138 
MAJOR P. A. MACAIAHOX OA" A CERTAIN CLASS OF 
Art. 39, This is an inversely symmetrical determinant of normal form involving the 
quantities In the compartment II, the elements, other than the units, 
have the denominator ys_i, s + i- The transformed of the minor is derived from the 
transformed complete determinant by deletion of the row and column, and the 
subsequent division of each y element in the compartment II by y^.j^g + i and multi¬ 
plication of each y element in compartment III by y 5 _i_ ^ + 1 . 
It is now obvious that if a minor be formed from the untransformed determinant 
by deletion of the 
and the 
•s*'" (s -j- 1)* . . . + cr)* columns. 
the transformed minor will be obtained from the transformed complete determinant 
by deletion of the aforesaid rows and columns, and subsequent division of all y 
elements which are at once above the I’ow and to the right of the (s -f column 
by ys_i^s + o- + i and corresponding multiplication of the inversely symmetrical elements 
by the same quantity. Or, as before, we may suppose the minor divided into four 
compartments and state the rule with reference to them. It will be convenient to 
allude to these compartments as I^, 10, IIO, IVj. 
In addition to the aforesaid rows and columns, suppose the 1)* . . . (^ + 
rows and columns deleted. 
In correspondence we have other four compartments, 0, 10, IIO, IV;; and there 
will be a certain extent of overlapping of compartments. 
Art. 40. The rule is (after deletion from transformed complete determinant) :— 
Divide y elements in 10 by y^.i, s + + 
5 ? 
10 
'Yt — 1, t + T + h 
with corresponding multiplication of the inversely symmetrical elements. 
If this be carried out it will be found that those y elements which are in both 10 
and 10 will he divided by y, _ i,, + ^ +1 y^ _ i, ^ ^ + 1 . 
The general rule guiding the formation of the minor when there are any number 
of sets of compartments arising from the deletions will be now perfectly clear. 
Art. 41. We are thus enabled to exhibit all the co-axial minors of the determinant 
as functions of the 
n — 1 
quantities y. 
So much of the theory of these interesting determinants suffices for present 
purposes. 
