MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
113 
comitants ; and it thus appears how Aq, A^, Ag, . . . are (§§ 9, 16) simultaneous solutions 
of the two characteristic equations in question. Moreover, since the Jacobian of two 
binary forms occurs in their concomitant-system, and therefore satisfies the charac¬ 
teristic equations, it is now evident that the quantities denoted by being 
9A; 9A,,, _ dM 9a„, ^ 
dp dq dq dp> 
must satisfy the equations ^ 
Hence, it appears that one method of obtaining the irreducible invariants, which 
are proper to the rank n and are additional to those proper to ranks less than n, is as 
follows :—(1) to obtain the concomitants of A,;_ 2 , and the simultaneous concomitants 
of A,;_ 2 , and of the concomitant-system of A,i_ 3 , . . . , A]^, Aq, viewed as binary 
forms ; (2) to frame the combinations of these concomitants which will satisfy the 
remaining characteristic equations = 0 = Agjf; (3) to select from among these 
combinations such as are, from the supposed known algebraical relations among the 
concomitants, found to be irreducible. 
35. Again, in the case of binary forms in two systems of variables, q and — p, 
q and — p', and with coefficients 
r, s, t, r\ s', t', 
a, b, c, d, a, h', c, d', 
the characteristic equations satisfied by their simultaneous concomitants are of the 
form 
that is, 
(Ag A g) t// _ 0 — (A^ -f- A 4 ) xjj, 
@^xjj = 0 ~ ®^xp. 
And every solution of these equations, with proper limitations as to degree and grade, 
is a concomitant. Hence, every functional invariant of the two dependent variables 
2 and z already considered can be expressed in terms of simultaneous ccncomitants of 
the set of quantities Ag, A'q ; A^, A\ ; . . . , viewed as binary forms in variables 
q and — q and — qd. 
Thus, for example, we have seen the simultaneous functional invariants, proper to 
the rank 2, are five in number, and they are—one, J, being the covariant p>q' — p'q in 
the variables alone ; two, Aq and A'q, being tlie quadratic forms ; and two, ^q and ^'q, 
which can be exhibited in the respective forms 
and 
i (« di' + P i') + (2 I + P' I) 
dq' ^ dp' 
Q 
MDCCCLXXXIX.— A. 
