42 
Transactions of the Boyal Society of South Africa. 
Further, it should be noted how the dropping of e and 4 in the second 
makes the first a quasi-deduction from it. 
Taki ng an additional column of ri's and an additional row of O's we 
doubtless would find that 
+ [a,;] . [y6//,/3c0] - [a,e].[yerj,ft^6^]l 
0. Returning now to Hesse's identity (Z) and using the result of § 3 
we obtain 
h^in]^[ay£,/3o4] = [a,/3] [a,o] [a,;] 
M] [yrc] [y,4] 
[ere] [^,4] 
This is the second identity of another series, the first of which is 
l ^'ml- [ay, /3c] 
I [y,/3j [y,c] [ 
the first identity of both series being in fact the same. The third 
identity, we may be sure, is 
\u,,,\3.[ayei],[3ci:6] = [a,/3] [a,c] [a,;] [a,6)] 
[7,/3] [y,c] [y,4] [y,e] 
[.,/3] [e,c] [^,^] 
[r;,/3] bhh] [^,4] [v,e] 
and by way of verification we note that the expanding of the determinant 
on the right in terms of the elements of its first row and their comple- 
mentaries gives 
[a,/3].|[y,c] [.,4] [rj,e]\ 
[«,c].|[y,/3] [.,4] 
+ [a,4] . i [y,/3J [n,e] I - [a,6] . I [y,/3] [^,5] [^,^] 1 
which by the use of (II) becomes 
[a,/3] . [yEf],ci:d] - [a,c] . [yf/y,/34^i] ] 
and thus reproduces the identity at the end of § 4. 
6. The four-line determinant 
I [a,/3] [y.J] [n,t)] I 
expanded in the preceding paragraph is manifestly a compound deter- 
minant with elements of the (7i + l)th order, and it is not difficult to see 
