108 
MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
Similarly, from the second pair 
from the third pair 
and from the fourth pair 
„ ... 
@1 ^ = dmY 
VJC 
^^ di/ ~ ^ 
f • 
3 V ^ 
(^o = 0 
ox 
©.T = 3 mV 
3 ?/ 
J 
^ sy _ 
dx ~~ 
0 V 
av"^ 
?iy 
© 
•’ a.y 
0 
= 0 
av .. 
By - - 
1 
av 
Y- I 
a. J 
And the general laws, of which these are particular examples, and which can be 
established by means of the successive educts of the invariant V, are 
a”V , , . a«-iv 
©, = 5 ( 3 m + n — 1 ) -r-r- 
a.'’* a?/” ^ 1 s 
a«v 3"-^^ 
" a.r" a?/ ^ ax” a^ i 
a”V _ _ a”A^ I ' 
dx^dif-’’ ~~ ^ ax’'“^a//“"’'+^ 
8 ”V _ ^ 0 ” V 
ax” “ 0y ” a.r ” “ ^ a^*" “ ^ 
From these the effect on Y of any combinations in any order of the operators 
a/a^;, a/ay, ©i, ©o, ©3, @4, can be deduced. 
31 . The following is another application of the theory of eduction. The index of 
IG) is 6, so that UgUo”® is an absolute invariant, and therefore 
(^3-,-x|;)u.U„- 
is an invariant, say 
V = Vo SU2 - 3U2 SUq. 
