Vol.. 7, 1921 
MATHEMATICS: 0. E. GLENN 
277 
It leads also to systems of certain new types which I designate as orthog- 
onal 1 and the extended orthogonal types of differential parameters, and 
methods of enumeration relating to these systems, both general and 
particular. 
In the paper of which the present article is an abstract, references to 
the literature are only incidental to the developments but mention may 
be given to memoirs by Christ off el, 2 Ricci and Levi-Civita, 3 to the tract 
of J. E. Wright, 4 containing bibliography, and to the symbolical theory of 
Maschke. 5 
2. The poles of the transformations on the differentials, 
are the roots of the linear forms 
where 
A = [Y^ 2 — — Y + 4 — 1 — 2 1* 
L\^2 tyi) dy 2 dyij 
and we may place h±i equal to exact differentials 
T : df +1 = h +lf = h-i, 
thus obtaining another transformation upon the differentials of the same 
degree of arbitrariness as T, since the functional determinant of will 
not vanish when A 0. The quantics df±i are relative differential co- 
variants appertaining to a domain R(i,T,A) whose defining quantities 
are the y\, y 2 derivatives of X\ and x 2 , and the expression A. The covariant 
relations are 
(3) df +1 = p+\df +1 , = pZ}J/_i, 
where 
1/dxi dx 2 , A 
are the factors in K(i,T, A) of the determinant D of T. 
3. Let 
dxi/dyi = oli, 'bxi/byi = a 2 , dx 2 b/yi = /3o, dx 2 /dy 2 = ft, 
Wfak = fi (* = 1, 2), 
and write, after Maschke, the quantic F as the m-th power of a symbolical 
exact differential, 
F = (df) m = +f 2 dx 2 ) m = (d<p) m . 
Then when F is transformed by the inverse of T, and the result expanded, 
the coefficients <p m ~2i (i ' = o, , m) of the terms in df+i, df—i are 
differential parameters of the domain R(i,T,A), forming, with d/ ±1 , a 
complete system in this domain. Their explicit form is 
