342 
Proceedings of the Royal Society of Edinburgh. [Sess. 
XXII. — Note on a Theorem of Frobenius’ connected with 
Invariant-Factors. By Sir Thomas Muir, F.R.S. 
(MS. received June 13, 1922. Read June 19, 1922.) 
(1) The theorem in question appears in his well-known paper of the year 
1894, “ Ueber die Elementartheiler der Determinanten.” * Save that 
English takes the place of German, the enunciation of it stands exactly 
as follows : — - 
// ^^r^s<n, and he any determinant of the n^^ order in the array 
C('k\ (^~ij • • • 5 • • • } 
being its complementary minor of the (n — order in the determinant 
of the n^^ order 
^kv 
afxk 
then 
In regard to it two preliminary remarks are necessary: (I) that in the 
original there is a very upsetting misprint of r for s in the third line ; 
(2) that the theorem is purely determinantal, any connection with the 
theory of invariant-factors that may be given to it being solely for the 
benefit of the latter. 
(2) Manifestly the equality predicated in the theorem may be viewed 
either as a summation of an aggregate of products of pairs of determinants, 
or as an expansion of a special Ti.-line determinant in terms of products of 
complementary minors of more general determinant. In the case of either 
view it is essential to take pointed note of the fact that the left-hand 
member is a grossly inflated representative of the member on the right, — 
a fact readily seen to be inevitable if it is observed that every first factor 
on the left with one exception is a function of elements not one of which 
is to be found on the right. This exception, too, is brought in as it were 
by a side-wind, being the case of where is 0 and where is taken to 
be I and its cofactor Df to be | a^^ l> thus giving us our only assurance that 
the right-hand member is represented on the left at all. Indeed, the exact 
state of matters is that the right-hand member forms a part of the first 
* Sitzungsh. . . . Akad. d. JViss. (Berlin), 1894, pp. 31-44. 
1, . . . , r ; /x = r-l- 1, . . . , 
A. = 1, . . . , .9 ; V = s + 1, . . . , n 
