78 
MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
0 = 
205 
22-105 0 
^Jl5 ^Iqi, ZiQ 
^025 2Zoi 
- 2 (201^2:20 + ^10^%) 
and, therefore, 
^0 — ^01^%0 “h ^10^^03 — (%05 2^115 ^OiX^Ol’ ^10)^ 
is an invariant. 
The integral of a partial^dififerential equation of the second order, which is most 
general so far as concerns the number of arbitrary constants, contains five such 
independent arbitrary constants ; and, therefore, a general integral of 
is 
Ao = 0 
a + hx + cy 
a' + h'x + c'lj 
It has already appeared that Aq is an invariant for arbitrary change of z; and 
therefore, an immediate corollary is that 
Z = (f) 
a hx + cy 
a' + h'x + c'y 
where ^ is arbitrary, is a general integral of the equation Aq = 0 . 
4 . As an invariant is self-reproductive after transformations have been effected, save 
as to a factor, it is necessary to obtain the form of this factor. For this purpose it 
will be sufficient to consider a simple case. 
Let Zi and be two functions, and suppose the transformations of the vpiables to 
be any whatever, say of the form 
Then we have 
and therefore 
x== 4 >{X,Y), y = rp{X,Y). 
T, , dy _ dx , dy 
' 0X ’ ^3 -2^3 fZ2 ^ 
Ql-AgY+?l 0Y’ 
ax 
aY 
Pl5 Ql 
— 
Pi5 qi 
P 25 Q 2 
lh> <lz 
'0X 
a (r, y) 
a(x, Y) 
Hence, in the present case, the factor is d (x, y)ld (X, Y) = J; and, by the analogy of 
all invariants, t]ie factor for any one will be some power of J. 
The invariants at present under consideration may, therefore, be defined as 
follows- 
