OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
385 
V v/^; 
dn L 
ys 
= y 5 J (^^’ 2 ’ ^^' 2 ) I («’ 2 , “ ^5 J (w^o, io\) I ?V,) 
V®?f^ ['^^3 J ('^^25 2) 2 J (^^2> ’*^^2)}- 
When substitution is made for the various invariants, and the reduction is effected, 
we find 
d (l\ dll 
2 \ 1 _ __ 2 dBl^ 
pip" Bd~S7' 
dn W 
±ll 
ds 
^ \pj pV 
H 
p'J B ds ds 
An 
dt \p' 
H_2\ 1 r/B rfH 
p) B ds ds ’ 
which are relations obtained earlier (§ 40). They show that ~ (K) and ~ ( i) can be 
expressed in terms of the other magnitudes. 
We also have 
ds \ 
d /I 
dn Vi 
V= v/«, IJ J , ,,y I (,,, ^ ^ . 
Wg 
y3 • 
All the covaiicints tliat occur in this relation are known j wlien we substitute their 
values and reduce tlie resulting expression, we find 
^ L dB __ j_ r/B 
ds dn dn ds B ds dn p" ~ds ' 
This result, and the corresponding result obtained for H in § 43, are special cases of 
the theorem, which can be established by using the invariantive forms : If IL denote 
any quantity, ivliich is connected nnth any point on the surface and the ex'pression of 
which is indeimident* of the curve <^ = 0 through the point, then 
1 dn _ 1 dn 
ds dn dn ds B ds dn p" ds ' 
48. Similarly 
Wo 
d_ 
ds 
J 
y5 
= i { hr, I («>„ tv\) - 0v\) S + 4. 
* The value of B is not independent of the curve; but B is one of the fundamental quantities for the 
expression of properties of the curve, and its expression is an irresoluble variable. 
VOL. CCI.—A. 
3 D 
