TRANSACTIONS. 
I.—The Law of Extensible Minors in Determinants. 
By Tuomas Mutr, M.A. 
(Received 21st February 1881.) 
§1. As a preliminary to the establishment of the law im question, it is 
necessary to state and exemplify another law to which I have elsewhere directed 
attention, viz., 
THe LAw or COMPLEMENTARIES.* 
To every general theorem which takes the form of an identical relation between 
a number of the minors of a determinant or between the determinant itself and a 
number of its minors, there corresponds another theorem derivable from the former 
by merely substituiing for every minor its cofactor in the determinant, and then 
multiplying any term by such a power of the determinant as will make the terms 
of the same degree. 
For example, taking the well-known identity employed by Hermire, 
| a bz | | @ab3 | | ab, | 
O05. Oe Dy | 
(Oetier) loeb 7 | Gatty | : : a) 
ey Guha 2 ep: 
la,do| | agd5| | asd, | 
2) eenmeroN aay By ard, 
* I do not know who was the first discoverer of this law. It presented itself to me when correct- 
ing the proof of my paper on ‘‘ General Theorems in Determinants” (Trans. Roy. Soc. Edin. 1879). 
But it must have been known to Professor Cayney before then, for in a note to a paper by Professor 
TanNER (Mess. of Math. 1878), he refers to it as a means by which Professor TANNzER’s corresponding 
law for Pfaffians might be established. 
VOL. XXX, PART I. A 
