OF A SUEFACE, AND THEIE GEOMETEIC SIGNIFICANCE. 
379 
regard to .s and to u, of the invariants already interpreted, identify tlie new forms by 
means of some member of the complete aggregate, and tlms we obtain the interpreta¬ 
tion of that member. Accordingly, we sliall usually state the results without the 
calculations, the laborious character of winch is greatly lightened by using the leadino" 
terms of the covai-iants oidy. 
We have 
rA 
iV 
i I 
a;.. 
{n>J (at., aA) — a Ad (ao, v\,)}. 
Inserting the cables of the invariants winch occur in this equation, and using the 
relation 
obtained in S 33. we have 
O 
P •' J P3 2P,= 1 
’O/x A rh\p'l + p’V( 
, 0-0.^ ,'/ / I 
B‘ (/ /1 
and this easilv leads to the relation 
d- 
d- 
A - 
2 <I / Bd\ 
3 d 1 
dr- 
p') 
ds- 
' A ■ 
P 
& d, \p'vl 
p" du ' 
P 
= 2A(-L') + -iFp I , 
^ ' p T 
P 
3 ' 
P 
H 
P 
/I 
on using the expression for obtained in ^ 40. The fact that the value of 
A/I\ A/1 
<h-\p'! <iAp\ 
is different from zei'o is another illustration of the remark in § 36. 
We also have 
^ ,h, (v-'f) ~ ^ aJ 
+ 
a.A) - ^ J (hu, w",) J (aq, 
when we substitute for the respective invariants and rerluce, we obtain an expression 
for J (a'.,, aq) in the form 
J ('"-dfil /1 I 9 / 1 3 K , 3 i/B r d /1 
P"t'J ^ / ' B <Is [drdp'J Br' ds J’ 
2 dB 
3 o 2 
