OF A SUEFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
399 
cm 
equivalent to the rejected while the second and the third may be equivalent 
to one another. Similarly, the only derivatives of to be retained are 
P 
ds^ Iph 
.p 
.jp ('M 
d dn ds dn 
\pT 
clnds~\p'r 
four in number; and these may reduce to two. There are six derivatives of H, viz. : 
dm #H 
cm 
dm 
ds^ ds~ dn cln ds^ chd ch ds drr 
drd 
which may reduce to four; and there are four derivatives of -\j, viz. : 
P 
d“ / 1 \ d~ / 1 \ d' / 1 \ d- /' I \ 
d.s^ [pdl ’ dsdn ^ p"! ’ dnds [pdl ’ 7hd\p") ’ 
which may reduce to three. Hence there are, in all, eighteen new geometrical 
quantities arising through the inclusion of derivatives of the next higher order; aud 
these eighteen quantities may reduce to eleven. 
Now the number of differential invariants, which involve derivatives of cf) ujd to 
order 7i and tlie corresponding quantities of proper order, is {Sn + 5) by § 27 ; 
and this number is certainly subject to diminution by 1 unit, as explained at the 
beginning of § 29, so that it is (3n + 5) — 1. When n = 4, this is 33 ; and we 
know that there are 20 invariants for n = 3; so that 13 new invariants are 
introduced by the differential equations for the new order. It has been indicated 
that there may be only 11 new geometrical quantities available for their expression; 
if so, the inference Avould Ije that there are two algebraic relations among these 13. 
These relations are outside the differential equations; and the only cause from 
which they could arise would be owing’ to the intrinsic significance of the magnitudes. 
As there actually is one* differential invariant of deformation of this order (that is, 
a function involving E, F, G and their derivatives np to the third order, and no 
other quantities), the obvious suggestion is that it would laehave like the invariant 
of the lower order, due to Gauss, and would be exjaressible in terms of invariants in 
the binariant system composed of the fundamental magnitudes ; but this inference is 
only a suggestion, and cannot be regarded as an established result. 
[Note; added, 12 Mccy, 1903. 
After the manuscript of this memoir had been sent to the Royal Society but before 
the memoir itself had been read, I succeeded in definitely establishing the inference 
suggested at the end of §56. The necessary calculations are long and are of the 
* 2 orawski, I.C., p. 31. 
