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II.—On some Transformations connecting General Determinants with 
Continuants. By Tuomas Murr, M.A. 
(Read 21st February 1881.) 
- §1.-It is well known that by a simple transformation of a determinant we 
may cause a zero to take the place of any one of the elements. The theorems 
of HerMiTE* and Horner,t for example, for depressing the order of a determi- 
nant may each of them be viewed as the result of repeated transformations of 
this kind, the operation being continued until all the elements of a row or 
column except one are replaced by zeros. 
With these facts in view, it occurred to me about a year ago to test the 
possibility of transforming a general determinant so as to have zeros in every 
one of the positions held by them in SYLVESTER’s continued-fraction determinant, 
viz., everywhere except in the principal diagonal and the two bordering minor 
diagonals. The transformations to which I was then led form the subject of 
the present short paper. 
§ 2. Beginning with the determinant of the fourth order |@,b,c,d,| we have 
as the result of a first transformation 
[a@,d,| |a.d,| a, a, 
| bd, | |Od5| b, 
|edy| |e,d3| cy ey 
0 0. ad, d, 
| a,b,c,d, | = + dads , 
and multiplying each element of the first column here by |¢,d,| and diminishing 
the result by |c,d,| times the corresponding element of the second column, we 
have 
d,| A,Co1,| |Ael,| A, Hy 
d,| bea, | | yd, | b; by 
0 led, | Cy C, 
0 OF ded, 
| byC30, | = > d,d.| Coll, | ? 
where on one side of the principal diagonal the resulting determinant is of 
* Liouville’s Journal, xiv. p. 26. 
+ Quart. Journ. of Math., viii. pp. 157-162. 
VOL. XXX. PART I. B 
