OF A SUPtFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
‘S‘31 
which involve the invariables is said to be a relative invariant when, if the same 
function F of new independent variables X and Y and of corresponding new deriva¬ 
tives of the transformed functions be constructed, the relation 
is satisfied, where 
/= OAF 
ax 0Y _ 0X a Y 
dx dy dij dx 
The invariants actually considered are rational, so that p, is an integer. The 
invariant is said to be absolute where p = 0. 
Now it is known, by Lie’s theory, that the property of invariance will be estab¬ 
lished if it is possessed for the most general infinitesimal transformation of x and y ; 
accordingly, we shall take 
X = X -jr i(x , y) dt, Y y rj {x , 7j) dt, 
where f and y are arbitrary integral functions of x and y. Derivatives with 
to x and y are required ; we write 
U 
mil — 
regard 
for all values of m and n. Thus, as only the first power of dt is retained, we have 
H — 1 -f -f 1701) 
The possible Arguments in the Invariants. 
3. Next, we have to consider the possible arguments of a differential invariant 
of a surface. Broadly speaking, these may belong to one or other of three classes :— 
(i) the fundamental magnitudes associated with the surface, and their derivatives 
of any order with respect to x and y ; 
(ii) functions ^ (x, y), i/; (x, y), . . . and their derivatives of any order with 
respect to x and y ; 
(iii) the variables x and y, and the derivatives of y of any order with regard 
to X. 
We consider them briefly in turn. 
4. Firstly, as regards the fundamental magnitudes : by a known theorem, a surface 
is defined uniquely (save only as to position and orientation) by the three magnitudes 
of the first order, usually denoted by E, F, G, and the tliree magnitudes of the secontl 
order, denoted by L, M, N. (If only E, F, G be given, the surface is defined as 
above, subject also to any deformation that does not involve tearing or stretching.) 
2 u 2 
