MR. A. R. FORSYTH OH A CLASS OF FUNCTIONAL INVARIANTS. 
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z and z are independent of one another ; but, if a third dependent variable be 
introduced, it can, by the elimination of x and y, be expressed in terms of 2 and z' 
alone, and its invariants will be expressible partly in terms of the invariants of 2 and 
of z, and partly in terms of functions arising in connexion with the transformation 
of s and z. It is thus sufficient to consider two, and not more than two, dependent 
variables when there are two independent variables. 
21. In addition to the invariants possessed by each of the dependent variables 
separately, there will be simultaneous invariants which involve differential coefficients 
of both the variables ; such a simultaneous invariant is 
J = 'pq' - p'q, 
which we have already obtained in § 4. 
When the characteristic differential equations of simultaneous invariants are formed 
by the method already (§ 8) adopted for invariants of a single function, tliey are as 
follows :—Let F' generally denote the same function associated with z that F denotes 
associated with z. Then the equations satisfied by a simultaneous invariant \jj of 
two functions 2 and z' are 
^ 0 ^ = (f^o + fi'o) ^ = SXt/;, 
— (1^1 “h ^1) 'A — 
= (^1 + ^'1) ^ = 0 , 
= (Ag -h A'g) xIj = 0, 
©gl// = (Ag + A'g) xjj = 0, 
©i.'A = (^4 "h ^ 4 ) ^ = 0. 
It is easy to verify that J satisfies these equations, its index X being unity ; and it is 
evident that the invariants of 2 alone, and those of z alone, all satisfy these equations. 
As in I 11, it is easy to prove that every simultaneous invariant must involve 
p and q; or p and q ; or p, q, p', and q. 
22. The only simultaneous invariant so far obtained is J ; we proceed to obtain all 
the simultaneous invariants which involve no differential coefficients of 2 and of z! 
which are of order higher than the second, using for this purpose the method adopted 
in §§ 12, 17. Among these invariants there must evidently occur 
J =pc[ - p'q, 
Aq = (fr — '^pqs + pH, 
Q — q-r — ‘Ip qs + P , 
the two latter being invariants each in one dependent variable only, which necessai’ily 
satisfy all the equations. 
MDCCCLXXXIX.-A. 
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