6 THOMAS MUIR ON SOME TRANSFORMATIONS CONNECTING 
the form desired. Operating in an exactly similar way on the other side, we at 
once obtain the kind of result which was hoped for, viz., 
[Cos | |@o@,| 0 0 
| Dyco@s| | Odg| |@_b3| 0 
DO | ads | [M05] | Aabscy | 
0 0 Jagd | | debs, | 
(1) | ayboegd, | = +| ab, | | @otlg| | Co, | , 
the non-zero elements making their appearance as determinants in virtue of the 
well-known theorem 
| tate | | vee, | 
=2p| LaY/pry |. 
lyate! |4p%y | 
For the case of the determinant of the fifth order the corresponding identity is 
| @,C,d,¢,| |ad,e,| 0 0 0 
| DyCGlye, | | Dods@,| |ab,¢,| 0 0 
0 | Colby | | Ayla, | | A250, | 0 
0 0 lande,| | agbsd,| | aab,c,0s | 
0 0 0 =| aabge, | | a.b9¢,€, | 
(1’) | @b,¢304€5 | = A$$ a$$" 
| tab504| |dqbses| | ose, | | Colge, | 
the general law of formation of the right-hand member being contained in the 
following rule :—To obtain the first part, viz., the continuant, take the original 
determinant | a,b,c, ....2,|; from the first column delete the elements which 
in a continuant are zeros, and replace them by zeros, writing all the deleted 
letters in order alongside each of the remaining elements of the column ; treat 
the other columns in the same way ; affix such suffixes to these added letters 
that the suffixes of the first column may be 1, 2, 3,..., »—1, of the last column 
2, 3, 4,...,”, and of each of the intervening columns 2, 3, 4, ...,2—1; enclose 
each set of suffixed letters in determinant brackets. To obtain the second 
part, viz., the divisor, take the product of all the elements of the continuant 
which border its principal diagonal, excepting those in the first and last columns, 
and rejecting duplicates, 
§ 3. Let us now return to our first determinant | @,b,c,d,|. Multiplying each 
element of the fourth column by m, and diminishing the result by m, times the 
corresponding element of the third column, and treating the third and second 
columns in a similar manner, we have 
