PROFESSOR A. R. FORSYTH OX THE DIFFERENTIAL INVARIANTS 
3 8‘2 
H ( w^) and I (iv^) have already Ijeen replaced by f)i and we replace J (w^) by Jg, 
leaving the modified aggregate still complete. Then we have 
d / 
oyc 
1 
—- V / ( .T 
which, after substitution of the known invariants and srmie reduction, leads to an 
expression for hg in tlie form 
T 
^6 
P 
1 dB(/Hl 
B ds ds }'• 
giving also the value of as an invariant. 
45. The expressions for 
. d-H 
did 
c/fH dm d-H dm 
ds~’ dsdn' duds' du 
r, can 
he obtained in another wav: 
it will be sufficiently illustrated by constructing tlie first of tliem. From the 
0H 0H 
expressions for V- —, V' - — in 38, we find the following bv differentiation : 
ox cy ' " - 
- h : E (EN - 2F^r + GL) - 2 V-l;; 
= Gol- 2 F ,8 + Ey + PGr + f) (Ga - 2 Fn + U (Ef - 2 Fa) + ESa, 
LN - I\F 
V-FT — T 
\ riji 
V- 
{F (EN - 2FM + GL) - 2VLM1 
= G^ — 2Fy + ES + PGr' + Q (Ga' - 2Fr') + B (Er' - 2 Fa') + ESa'. 
1 LN - M- < 
[G (EN - 2FM + GL) - 2Y-N ' 
01 - yi 
= Gy — 2F8 + Ee + PGr" + (} (Ga" - 2Fr") + B (Er" - 2 Fa") + ESa", 
where {§3) 
2V-T == GE 
10 
F(2F„-E„) 1 
2Y^A E (2Fio - Eoi) - FE 
10 
2YG" = GE,i - FG,, 
r 
2Y^A' = EG,o - FE,, 
I 
2YH"' = G (2F,, - G,,) - FG,„ J 
Knowing the values of x' and i/, we form 
2Y^A" = EGo,-F(2F„-G,oy 
dx' dx' du' du' T 1 
, - , , , , d -. and then we have 
dx (hj (fx dy 
