202 Proceedings of Royal Society of Edinburgh, [april 18, 
where, in general, ^ consists of m _ 3 C m terms in each of which there 
are ^ indices of the form X + 1. 
To deduce the Extensions involving n - 2 general indices 
q, r, s, ... z from these n - 4 results, — we have 
! a°b l c q d r . . . l z \ = \ (q- 2), (r-3), . . . , (z-n+ 1)|£, 
that is, in the above notation, 
b, r, «,...«] = I 2, 3, 4, ... w - 1 i ? . . (D). 
Then, if 
3 [g + 1, r, s, . . . z] or S]_ = S^- , 
on writing down the determinants in S x and expanding them in 
terms of the elements of the first column in each, it may he shown 
by equation (1)' that 
S x = | 1, 3, 4, . . . n - 1 1 + j 2, 2, 4, 5, . , . n - 1 | , 
and thus 
Again, if 
*i = \l, 3, 4, . . . n-l\£ .... (1). 
2[? + l, r+l, s, . . . z] or 2 2 = S 2 £s 
we shall find in like manner by equations (1)' and (2)' that 
S 2 = | 1, 2, 4, 5, . . . n — 1 | + 1 2, 2, 3 , 5, . . . n - 1 1 , 
and consequently 
S 2 = | 1, 2, 4, 5, ...»-]]£» . . . . (2), 
while by the application of (2)' and (3)' it may further be shown 
that 
+ 1, r+ 1, s + l, t, . . . z] or S 3 = 1 1, 2, 3, 5, . . . n - 1 1 £* . (3). 
Generally, to find the corresponding value of having terms 
in each of which there are /x indices of the form A.+ 1, — suppose 
Now since in ^ there are terms with the index q+1, and 
n-fd/x. terms with the index q, on writing down and expanding the 
determinants in as before, it will be seen that the coefficients of 
q-l consists of the sum of w _ 3 C /a _ 1 determinants which by equation 
(fx - 1)' is equal to | 2 , 3, 4, . . . /x, /x + 2 , . . . n - 1 1 , while the co- 
efficient of q - 2 consists of the sum of W _ 3 CV determinants which by 
equation (/x)' is equal to | 2, 3, 4, . . . /x + 1, /x + 3, . . . n - 1 | ; and 
