no ilR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
Again, we have 
Qi2 — H" 81m/Pg 2Pg® 
= 3w^X + 81 w/Pg - 2P83; 
so that X, denoting fi^^ 4 ,"P ;^3 + P^P^g, includes all terms proper to the rank 4. When 
we substitute for Pg, P^,’ ^is values we have 
X = — 12«3^ — fuf^Vg) + (wi^'^Us + ISwpfgWg + SGWgh^^,) 
+ X', 
w 
here 
X' = {^u-{n,^Urj — — -^^Vg) + '""'7 + '’^u^~u^). 
U3 
Now, for the hrst part of X we have 
-X-= Vl?, + AT (^^->^13 + %^13) i 
and 
’V<B + «3«i3 = -H (?> ^ + ? -0 J g^, + ( <l a;; + V 
Cp 
dq 
a« \ /_ duj 
dp /V dp 
— 8'*'hJo2J 
the former of the two last lines being obtained partly from the forms of and 
and partly because and w^g are homogeneous in p and q. Hence 
X = X' + + f %Jo2 
— X X;^A3 AqJq2, 
and X' includes terms of rank not greater than 3. It thus appears that the aggregate 
of the terms proper to the rank 4 are functionally the same in Z as in Q^g; and we 
have 
Qi2 _ <27 u ^P — 
oil, 
‘>"4 "4 
X' + 27t*/Po - I A’ + 7 + I 
Now, from § 13 we have 
Joi = - (^^Wg + upx..) ; 
and from the values of Py and Pg it follows that 
Ua ■= 
6^ 
-^8 
Vr. 
UM. , 2%, 
