372 
rKOFESSOR A. E. FORSYTH ON THE DIFFERENTL4L INVARIANTS 
and consequently 
v/ an 
i _ J (»’ o, 
ds \yv 
Secondly, let /, r/, h = L, M, N, so that u becomes iv'. ; then, on reduction, we 
find 
= v:-i + ^,3- ■ ( au, u-'h) — n-'hJ (wu, ich)} 
+ [(EG,o - 2 FF,o + GE,o) <^01 + . . . 
and consequently 
= ys + (^G> ^^" 2 ) - 
Thirdly, let /, g, h = a, h, c, so that it becomes w'\, ; then, on reduction, we hud 
= J v’ + wt * - 2FF,„ + GE„) <Aoi + . . . ) S. 
and consequently 
The hrst of these gives 
^ ■ VV “ ^ ye' 
J {w., w”d) _ 2 go f/B 
ye ds' 
and the third of them, taking account of the value of ir'k which has already been 
obtained, gives 
y? 
ds \p'' 
The second of them can also be used to identify J (an, id'. 2 ), because all the other 
quantities occurring in the relation have been identihed ; the value is 
J (?r,, ?F'o) _ p , d /B- 
. 1 d / 1 
= B,'y{-)-BvE"(y)- 
P ' 
dt 
P 
P ^ 
Substituting the earlier value on the left-hand side, we have (after a slight 
reduction) 
d (1 
dt[p' 
d / 1 
ds 
// /’ 
P T 
being an illustration of the remark in § 31, and showing that in general the rate 
ol cliange of the curvature of a normal section is not the same along the curve (f) — 0 
and the geodesic, both of which touch that section. The result can also be written 
in the form 
