31)7 
OF A SURFACE, AND TilEIR GEOMETRIC SIGNIFICANCE. 
I 
(I / I \ 
, f ^ 
V 
, (' 1 
p' 
ds [p'J 
(In \ p ! 
' dsAp'> 
' dsdn\p'/ 
' (In(.Is[p'J 
(hP \ p' 
H, 
(IR 
f/H 
cFH 
dm 
(PH 
(PH 
ds 
dn 
’ dP 
dsdn 
(bids 
(hP 
B, 
(IB 
dB 
(7'B 
(PB 
(PB 
(PB 
ds 
(In 
ds- 
(Is (hi 
du ds 
(hP 
(Is \p 
I ,J / I 
1 
/ 
T 
<1 / 1 
(Jii \ p". 
and the derivatives of But not all of these can he retained as independent magni¬ 
tudes. In § -to it was proved tliat 
' A 
- - _ 
('.H - 
ds ' 
r'B (Is 
d 
'' 1 1 
II 
1 
1 (/B 
du ^ 
l ' ^ 
\ — ' 
\ p / 
B (Is 
r/ / 1 \ 
P > 9 
ihi 
P ! 
+ W (—7 ) + --TT' 
(/.'< ' p p T 
f/H 
<h 
1 
S) tliat the first derivatives of and conseipiently also the second (and liigher) 
T 
derivatives, are exjiressihle in terms of the derivatives of the other (piantities 
retained.Again, in §§ 41, 43, 47 it has lieen shown that the (piantities 
(Is (In. 
- V ) 
>\p'^ 
(In (Is ' p / 
(PH 
_ (/'E 
ds du 
(In (Is 
(PB 
_ (PB ^ 
ds (In 
(In (Is 
are expressible in terms of the derivatives of the first order; so that it is sufficient to 
(P / 1 \ (/'H (/-B , . , (/- / 1 . <PR #B 
( 7.1 ' 
retain 
(Is (In 
/’.Vi A' 
' ’ duds’ duds 
Further, in 
dsdn " duds' p duds on (is 
§ 41, it avas proved tliat 
* It is proved in D.vr.aoux’s ‘Theorie gOierale des Surfaces,’ vol. 2, p. 3(10, that the quantity 
d /I 
(U \t / \ p! p 
which occurs in the first of the two equations, is the same for two curves tliat ha\ c the same tangent. 
