MAJOR P. A. MAC]\tAHOR ON A CERTAIN CLASS OF 
1 16 
since, as will be seen presently, the determinant 
1 (X] (X ;3 ^3^2) (^3 <^3*3) ! 
vanishes ide?itically. 
The rightdiand side of their identity does not, 011 expansion, contain any terms 
which are functions of s^x^, and of the coelEcients a, b, c only. 
Art. 7 . Before proceeding to establish this, it may be remarked that the above 
identity may be written in the determinant form :— 
c 
no.s'^a'i. 
Ug.s'ia’i 
'A 
— 1, 
1 - 
0, 
0 
0, 
1 - 
0 
0, 
0, 
1 AX3 
-Xd 
1 
1 
1 - ^ 
1 — s 
iXi 
iSoA’j 
Sjj Xg) , 
& 3 « 3 ' 
*3 
1 - 
V ’ 
1 - S3X.3 
1 — S 
3X.3 
c 
CgSg^-’g 
(bA X3) 
- S3X.5 ’ 
1 - ..3X3 ’ 
1 - SoXg 
0 0 
and, in this form, is very easily established. 
Art. 8 . Consider, in regard to the order n, the algebraic fraction 
... 3/ [ (Nj (A.^ * • • (A-^ ( 
(1 — SjAi) (1 — S0X3) ... (1 — s^Xd 
wherein t has an integer value not superior to n. This fraction is specified by the 
first t natural numbers, but this is merely for convenience, as what follo^vs can 
be readily modified to meet the case of a fraction specified by any selection of 
t natural numbers, which are unequal and not superior to n. 
To show that this fraction contains, on expansion, no terms which are functions of 
50X0, . . . s„x,^ only, it is merely necessary to show that every term in the develop¬ 
ment of the determinant 
I (^1 rqaq) (Xo 1 ? 
