389 
OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
and therefore 
Also 
and therefore 
J IVo) 
Y2 
= AB cos X. 
sin X = V 
UIx dy 
\dd ds 
dy dx\ 
ds' ds) 
V'wM: 
= AB sin X. 
We can regard the quotient of the last two invariants as giving the angle X ; and 
we can regard the sum of tlieir squares as defining the magnitude A. Clearly 
J' (a’l, IV,) + = VCX^B'^ 
= iv^Vl,, 
a relation already used ; it may be further used to replace J iv,) by W„. 
51. The general theory shows that all other invariants, which involve deiivatives 
of cf> and xp up to the second order inclusive, derivatives of E, F, G of the first order, 
and the fundamental magnitudes of the first three orders, can be expressed in terms 
of the aggregate already retained, composed of the eleven invariants selected in § 39 
and the five just identified, viz. :— 
u\ 
V 
J (W^, IV,) 
y2 
or 
Wo 
V2 
r/// 
J (W.0, W^A) I (Wo, W%) 
It is not without interest to illustrate the property b}^ one or two simple examples. 
Consider the circular curvature of the geodesic tangeiit to xp = 0 ; after the result 
in § 34, it manifestly will be given by 
W'o ^ A2. 
n' ’ 
V p ^ 
according to the theory, it ought to l^e expressilfie iii terms of the invariants retained. 
Take 
Vi = L(^oi^()i — M ((/joi'Aio + '^lu'Aoi) + > 
then we have the equations 
= to'.,W', - IV,m (w',), 
tv,V, = iv',J ('iv,, u>,) — ^v,J (iv„ iv ',); 
and therefore 
IV,^ {w',Yd', — U?i^H {w',)\ = {w',J {W„ UK,) — IV, J {-w„ 
