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IX. The Dlfierential Invariants of a Surface, and their Geometric 
Significance. 
By A. E. Foesyth, M.A., Sc.D., F.R.S., Sadkrian Professor of Pure Mathematics 
%n the University of Cambridge. 
Received February 14,—Read March 5, 1903. 
The present memoir is devoted to the consideration of the differential invariants of 
a surface; and these are defined as the functions of the fundamental magnitudes of 
the surface and of quantities connected with curves upon the surface which remain 
unchanged m value through all changes of the variables of position on the surface. 
Ihe idea of differential parameters for relations of space appears to have been 
introduced^ by Lame ; it is to Beltrami^ that the earliest investigations of the 
corresponding quantities in the theory of surfaces are due, as well as many detailed 
results, t 
It IS natural to expect that these differential invariants would belong to the 
general class of differential invariants which constitute Lie’s important generalisation 
0 tie original theory of invariants and covariants of homogeneous forms. This 
association has been effected]: for some classes of differential invariants by Professor 
ZoRAwsKi, and lie has obtained the explicit expression of several of tlie individual 
functions. 
Professor .^orawskTs method is used in tlie present memoir. In applyiim it a 
considerable simplification proves to lie possible ; for it appears that, at a cmfain 
stage^ 111 the solution of the partial differential equations cliaracteristic of the 
invariance, the eipiations which then remain unsolved can be transformed so that tliey 
become the partial differential equations of the system of concomitants of a set of 
simultaneous binary forms. The known results of the latter theory can therefore lie 
used to complete the solution of the partial differential equations, and the result gives 
the algebraic aggregate of the differential invariants. 
This memoir consists of Lvo parts. In the first, the investigation just indicated is 
carried out; and the explicit expressions of the members of an aggregate, algebraically 
^ In Ins memoir, “Sulla teorica geiierale del parametri differenziali,” ‘Mem. Acc. Bologna,’ 2nd Series, 
vol. 8 (1869), pp. 549-590, Beltrami gives a sketch of the early history of the sul.ject. 
T An account of the theory, developed on the basis of Beltrami’s researches, is given by Dareoux 
iheorie g^n^rale des surfaces,’ vol. 3, pp. 193-217 ; he also gives references to Bonnet and Laguerre ’ 
+ In a memoir hereafter quoted (§ 1). 
VOL. cci. —A 339. 2 u 
24.G.03 
