OF A SUKFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
333 
taiigent-liiie defined by dx : dy, and tlie arc derivatives are efiected along the geodesic 
tangent."^ 
Accordingly, the quantities of the class under consideration that may occur are 
E, F, G and their derivatives up to any order, together with the fundamental 
magnitudes of any order above the first, but without any derivatives of these 
fundamental magnitudes, t 
5. Secondly, as regards functions (f) {x, y), xfj (a’, y), . . . and their derivatives : we 
do not retain the functions themselves, but only their derivatives, for the following 
reason. The invariantive property is usually some intrinsic geometric property 
connected with a curve on the surface represented by = constant or zero, 
xjj = constant or zero, and the like. Accordingly, we retain only derivatives of these 
functions up to any order; the equations ol transformation will show the connection 
of the order of these derivatives with the order of the derivatives of E, F, G retained. 
6. Thirdly, as regards x, y, and the derivatives of y with respect to x up to any 
order : it is clear that x and y will not occur explicitly, for their presence cannot 
contribute any element to the factor 12 ; it is also clear that they will not occur 
explicitly, for the further reason that their increments involve ^ and y but not 
derivatives of f or y, whereas all other increments involve derivatives of £ or y, but 
neither ^ nor y themselves. Further, after the retention of quantities of the second 
class, we shall not retain y. For let the value of y' belong to a curve x// =0 on the 
surface, so that 
We know that 
'/'iO + // ^01 ~ d- 
EG - F3 ~ 
where I is an absolute invariant; if then we have a difierential invariant involvinir 
O 
y , we turn it into one involving and xpQ-^, by writing 
?/ = - ; 
Vtji 
while if we have one involving and we turn it into one involving y', by 
writing 
^ ^10 ^ [t EG-F^ 
I y’ I E + + G^'^j • 
It would therefore be unnecessary to retain y', when we retain first derivatives 
01 any number of functions in an earlier class. 
Similarly, it can be shown to be unnecessary to retain y'', when we retain second 
derivatives of any number of functions in an earlier class ; and so for other 
derivatives of y with respect to x. 
* See § 31, post. 
t It will appear that the iutrocluctioii of these magnitudes not merely avoids the difficulty as regards 
the derivatives of L, 21, N, but also secures a substantial simplification of the expressions of the (htlereutial 
invariants. 
