48 THOMAS MUIR ON GENERAL THEOREMS ON DETERMINANTS. 
In a determinant of the un degree, if the rows from the 1% to the q" inclusive 
and the rows from the p” to the n™ inclusive be taken; and if a minor of the 
(q—p +1)” degree be chosen from the rows common to these two sets; and if from 
the first set each minor of the q* degree containing the chosen minor be multiplied 
by the minor which contains both the complementary of the former and the chosen 
minor; and tf the sign (—1)° be preiaed to the product, s being the sum of the 
numbers indicating the rows and columns from which the first factor is formed 
increased by qa—p +1 for every such number greater than q : then the sum of the 
products thus obtained is equal to the product of the chosen minor and the 
original determinant. 
Let 
44 9 Ars . . . A» . . . Ayq . . . Un 
Wg My Migs was ee Opa ee lan a Onn 
A, Aso Uz . . . Asn . . . Aq . . . Azgn 
iy Cop clnsheara- ape ee Ang + 8s Eon Wen a Oa) 
Lay Uo Ag i Lap aq Lan 
Ga Gaede ce ae Oa 
be the given determinant, and 
yp : Ung 
ee tae F or 6 
Lg - Wag 
the chosen minor. 
From the elements of D(a,,) we form a determinant 
CAREER Cp Oyq 9 0 Bn 
ON EET Tipe Aq 0 Gen 
@31 Azo 33 Asp Azq 9 0 W3n 
Gis, Oye “Os Ap | Gy, 0 0 Bi, 
Beso ai cake es ee a Pha MeL ccomauiegite ie 
Qy Lae Qa Lp iq ag 0 0 Zz Lan 
Up Ang Ug 0 > 0 Lp Lng Cnn 
Aq, Uqq Ags 0 0 Ag Cag gn 
On Ane Ans 0 : 0 np ’ Ung Onn 
which is of the (1 +g—p+1)™ degree, its rows from the 1% to the g™ inclusive 
