GENERAL DETERMINANTS WITH CONTINUANTS. 13 
Note on Mr Mutr’s Transformation of a Determinant into a Continuant. 
By Professor CHRYSTAL. . 
(Read 21st February 1881.) 
I. The following way of arriving at some of Mr Murr’s elegant theorems may be of some 
interest :— 
Consider the system of equations, 
(11)x, + (12)a,+ ... +(1n)e,=90, 
(21)z,+(22)a,+ ... +(Qn)a,=0, a) 
@ le) Je 6 0 ee 6 se) 6 “8! 0 oe ei “eye “aT oe 
(n1)a,+(n2)a,+ ... +(rn)a,=1, 
the left hand sides being zero in all but the last. Let A be the determinant of this system. 
From the first » — 1 of these equations we can eliminate all the variables but #, and #, in one way ; 
and all but #,_,, z,, and #,,, in m-1 ways; also from all the » equations (7.¢., from any x — 1 of them, 
the last being always included) all the variables but x,_, and «, in ~»-1 ways. We thus get 
Ex, + Fie, =9 
D,2, + E,v, + Fw, =0 
(2). 
oO ey) ee: a) ow! ote 
D nBy— 1 a E,2, aa F, 
Where the determinants D, E, F are derived from A as follows, by omitting 
‘ie etal ay Columns. Rows. 
——h | — — = a — 
Ky, | 2d | ro | | 
| 
¥, Ist | we | 
—_— -— | Z | SS 
} | 
D,. peand7—1/* | s*andn* | s is any number | 
pace Saale 
E, r—l\*andr+1* | 5s and n® common to these | 
Sap | | 
Hie we r andr—1\" | sand ni three except 7 
} : ' 
La) a ¢. | # is any number 
E, n —1\"* / ee | common to these 
. — | 
Poth mand aw— 3" | 2 and n™ | three except 7 
1 
| 
—————s 
VOL. XXX, PART I, C 
