146 
MATHEMATICS: 0. E. GLENN 
f m of order m is subjected has not been formulated, although a paper on 
invariants 1 published in 1843 by Boole treated certain functions of this type. 
These were the concomitants of forms under transformations which rotate 
cartesian axes inclined at an angle w into another set with inclination w'\ 
invariants which contain the parameter w. Orthogonal concomitants, which 
are special cases with w = J -ir, were made the subject of a number of later 
papers 2 , notably by Elliott and by MacMahon. 
I have considered a general doctrine of such concomitants for the transfor- 
mation with four parameters 
T: %i = aiXi + c^', #2 = Po%i + fttfa'j D = aift — 0:2ft 4= 0. 
The elements of the methods are based upon the two forms 
£ = 2/3 0 *i + (ft -ai-f- A)* 2 , 77 = 2(3 0 Xi - (ft - <*i - A)^ 2 ; 
whose roots are the poles of T, and the expansion of f m in terms of £, 77 as 
arguments. The quantity A employed here is the square root of the dis- 
criminant of the form 
J: ft*i 2 + (ft — ai)xiX2 — q 2 % 2 . 
The coefficients (p m -2i (i=0, . . . . , m) in the expansion of f m are invariants of 
a new type 3 belonging to the domain 22(1, T, A) of rational polynomials in the 
coefficients of / m and those of T y increased by adjunction of A. These inva- 
riants compose a fundamental system in R. They satisfy the invariant 
relations 
<Pm-u = P w ~ 2 **£>V w -2i (* = 0, . . , m), (1) 
in which p is one of the two factors of D in R : 
P = K«i + ft+A). (2) 
When one seeks complete systems for the given domain R(l, T, 0), free 
from A but including rationally the coefficients of T, it is found that the con- 
comitants are in one to one correspondence with those invariantive products 
m 
for which the exponent of p in the invariant relation P' = p a D h P, is zero. 
The conclusion is then drawn that concomitants in 22(1, T, 0) are in one to 
one correspondence with the solutions of the diophantine equation 
m 
is'2*<(«- 2i) + p - a = 0. 
i=0 
(3) 
