DIFFERENTIAL INVARIANTS OF SPACE. 
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Property of In variance. 
2. A combination F of any number of functions of u, v, w and of the derivatives ot 
these functions is said to be a relative invariant if, when any new independent 
variables if v', v/ are introduced, and the same combination F' of the modified 
functions and of their derivatives is formed, the relation 
F = n-^F' 
is satisfied, where 
0 (u, V, iv) 
The invariants actually considered are rational, so that /x is an integer. The invariant 
is called absolute when /x = 0. 
It is a known theorem that the property of invariance is possessed in general, if it 
is possessed for the most general infinitesimal transformation ; we shall therefore take 
u' = w + ^ ( t/,, V, iv) dt, 
v' z= V Tj {u, c, w) dt, 
iv' = ly -f { (m, V, -w) dl, 
where y, C ^^re arbitrary integral functions of u, v, w, and dt is an infinitesimal 
rjuantity of which only the first power is retained. 
Derivatives with regard to u, v, w are required ; we write 
dv"‘ d'W'‘ 
for any quantity 0 and for all values of /, m, n. With this notation, we at once have 
the retained value of H in the form 
= 1 + (^100 + 'l?010 + ^ooi) 
Arguments of the Invariants and their Increments. 
3. As regards the possible arguments of a differential invariant of space, we shall 
have the fundamental magnitudes of the first kind and their derivatives. It is 
conceivable that properties of surfaces in the space and of curves in the space will be 
involved; provision for the possibility will be made by the introduction of functions 
such as {u, V, iv). One of these, equated to zero or to a constant, will give a 
surface : two such surfaces will give a curve or curves : three such surfaces will give 
a point or points. 
It is not difficult to see that u, v, iv will )iot occur explicitly in the expression of 
the invariant, nor will any function of them such as (f) {u, v, w) occur explicitly; but 
