87H PKOFESSOR A. R. FORSYTH ON THE DIFFERENTL4L TNYAEIANTS 
and let a,, denote tlie leading coefficients of a,, a., respectivelv. so that 
«, = GP - 2FO + EH, 
a, = E-S - 3EFH + (EG + 2F-) Q - FGP. 
Then it is not difficult to establish the identitv 
E/^ = La. - 2 (EM - FL) + (EN - 2FM + GL) (EQ - FP) 
- ('EG - F-) {L (EQ - FP> - P (EM - FL) 
Noting that all the quantities on the right-hand side are leading coefficients of 
covariants, we change the identity into a relation among covariants ; and the result, 
on division throughout by in.., is 
= — a. ~ J (umG) a,+ ~I (aW.) d (amr..) 
i/n ir. ■ /r. -/ i v _ . v . 
?r.v 
so that r/K/da is expressed in terms of the members of the retained ao-oreo-ate 
Substituting the values of the invariants in the equation and dividing out hv V'‘B 
after substitution, we find 
= (H — ^ ^ ^ ^ 1 4- ^ ' ~ 
<^1n p' (In \ p/dn^.p'J F dt Vp'J BpV' ds r ds BF ds ' 
Effecting tbe same transfoi'ination as lief'^in. liy taking 
we find 
d /I 
drGF 
d 
du \ 
( K 
'U ' 1 = 
^ f f- 
p 
p~! 
dn \ ' 
B-I 
d 
l\ \ 
2 ( IB 
dH , 
IT 
(IB 
dG 
•^p'J 
p'B ds 
“ (Is + 
B 
(Is 
d 
ds 
1 + 
pd +' 
' •' \ 
H - L 
' P 
1 
B 
i/B 
(Is 
dll 
ds • 
Identification of the remaining Invaidants obtained in § 23, n'ith some 
Modifications of the Spsteni. 
41. We proceed now to the identification of the invariants of the next hioffier order 
of derivatives ; these involve derivatives of F of the third order, derivatives ol xjj of 
the tliird order, and tlie fundamental magnitudes of the fourth order. The method 
used is similar to that adopted in the ])receding sections ; we form derivatives, with 
