1890-91.] Dr T. Muir on some unproved Theorems. 
73 
On some hitherto unproved Theorems in Determinants^ 
By Thomas Muir, LL.D. 
(Read January 19, 1891.) 
Most of the theorems in question occur at the outset of a paper * 
by Professor Cayley, entitled, “ Chapters on the Analytical Geometry 
of n Dimensions”; they constitute, in fact, Chapter I. The first 
theorem I should prefer, for the present, to enunciate as follows : — 
If m determinants of the n th order all have the same n - 1 columns 
in common , and all vanish , then every determinant of the n th order 
whose n columns are chosen from the m + n - 1 different columns 
must vanish likewise. 
Taking the case where m = 3 and n — 4, and where therefore we 
have 
[oq&2 C 3^4] = |^i^2 p 3^5l = !^1^2 C 3^6i = ^ » 
we are required to show that the twelve other determinants of the 
4th order formed from the array 
a 2 ct^ dg 
h i h 2 b 3 & 4 h 5 b 6 
C 1 C 2 C 3 C 4 C 5 C 6 
d x d 2 d 3 dy x dfy d'Q 
also vanish. To this end we note first that any two of the given 
three are connected with one of the twelve by a linear relation, in 
virtue of which the latter vanishes when the two former simultane- 
ously vanish. If we write the first two in the shorter form |1234|, 
|1235|, the relation in question is 
|1234| jl257| - |1235||1247| + |1237| |1245| = 0 , (A) 
7 being the suffix-number of any new arbitrary column. Inter- 
changing 2 and 3 we have also 
|1324j |1357| - |1325| jl347| + |1327| |1345| = 0 , (A') 
and interchanging 1 and 2 in this we have 
|2314| |2357| - |2315||2347| + |2317| |2345| = 0 . (A") 
* Cambridge Mathematical Journal, \o\. iv. pp. 119-127; or, Collected Math. 
Papers, vol. i. pp. 55-62. 
