30 Transactions of the Royal Society of South Africa. 
(9) Let us now consider a determinant differing from R„ in the last 
row, which instead of being 
(2a - (2/^ - 3),_2, , So, 3i, 1 
is 
(2ii + ni - 4)„_], (2/^ + m - 6)„-2, • • • , (m + 2)3, m^, 1. 
This determinant may appropriately be denoted by R„(m), which implies, 
of course, that what we have hitherto called R„ would now be denoted by 
R»(3). 
On account of the equality 
r,= (r- 1), - (r- 
the last row of may be partitioned into two, namely, into 
{2n -I- m - 5)„_i', {2ii + m — 7)„-2, • • • • , + l)o, (m — \)^, 1, 
(2n-^m-^)„-2,{2n,+ m-7),-:u 1, 0; 
so that we have at once the recurrent law of formation 
R„,(m - 1) - 2(n -1) + 1) . . (VI) 
If now we view E<„(3) as known, namely, 
E,(3) = (- (2/^. + 1) (2/? +3) (4?i - 3) 
we can readily evaluate For, when m is 1 the 71^^ row of the 
determinant is identical with the [n - 1)^^^ row save in the n^^ place, so that 
we have 
R,(l) = - {2n-?>) R«-i(3) 
whence bv substitution 
= (- (2/^ - 3) {2n - 1) . . . (4.. - 7) ; 
and, in the next place, putting m = 2 in (YI) we have 
R.(2) = R„(l) - 2(/. - 1) R,_i(3) 
v/hence by substitution 
R„C^) = ( - 1)"-' (2/^ - 1) {2n +1) ... (4^ - 5). 
We thus have a formula for R„(m) which holds for three consecutive values 
of m ; and, this being the case, the recurrence formula (YI) enables us to 
show that it holds generally, namely, 
R„(m) = (2A^ + 2m-5) (2n^2m-d) (4^?, + 2m-9) . (YII) 
(10) It is of importance, however, not to assume a knowledge of the 
value of E„(3), but to establish another recurrence-formula which is effective 
without it, namely, 
R,,(m) = - {2n + 2m - 5) E,-](^u + 2) . . (YIII) 
This is done by performing on E„(m) the operation 
{2n — 2) • row„— row„-i 
r 1 ] I 
— (m — 1 ! -j row,, -2 + 2 i'ni + 2) ^ • row,, _ 3 + ^ (m + 4)o • row„_4 + ...[ 
