200 
Proceedings of the Royal Society of Edinburgh. [Sess. 
for r=l, . . .,6. These equations are 
V6 
+ d 
)PA 
(f- 
y.+ 
(«--■ 
)Ps + ' 
(f 
(f ")'■• 
-0, 
(- 
Ve 
+ + 1 
)P3 + I 
)Ps + 
(!*'>• 
= 0, 
(~ 
\6 
+ 1 
)p,+i 
)P.-H 
)Pa + l 
= 0, 
/T 
V6 
-ft 
)Pi+( 
)?3 + ( 
)P3 + ( 
= 0, 
(- 
Ve 
^s' 
)p>+( 
)^2 + { 
■5^,' 
v(£ 
)?s + ( 
;i +!>>.*( 
= 0, 
(N. 
Ve 
+ n 
)Pi+( 
)P3+( 
[h\ 
)P3 + ( 
= 0. 
Hence there are six principal values of being the roots of the sextic 
equation 
A L T S N 
’ 6+’ 6+*’ ’ e 
+ m, 
+ 1 , 
+ t, 
!-■ 
which, when written in full, is 
□ Dn iD^n ^,D^n 
= 0, 
d:6 ‘ 
(£4 
+ 
All the coefficients in this equation are differential invariants ; and thus, 
if these principal curvatures are (^^^,(^2 5 • • •’ ^6’ have the significance 
of six of our differential invariants in terms of symmetric combinations of 
these principal curvatures. In particular, we have 
6,H-6, + 63 + e, + e, + 6,= 
ei®#3eAee=-^. 
Further, there are six principal pairs of directions at a place in the 
amplitude, respectively associated with the six principal curvatures ; each 
