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MATHEMATICS: 0. E. GLENN Proc. N. A. S. 
<Pm-2i = [(yi- A)/i-2/3 0 / 2 ]"- ? '[- (Ti+ A)/ 1 +2 i 8o/2] / (-4/3oA)|- w (^ = 0 ,. 
and the invariant relations are 
(4) <p' m -2i = P+T 2i D t <p m _2i (i = o, . . .., m). 
The functions <p m -2h df+i, df-i are the invariant elements. 
4. By differentiation of the equations (4) relations are obtained as 
follows : 
^ <Pm-2i _ r — sps m — 2i -r\i ^ <Pm - 2i , 
(r = o, 1, ,; 5 = 0, f r;i = o, , m). 
As a result, employing the transvectant symbol 5 
UxV % - U 2 V! = (U, V) 
we obtain a general category of relative differential parameters, 
#S?-« = (...((**-* Q (i) ),-.-<3 (i) ) (e w -^I*D0. 
the number of iterations being r, and 
(5) *2?_* = Pn 2i D i+r ^-2i; (*-*,.. .,m). 
We define 4$-2i to be p m _ 2 <. 
Write 
k = 0 
and let the conjugate product be P_i; 
k = 0 
then the parameters of the extended orthogonal type are defined to be 
those which can be generated in totality by forming such rational ex- 
pressions in ^m-2i, df±i as simplify by multiplication into functions 
appertaining to the domain R(i,T,0), free, that is, from the expression A. 
The essential forms from which to construct this totality are evidently 
P+i =*= P-i. When r = 0 the type is called orthogonal. 1 
Finite complete systems can be derived in this theory. In fact the 
products P form a Hilbert system of monomials whence it follows that a 
complete system of concomitants of the extended orthogonal type is 
furnished by the finite set of irreducible solutions of the diophantine 
equation 
r m 
(6) 2 2 (m-2l)X kl -(n+<T2 = o. 
k=0 1 = 0 
Particular systems of the orthogonal type have been constructed by the 
present writer in this and previous papers for the quantics of orders one 
to six inclusive, the system of the sextic for example being composed of 
31 parameters. For the case r — 1, m = 2 the system comprises eleven 
concomitants of the extended type as follows: 
<P0, ^O^, K = <P2^-2> o = ((p 2 , P + l) (<P-2> P-i)» 
P±l = <P2(<P-2, P-l) =*= tf>-2(<P2>P 2 +l)> 
