y88 PEOFESSOR A. R. FORSYTH ON THE DIFFERENTIAL INVARIANTS 
The five invariants that remained for interpretation were 
J {tv^, J (-zro, 1(147.2, xo'"^ 
V ’ V“ ’ ’ ys ’ y4 - ; 
after the changes that have been made, the five are 
XO^ J (447^, 447 . 3 ) J (Ws, W'"o) I [W., 447"b) 
V’ y^ ’ yi ’ ys ’ y4' ■ ’ 
of which the last may also be written I (Wg, V'f The interpretation of the 
first two of these is easily obtained ; for the interpretation of the remaining three, 
which involve derivatives of i/> but not of <^, the results of earlier interpretations 
can be used. 
50. For the purpose of the inteiyretation, we need certain geometric j^roperties 
of the curve xjj ^ 0. Let ds denote an elementary arc along the curve, and dii,' an 
element along the normal to the curve; and let 
A — 
dn'‘ 
Further, let — denote the circular curvature of the geodesic tangent to if/ = 0, 
P 
and ~ the curvature of torsion of that tangent; also, let 4- denote the geodesic 
P ^ 
curvature of xfj = 0. Then Wg, I (Wg, W'L), J (Wg, W"b), stand to xf/= 0 in 
precisely the same relation as ?4’2, xi/'„, I (^t^g, xv"„), J (w.^, xv",) to (j) = 0 ; and therefore 
Wo _ *0 
y2 — ^ > 
W'"o _ _ 2 A3 
V p ^ 
IIWjaJWA) _ 9 /_ _a \ 
Ua p"J’ 
J W^A) _ „ 4/a 
^ ds'- 
Moreover, we have 
^] j / _ ^10. 
yWg’ ds 
so that, if X be the angle at which <f) = 0 and if/ = 0 intersect, we have 
