MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
83 
Invariants in the Second Order. 
10. First, let us consider invariants which involve no partial differential coefficients 
of order higher than the second. The differential equations to be satisfied are, in the 
non-subscript notation. 
("*•) 
dp 
a/ 
{n) + 2s g; +1 
dt 
3s 
0 , 
so far as concerns the form of the function. From these equations we have 
df df ^ W 
dr _ 3s _ di _ dp _ Sj 
q“ — 2pq p^ 2 {q)t — qs) 2 (qr — sp) 
When @ is taken to be the common value of these fractions, it follows that 
d/= til, + fcls+ fdt + f dp + dq 
or as dt op dq 
= @ d {qh' — 2pqs + ^9®^). 
Now, df is a perfect differential, and therefore © is some function of q^r — 2pqs-\-pH ; 
hence,/’ also is some function of qdr — 2pqs pit, and therefore the only irreducible 
invariant loliich contains differential coefficients of order not higher than the second is 
(fr ~ 2pqs + pH. 
This is the function Ay, already (§3) considered ; the integral equation corresponding 
to the vanishing of this invariant is known. 
Invariants in the Third Order. 
II. When we come to consider invariants which involve difterential coefficients of 
higher order, the method just used is no longer available, because the four differential 
equations are not sufficient to determine the ratios of the differential coefficients which 
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