384 
rKOFp]SSOR A. i;. FOIJSYTII OX THP: DIFFERENTIAL INVARIANTS 
another illustration of the leinark in § 36. It may similarly he proved that 
r/^K _ r/'^K _ 1 (IK 
(W p"<Jn' 
These are particnlar cases of the theorem, which can similarly be established : If 
n (Jenote (uiji (jiKrntitp, trlnch is cdunecfed irith any point on the surface and the 
(xpression of which is independont of the curve (f) = 0 through the point, then 
ddJ. _ d'n __ I do 
ds~ dd ~ p"dn' 
46. Proceeding’ to the identification of the two invariants Hla-'g) and which 
involve the coefiicients of a-' we constrnct I (iv.,, ir"f) and I (-w„ id'd, and 
as ~ ' dn ' 
we find 
v/ w. 
' i = A'C he™ - liF/ + Ok )+ 
2V 
• > 
^''N/!;rTi"'*}=2v4!E(E<.-2K.m + G0-F(Em-2n+G/ai^„,+..^^ 
Let these covariajits be denoted by eg, y. respectively, so that 
ig = {Km — 21/ + G/.’) (^,,1 + . . • 
y = {E (Ea - 2lLa + G/) - F (E/h. - 2F/ + Gfl-)} + . . . 
Then C| can he used to replace H (a/g) in the aggregate as aj replaced H and 
Co can he used to replace ft> {up) in the aggregate as a., replaced <i> (ag), in each case 
without affecting the comjjleteness of the aggregate. The index of ig is 7 ; that of 
Co is 8. Their sionificance is niveii hv 
~ O O » 
4/. We have 
v'Ga.o _ HKwIIA 
Kis 
and 
yl,, J J {^K, w") - a’"o I {up idf) + - . 
' tl r, w V V M.Lt 
