278 PROFESSOR A. R. FORSYTH OX THE 
the invariants proper to the third order; there is reason to suppose that, when they 
are completely identified in association with even a single surface in space, they can 
be used to establish two fundamental geometrical relations affecting the suiface. 
The Fundaiaeatal Moymtudes. 
1. The independent variables of position in space are taken to be u, v, vj. That 
position is also defined in the customary manner by rectangular Cartesian co-ordinates 
X, y, z ; and then u, v, iv may be regarded as three independent functions of sc, y, z. 
Conversely, we shall assume that x, y, z are expressible as functions of u, v, w, which 
are regular in the vicinity of any assigned position, save for singular lines or points 
with which we are not concerned. Moreover, if 
u = u {x, y, z), V = V {x, y, z), n' = w (x, y, z), 
the surfaces u = constant, v = constant, tv = constant, form a triple family; it v ill 
not be assumed that the triple family is orthogonal. The parameters u, v, w of the 
surfaces are sometimes called curvilinear co-ordinates. 
Fundamental magnitudes for space arise in the expression of a distance-element in 
terms of u, v, tv, du, dv, dw. Denoting this element by ds, we have 
</.s' = dF -b dy" -f dz^ 
— a du~ + 27i du dv + 2g du dtv + h d F -f 2 /dv dtv fi- c dw~ 
= {a, h, c, /, g, h\du, dv, divf, 
in the usual notation, where 
^dxdx _ V 7. _ V 
~^dudv’ 
the summation being taken over the three variables x, y, z in each case. 
The quantities ct, b, c, f, g, h may be called the tundamental magnitudes of the fiist 
kind, as involving derivatives of only the first order. Other fundamental magnitudes 
may exist in association with derivatives of higher orders they are ignored for the 
purposes of the pre.sent investigation. 
* Those magnitudes, which are the natural generalisation of the magnitudes L, M, X in the Gaussian 
theory of surfaces, are not independent quantities; they are proved by Cayley, ‘ Coll. Math. Papers, 
vol. 12, p. 4, to be expressible in terms of derivatives of a, b, c,t, g, h. 
It may be added that the memoir by Cayley, which has just been quoted, contains the establishment 
of the six intrinsic equations mentioned in the introductory remarks. For reasons which will appear in 
the course of the memoir, I have found it desirable to deviate to some extent from Cayley s notation. 
