 
functions (complete in the sense that they are algebraically independent 
of one another), I decided to adopt the method of Lie’s theory of trans- 
formation groups. It is quite general; it provides a system of partial 
differential equations of the complete Jacobian type, and so furnishes all 
the tests necessary and sufficient for the construction of an algebraically 
complete system of covariantive forms ; and the only difficulties, which 
occur in using it in the shape adopted in this paper, are of the kind 
that always arise in the integration of simultaneous partial differential 
equations of the first order. The first application of the theory to 
differential invariants was made by Lie himself J to the construction of 
the invariant which is the Gauss measure of the curvature of a surface ; 
and there have been later extensions and modifications of this application. § 
In order that the sequence and the arrangement of the following 
analysis may become reasoned and clear, a brief statement of the essential 
elements of the theory as it is used here is prefixed, without any proofs.|| 
The main results, stated for four variables subject to transformations 
•A ~ ^ (‘^1 5 ^2 ’ ^3 ’ ^4 )’ 
./‘o = G {x^ , ), 
x^ = , x^ ^ x^, ), 
■^4 — , x^, ^3 , *^4 ), 
* Grelle, t. Ixx (1869), pp. 46-70, 241-5. An account is also given in Bianchi’s Lezirmi 
di geometria differenziale, vol. i, chaps, ii, xi. 
t Ges. JVerke, pp. 384 et seq. Riemann died in 1866. 
I Math. Ami., t. xxiv (1884), pp. 537-578. 
§ In particular, reference may be made to a paper by ^orawski, 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. 
II 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 ; (6) the paper quoted in last 
note but one ; (c) Theorie der Ti'ansformationsgruppen, Bd. i, Kap. 25. 
(2) Campbell, Theory of Continuous Groups, chaps, iii, iv, v, 
(3) Wright (J. E.), Invariants of Quadratic Differential Forms. 
