OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
3G5 
— fn-' — -|- 1 ; and (h) tlie Jacobian of any one of the qualities with each of the 
rest, being 3/i — I in all. Tims the tale of the concomitants of the binary forms 
= hi + (fri" — + I) + Sn — 1 
= f + ~^7l. 
But these are relative invariants ; each of them must he divided by the aiDproprlate 
power of \, so that, as one of them is V", and the quotient is unity, thus making 
the function no longer an invariant of the surface, the number of absolute Invariants 
from this source is — 1. Thus tlie required aggregate of invariants of 
the kind specified up to order n is, in all, equal to 
^ — 1) {n — 2) + — 1 
= 2yr -h 4». 
29. But all these numbers are subject to diminution by as many units as there are 
algebraically independent relations among the invariants, which do not occur merely 
through algebraical forms, hut arise through intrinsic relations associated with the 
general theory of surfaces. One such relation, being Gauss’s equation, has already 
(§ 23) been mentioned ; so that the number 2n® + An would certainly be diminished 
by unity. It might happen that certain other combinations of the fundamental 
magnitudes of the various orders could be expressed in terms of E, F, G and their 
derivatives, the combinations being invariants of the set of binary forms, and the 
exjiressions in terms of E, F, G, and their derivatives being invariants of deformation. 
Each such relation would diminish the number 2n^ fi- by a single unit. 
So far as I am aware. Gauss’s equation is the only relation of the type indicated 
which has already been established ; but there is reason (§ 5G) for surmising that 
other relations of that type do actually subsist. 
PART II. 
Geometric Significaxce of the Invariants. 
30. The algebraically complete aggregate of the invariants of a given surface and 
of any two curves drawn upon it has been proved to be determinable by the develop¬ 
ment of Lie’s method, as used by Professor 2orawski for the invariants of deformation, 
dhe actual determination of the members of those affoueffates, which belone; to the 
lowest oi'ders, has been made. Each such invariant has a geometric significance. 
