OF A SUEFACE, AND THEIE GEOMETRIC SIGNIFICANCE. 
361 
’ ys ’ yis 
I («t) J vu) 
TI 16 S 6 thiity difFcrGiitial invariants constitntG the filg€Jjvciicfill}j coiivplete acjQveQeite 
m terms of ivhich all invariants, involving (i.) some or all of the derivatives of the 
finidamental magnitudes E, F, G, L, M, N, up to the second order inclusive, as well 
as the magnitudes themselves, (ii.) the magnitudes P, Q, R, S, a, /3, y, 8, e, (iii.) and 
the derivatives of two functions (f> and xfj up to the third order inclusive, can he 
expressed alr/ehraically. But it is to be noted that this inference is concerned solely 
with the partial differential equations, and it assumes that the various quantities 
E, F, G, L, M, N, and their derivatives are independent of one another; if any 
relations should subsist, owing to the intrinsic nature of the magnitudes, then the 
niimher of invariants in the above complete aggregate will be diminished by the 
number of relations. 
Now one such relation is known; it is the relation commonly associated with 
Gauss’s name, and it expresses LN — AP in terms of E, F, G, and their derivatives 
lip to the second order inclusive. But LN — AI® is H in the foregoing set; and, 
as will he seen later (§ 35) in the course of the geometrical interpretation, we have 
V = 411 (m.) H (nL), 
so that the nnmher must he diminished hy unity. Accordingly, the algehraically 
complete aggregate of differential invariants, involving the magnitudes vp to the 
specif ed order of derivation, contains 29 memhers; in terms of these memhers, every 
other invariant, involving the same magnitudes up to the specif ed order of derivation, 
can he expressed edgehraiealhj. 
24. As an illustration of the remark in § G, we can obtain Minding’s ex})ression for 
the geodesic curvature, quoted* hy Professor Zorawski as an invariant. Let i/) = 0 
he the equation of the curve, then 
^10 + ^oiV' — 0 , 
so that 
= 0 , 
VOL. CCI.—A. 
* Lor. cit., p. 6.3. 
3 A 
