52 THOMAS MUIR ON GENERAL THEOREMS ON DETERMINANTS. 
the multiplication theorem. But now applying to the determinant of reduced 
order the theorem by which it itself was obtained, and putting 
| [441 a9] |My2 Cos! | 
|= Qo | My Agq Mog, 
| 411 Ag9] |Qyq Aa! | 
l@12 U3] [A213 ol 
l@y2 Ug] [413 Ayal 
|= yg | A Mog gq], 
as the identity in the opening paragraph entitles us to do, we find 
Ayo |Ay1Agoltgg] Cyg|@ yo g3g4l - - - | 
| 
1 il Ay9|My4Agobgg] A yg|My9A og 44) - - - 
D(a,)= 
(Gin) M913 eae Ont |@49Mo3| \d134o4| cee lan aon —1| 4 
Qi lO 1Oootns! Taito tosa a mee 
and therefore 
Ga Ging Cac) \Oio Cog Gay\» + amo Con ay Oem 
D(a \= 1 yy 99 Ay 49 52 Giale sell ales Aon —1 Usn| (A ) 
1 eee 
‘i [Oia Gog) ig Gp,l = [an ke Con jie eet ee ; 
7 ign, One| \Bin Gog Onal- - na Co, One 
In exactly similar fashion we can next show that 
at Moo As Ay4| » ++ |Un—3 Cen—2 U3n-1 Us| 
| 
[dit (io ht Wives pes ee eee a 
Hay 99 Ass gal. - |Un—3 Con—2 C3zn-1 Asn (A ) 
eee 3 
D(4in) = 
G11 Uqq Agg Anal «+ - |Cin—g Fon—2 Ton Ann 
and so on, the extreme case being, as it is curious to note, the extreme case 
also of the theorem of § 1., viz., that exemplified by (a). 
(A,) is practically useful in evaluating a determinant whose elements are 
given in figures. It suggests, however, another identity closely ce it 
and established in the same way, viz., 
Ay, Gon G13 -- - Gn dite aleea acre ee Br Gon 
+ Wixi Won Coy. » « ap 
itis ny ty ARES (11 Ua) [yy Ags] - «+ (qa Corl Ne ee 
Qn Une Ang - - + Onn 1 Aa ee Ara oval a hae 
which is still better adapted for this purpose, more especially if @,,=1 or a low 
