362 
PROFESSOR A. R. FORSYTH ON THE DIFFERENTLVL INVARIANTS 
Now is an invariant, as also is hence 
is an invariant, say U, so that 
V — 2F(^oi^io + 
— 2F(^oi(/)io + G(f>wf 
[{2V^^oq + (Eq;^ 2Fjq) r Ej^q.s} 
2 {2Y~(f)i^ ^10^' ~ Eoi'?} <^01^10 
+ {2V~(^o3 — GoP’ + (Gio — 2Foi) s}(f)^Q^] 
V (E 4 - 2Fi/ + ^ + (2GF01 ErG^o FGqi);^'-^ 
' +(2GE„ + 2FF..-3FG,„-EG„)2/^ 
— {2EG,|) + 2FF,„ — 3FEq, — GEjo)?/' 
- (2EF„ - EE„, - FE,„)] 
O 
77 5 
P 
according to Minding’s expression for the geodesic curvature ; or the geodesic 
curvature of the curve <^ = 0 is the invariant 
1 o 
~ ^ v7/ 
25. It is possible to make further inferences from the results. Thus we can settle 
the algebraically complete aggregate of invariants up to the order of derivatives 
retained, when those invariants are required wliich involve derivatives of E, F, G, and 
only one function, say (j). They manifestly constitute the aggregate, complete up to 
tlie order specified, of all the functions that remain invariant when the surface is 
deformed in any way without tearing or stretching, account being taken of a particular 
curve (j) = 0, and the invariance persisting through all changes of the independent 
variables of the surface. This aggregate, algebraically complete up to the order 
specified, consists of the nine members 
Wo, V 
’ yr ’ 
w", l{wo,w^o) 
J {h'o, w"o) 
y5 
yy ’ 
H 
yi2 ’ 
ys ’ 
and 
yzi ’ 
the first five of which were given by Professor Zorawski, who considered the specific 
aggregate only up to one order lower. 
