GP]NERATIN'G FUNCTIONS IN THE THEORY OF NUMBERS. 
147 
whence 
(- 1 ) = ■#> (4) = 2, 
and 
^{n) = 6 (n) - (/> (n - 1) + Q] cf> {n - 2) - . . . 
and, by summation, we obtain the result 
xjj [n) — 1 1)«; {71 2) 
shewing that 
xjj (2m) = 2. (in > 1) 
xfj {2771 + 1) = 0. 
Hence, when the determinant is of even order greater than two, there are two 
special relations between the coaxial minors and these two relations can each be 
thrown into a form which exhil3its the determinant as an irrational function of its 
coaxial minors. 
In the case of a determinant of uneven order no special relations exist between the 
coaxial minors, and it is not possible to express the determinant as a function of its 
coaxial minors.* 
Art. 52. In the investigation we met with 
equations 
P- 
1 
! 
<lry 
P’J 
1 
- q:r. 
7i/-- 
qr- 
P= 
involving the quantities y^y and the coaxial minors of the first three orders of 
the determinant | | . Plence, by elimination, we find 
between such coaxial minors. 
Also we found 
n — 1 
identical relations 
* It is evident that these relations must occur in pairs in accordance with the ‘ Law of Comple- 
mentaries' which is so important in the general theory of determinants. 
u 2 
