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PROFESSOR A. R. FORSYTH ON THE DIFFERENTIAL INVARIANTS 
When the geometric values of all the invariants are substituted, the preceding 
relation (after mere simplification and division throughout by A~B®V®) becomes 
1 _ sin^ A. /cos X . sin kV 
PP^ PiPi \ P r ' 
a relation whicli can be verified independently by means of Euler’s theorem on the 
curvature of a normal section and of the expression in § 35 for the torsion of the 
geodesic tangent. 
Similarly for the curvature of torsion of the geodesic tangent to li; = 0 ; after the 
result in § 34, it manifestly will he given by 
J (W„ Wh) _ A2 
According to the theory, it also ought to be expressible in terms of the invariants 
retained. Take 
* = 2 (EM - FL) - (EN - GL) + 2 (FN - GM) ; 
then we have tlie equations 
cp- = 4J {w., w'.J J (Wo, W'o) + R.’pW (— — —y, 
\pi pJ 
= 2J (itq, iCo) J (iro, irh) — Ript'oI(R’o, w/o) + 2Y~w^w'„, 
and therefore 
w/14,1 (iv ,,!(/,,) J (Wj, wy + ic,=v' (1 - IJI 
= {2J {ivj, «q) J (w,, iv'o) — iv^wj. {iv,, w'„) + 2Vh('p<6’h}p 
which gives an expression m terms of the invariants. When we substitute the values 
of all the invariants and divide out by A~B®V^°, we find 
- - (1 1V • o ^ 1 
, , — —-siir X + 
Vi Pi/ 
rr^ 
cos X 2 sin X 
+ (— + —) sin X 
\pi Fr 
That some results of this kind, connecting p and p^, should exist, can easih* he 
seen. Wlien p^ and p. are given, p' is determined by the inclination of <^ = 0 to a 
hue of curvature ; X being given, we then know the inclination of xp = 0 to that line 
of cuivature, and so is known. Similarly for some result connectino- E and rfi 
As a last illustration of this kind, consider the invariantive expressions for and 
ds' 
m 
dn' 
Let and fq be the invariants corresponding to iq and a.^, so that 
