3-28 
rROFESSOPv A. ]l. FORRVTH OX T?IE 
say. Now 
so that 
IFeiice 
(hi ■= cos SN . r/s, 
(Iff) clfb QAT 
' = / cos oX . 
(Is (In 
1 = L - fl cos SN sin o 
an fhi an 
~~ dndn' dn"' 
tlie skew symmetric expression required: 
It is easy to verify that the values obtained for the invariants satisfy the relation 
in § 10. 
Signijicance of the Invariants of the Second Order. 
-32. Passing; now to the differential invariants of the second order associated with a 
single surface, we shall adopt anotlier metliod of identifying them geometrically. The 
functions are invariantive through all changes of the independent variables and 
therefore possess the value given Ijy any particular selection of variables. Now one 
simple transformation of the vaiiables is that wliicli makes x, y, z respectively equal 
to tlie independent variables, so that we take 
X — V, 
V 
V. 
— v\ 
Witli these values, we have 
and 
Also 
o, = 1 , 6 = 1 , r = 1 , ./’ = .7 = 0 - 6 = 0 ; 
A = 1 , B = 1, f ^ = 1 , F = 0 , G = 0 , H = 0 ; 
Id = 1 , 
__ 
y*/1/1 n — ■?; TV • 
ex' nf C:" 
a = 2(f).2(^Q = f — *^^011 — 
b = = '2(f)yy. g ~ 
C = '2ffyy2 — '2(f);:, ll = '2(f>]H) = 
" ‘ = R 
^1 
A ^ = (</>yy + </>.---) fkf + {4>-^ + 
— 2<f)j.(j)j^(f)j-y — 2(f)j.(f);(f)r-. — 2(f)y(f);(f)y;, 
A + Ayy "A 
-Ai"' 
Hence 
