Il.— General Theorems on Determinants. By Tuomas Murr, M.A. 
(Received 6th March 1879.) 
§ 1. The rows of a determinant of the n™ order having been separated into 
two sets, one containing the first p rows and the other the rest, if each minor of 
the p degree formed from the first set be multiplied by a minor, called its com- 
plementary, formed from the second set, and the result have its sign chosen in 
accordance with a certain law, it is well known as an elementary theorem that 
the aggregate of the products thus obtained is equal to the original deter- 
minant. 
This suggests the inquiry as to the possible existence of a corresponding 
theorem in the case where the two sets of rows, instead of being contiguous, 
overlap each other. On a review of the properties of determinants it is found 
that what may be considered one case of such a theorem is already known, viz., 
the case in which the first set includes all the rows except the last, and the 
second set all the rows except the first. Taking a determinant of the fifth 
order— 
yO ia Onis 
Go Con Won Cox (Was 
31 Ago Agg Ag, Ae | Or D(a) 
Uy Ugg Ug Uy Uy 
Qs; Uso 53 A54 55 
this special theorem is 
(@q3 “29 M33 44) (oa Ugg 44 U55) — (Myo Mog Ug4 Uys) (a1 Ugo 4g U4) =| go Ugg Uy4 | D(A), 
B49, 443 V 44 
or, in its usual form, 
Azo M33 Azq | D(a45) Sole) 
| Ugo Vg Faq 
Ay Ays 
Na oN 
51 55 
where it has to be observed that in the right-hand member the original deter- 
minant is now accompanied by a factor, and that this factor is the minor 
common to all the determinants on the left. 
The general theorem which has been found to include this is as follows :-— 
VOL. XXIX. PART I. N 
