270 
P 
of their ideal factors are given, then Fm is singly determiffed by 
P 
m. For the sake of simplicity we shall choose this order in such way that 
P Pq P 
r —x’y’..... In order to be able to determine also singly m, we 
shall first prove the following theorem : 
Principal theorem A. Every algebraic form of the total degree P, 
homogeneous and of the degrees 9,0,.., in different sets of variables 
kP Y n . 
x,y,... each of which may be regarded as coefficients of variable 
P q 
forms Fx’, Fy’,..., linear and of the degrees p,q,... in sets of n 
different covariant variables, can be written as a productof P ideal 
linear forms. When for the sets of variables x’, di is prescribed, 
that x) 7 ... are separately annihilated by definitely indicated ordered 
elementary operators either of the first or of the second kind *), as 
for the rest the variables being able to obtain all values, then one 
single definite affinor of degree P belongs to the given form for a 
definite choice of the order of the sets. 
When the characteristic numbers of the sets are a’,,....,2",; 
Ys +++5Ypi---., « being nr, Bbeing n? etc. then every term of F 
has the shape: 
P 
p 2 7 7 
ee A a © Dt da de rebels get a 
Of ake aioe 
Od. Os= 6, 
When 
5 ’ i q , q ’ 
Eh het EEE ie. Jtd nb de 
are the products of p,q,.... of the fundamental units belonging to 
Pg 
the characteristic numbers of xX’, y’,.... and 
D P q q 
Cire cee Cy ; Ors + + sin CB ge ce 
the products formed in the same way from Cn, Oe en then 
the affinor 
P P Ln de q 
eae d 
n=x? Zn, Keent Py? Pyrite ores se €, 1.0.02 tE, 1...0¢8... 
may be formed. 
