363 
OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
26. If we lequire the aggregate of invariants of this class involving derivatives of 
E, F, G up to order n — 1 and derivatives of ^ np to order n, the number of 
members in that algebraically complete aggregate can be obtained. The total numher 
of members is 
o 
n-; 
it IS composed of i (w — I) {n — 2) quantities which do not involve the derivatives of 
(f), these quantities being called Gaussian invariants of deformation, and their number 
haGiig- been determined'* by Zorawski ; and of in (r. + 3) - I quantities, each of 
which involves derivatives of f To make up the latter aggregate of in (n + 3) - 1 
quantities, we need (m addition to the binary forms already used) otlier binary forms 
of orders 4,5,..., n ; among these, the binary form of order m (for all values of m) 
has and — <f)^Q for its variables, and its coefficients are linear in the derivatives of 
(f) up to order m inclusive; and the members, that would occur in the simplest 
expression of the aggregate through the existence of the binary form of order m, 
would be the quotients (by proper powers of V) of the binary form itself, of the 
m — I (Hermite’s) associated covariants, and of the Jacobian of w, and the Iniiary 
form, making m + I in all. Thus the total numberf up to order n is 
1 d" 3 + 4 “b . . + R 
. . ~ “k J) — Ij 
the number in question. 
27. If we require the aggregate of differential invariants, which involve derivatives 
of E, F, G, L, M, N up to order n — I and derivatives of a single function up to 
order r, the number in that algebraically complete aggregate can lie obtained as 
follows. We can replace the derivatives of L, M, N of the specified orders Ijy the 
introduction of the fundamental magnitudes of orders 3, 4., n I defined as 
the coefficients in the various powers of and in the complete expression of the 
quantities 
^Ll\\ (l \ 
ds\pr ds^\pf'' 
where p is the radius of curvature of the normal section through the tangent defined 
fs’ arc-differentiation of is taken along the geodesic 
tangent|. 
When n = 2, the system of bmariants is composed of three quadratic forms with 
theii three disci iminants, a cubic form with its set of two associated covariants, and 
* In his memoir, § 13. 
^ T It will be noted that VV-^ in the aggregate in § 22 is a Gaussian invariant of deformation, and so is 
included among the (ft - 1) {n - 2) quantities which do not involve 
I For the significance of this remark, see § 31, post. 
3 A 2 
