90 
MR. A. R. FORSYTH OR A CLASS OF FUNCTIONAL INVARIANTS. 
By means of these results we at once find 
= 18Q; = ; 
^oQg “ 2 /Qg = tliQe ! 
= 12 Q, = n,Q,. 
Hence Ag, Q 5 , Qg, Q 7 satisfy all the necessary equations, and they are therefore 
invariants ; their respective indices are 2, 4, 9, 6 ; and, therefore, every invariant 
ivhich involve!^ differential coefficients of z of order not higher than 3 can he exg)ressed 
as an cdgehraical function of Kq, Qg, Qg, Q^, where (changing the sign of Qg from § 13) 
Q,g = H,+ 2L,-4Ho^ 
Qg = Aj® + 3AQA;^Jg^ — 72Ag"AjHQ + 
Q 7 = 2 AgjQi + 12 Ag“Hg, 
and the qucmtities Aq, Aj, Jg^, Hg, H^, Hg^, H cere given by the equations 
Ag = 
T _ 
Jq] — 
(%.o> ^in ^ 10 )"! 
(^30> ^ 12 ' ; 
0 Af) 0Aj 0Ag 0A^ 
0 ?ig 0^01 
0^01 9^0 
Hn — 
'11 ’ 
H,n = 
1'^ (^01 (^30^13 H“ h)3A3o) ^10 (~30~03 '^2^11^13 “h ^02^3l)J 
0 ^Ao 0fi4i 
ri-v 3 Pjv 2 
u-io 
— 9, 
0-Ag 0'Aj 02 Ag 05A;^ 
05,n03;m 02,n03:Ai 0 A ® ’ 
10 ^■'01 
(h(di2 %o^o3 Hd\2’ %fi'n3 ^lo)'' 5 
Hi 
^3 — ^01 {^30^'^03 + (^^ifi ^30^03) % — 
O' 'y ‘7 t 
"11^03^30) 
^10 {^30~11^03 (^^ifi “h ^30^03) ^13 “h 2-11^03^31 ^0 
3%o} 
_ ^ /0Ag^i 0A„0H 
“ \ 0 L„ 0 - ' 
'111 '^~oi 
11 
9% 9,?,,, 
The invariant Ag is that which was obtained before, and it may be called the irre¬ 
ducible invariant of the second order ; the invariants Qg, Qg, Q^ may be called the 
irreducible invariants of the third order. 
15. It may be remarked that the quantities additional to A^ and Ag which are 
necessary for the expression of Q., Qg, Q^ all belong to the simultaneous concomitant 
