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VIIL The Dijferential Invariants of Space. 
By A, K Foesyth, Sc.D., LL.D., F.R.S., Sadlerian Professor of Pure Mathematics 
in the Uiiiversity of Camhridge. 
Eeeeived June 18,—Eead June 18, 1903. 
The present memoir is iiitendefl to carry out investigations, concerned with the 
difierential invariants of ordinary space and of surfaces in that sjjace, similar to those 
in a former memoir,"^ concerned with the differential invariants of a surface and of 
curves upon that surface. 
The method used in the former memoir is used here in Avhat is the obviously 
natural development. It is based upon the method,! Avhicli was originated by Lie 
and amplified by Professor Zorawski, When applied to two-dimensional invariants, 
it proved possible to modify and simplify the later stages of the calculations by 
making them dependent upon the concomitants of a system C)f simultaneous binary 
forms. When applied to three-dimensional invariants, it proves })Ossible to effect a 
corresponding simplification in the later stages of the calculations; the required 
functions are found to be the Invariants and the contra variants of a system of 
simultaneous ternary forms. 
The expressions for an algebraically complete aggregate of invariants up to the 
third order inclusive have been olitained. The calculations necessary for the con¬ 
struction of these invariants were laborious ; indeed, the calculations for tlie invariants 
of the third order are so long that in this memoir they have been suppressed, and 
only the results are given. It may be mentioned Incidentally tliat, among the 
invariants of the third order, six (in particular) occur possessing a special property. 
They can be so taken in the algebraically complete aggregate as to coincide with six 
quantities which were proved by Cayley to vanish on account of the intrinsic 
significance of the fundamental magnitudes. These six eipiations are the generalisa¬ 
tion, to surfaces not orthogonal, of Lame'S six equations for triply orthogonal 
surfaces. 
The geometric significance of practically all the differential invariants of the first 
order and the second order has been obtained. I have not yet attempted to identify 
* ‘ Phil. Trans.,’ A, vol. 201 (1903), pp. 329-402. 
t Eeferences are given in the memoir just quoted. 
VOL. CCH.-A 353. 
26.11.03 
