GENERAL DETERMINANTS WITH CONTINUANTS. G 
Ay |MyAq| |Mgey| | M30, | 
by |myzby1 | mabg 1 | m5, | 
| a,b,c,d, | = +M3zNM, , 
Cy | MyCy | | mcg | | mee, | 
d, |m,d,| |m,ds| |med,| 
and subjecting this new determinant to the set of operations to which | @,,c.d, | 
itself was subjected in § 2, we finally obtain 
| acy | | mands | 0 0 
| dyey45| | mydyds | | mabe | 0 
0 | myeods | | myao¢3| | MAbs, | 
0 0 | mad, | | my a.b5d,| 
| ayb,e,0), | =—?——________________f  , , , (2) 
| m4@b3| |myaod5| | m,c2ds | 
A comparison of this with (1) brings out the fact that the right-hand member 
there is not altered if we change the elements of the last three columns, and 
the factors of the divisor, all into determinants of the third order by inserting 
an m, in each. 
§ 4. Making use now of the Law of Complementaries we return to (1), and 
substitute for each determinant its complementary minor in |@,),¢,d,|. This 
gives us the new identity 
b [diel 0 0 
Uy | aye, | |e,d,| 0 
|a,b,| |dye,} led.) = . | 
vd4| |Dye4| led, | 0 |a,b,| |d,d,| d, * 
0 Oy | de, 
a relation connecting only the elements 
a6,¢,d, 
Ab yCydy , 
so that as there are six pairs of these lines, we can at once write six 
identities like (3), and thence find by the Law of Complementaries six 
identities like (1). 
