OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE. .371 
system, when H (h^), w'o, J {vj.2, w'.^) are retained; as a matter of fact, the 
relation 
w'o) = I {ui 2 , w'2) Wow'2 — H (« 3 ) w'o' — H {w'o)uY 
subsists. Now the signiticance of I [iVo, iv'^) is known : we have 
I {iVo, iv'o) _ 1 , 1 
Pi P 2 
Accordingly, we substitute the values that have been obtained, and we find 
'2 
' - + M - 1 
^'\pl piJ p^ plpi 
1 _ I 
Pi p' 
again the well-known relation giving the torsion of a geodesic at any point, 
torsion vanishes when the geodesic is a tangent to a line of curvature. 
' - ' ) 
/ h 
P Pi/ 
This 
Interpretation of the Remaining Invariants Associated with Wo, w'o, up 
36. We require the derivatives of iVo, w'o, w"o with respect to the arc; for this 
puipose we shall use the property already quoted (§ 30)—that a binariant is uniquely 
determined by its leading term which, in the present instance, is the term involving 
the highest power of Writing generally 
u -f h(f 
we have 
ds ~ ~ + <kiiy') + • . . 
, + +/od/') + . . . , 
so til at 
( !') / 
gg ~ (^./V*ii ^9^20) + • • • 
+ ^ofVio + • • • 
= Y2 - //«) ^01 + • • • !■ 
+4^ [{/(EGio-FEoi)+^ (EEoi-2EFio+rEio)+Vyio} • • ]• 
Firstly, let J, g, h _ E, F, G, so that u becomes ; then, on reduction, we find 
/ _ J {w^, IV g) . w^ (/-ppi ORR _L PR \ 
^ 2 ds V~ ■ ^ 10 + GEjo) (p^^2 + • • 
3 B 2 
'} 
• )■ 9 
