76 
MR. A. R. FORSYTH OH A CLASS OF FUNCTIONAL INVARIANTS. 
additional invariants will be introduced. For instance, we then have 
_ ^30 ho '1-20^ ^ JB ( 2 ) ^03^01 ^^02 ^ 
both absolute invariants; in the former the variation of y, and in the latter the 
variation of x, do not come into consideration. From these v/e can derive a series of 
educts by the application of combinations of the absolute invariantive operators 
Ij'P djdx and l/q d/dy. When any such educt is invariantive, say 
we may obtain from it other invariants by taking as the function, the differential 
coefficients of which are to enter, not 2 ;, but any educt which is an absolute invariant. 
All such invariants, however, thus obtained are expressible in terms of the educts 
obtained from A( 2 ;) and B (z) by repeated application of Ijp djdx and Ijq djdy in all 
possible combinations. Thus, it is easy to verify that if I be any absolute invariant, 
and Ij, lo, I 3 its first, second, and third educts due to successive operations on 
I by Ijp didx, the equation 
A(I) = 
Id 
A(y) 
Id 
is satisfied ; and the law is general. 
2 . Nor is it necessary to consider in any detail functions of the differential coeffi¬ 
cients of z, which are invariantive for isolated transformation of the dependent 
variable; that is, for a transformation which connects z with a new variable 
without regard to the dependent variables. Such a transformation can be effected by 
means of an equation, 
(z, C) = 0 ; 
and we then have 
‘-01 
clz 
01 df- 
no 
= £ -■ 
Since dz/c/^ is determinate from the transforming equation, it follows that 
‘01 
no 
is an absolute invariant for the transformations at present under consideration. 
Moreover, in the present case dfx and d/dy are absolute invariantive operators ; and 
therefore 
