THOMAS MUIR ON GENERAL THEOREMS ON DETERMINANTS. 53 
integer. By the continued application of (B,) there results a series of identities 
corresponding to (A,), (A;) . . . Viz., 
11 Uz Ugg] |@q1 Cog Ugg - - + U1 Mog Gn 
Dia;,)— 1 Coa Uys] [O41 Con Gay, - ~~ G1 Lop Un Peta ha unre By) 
‘ | 
Britny Onn Gry Cag lOna x < «iy Don Din| 
| (el Opa Can Gaul ® 2 «yy Ope Onm Den Gin 
Tia.) | GiGi Gas idig\l ao AA Ogi Opa Wen Lea abe! OURS 
Oia One Gay Ousl 62's 2 \@yy Cog Bay Gaal 
and so on. 
§ 3. PRODUCT OF A DETERMINANT AND A POLYNOMIAL.—The product of a 
determinant of the n” order by an expression of n terms is equal to the sum of 0 
determinants, the first of which is got from the given determinant by multiplying 
each element of the first row by the corresponding term of the given expression ; 
the second by multiplying similarly each element of the second row, the third by 
multiplying similarly each element of the third row, and so on. 
Let the given determinant be 
Gidat. ... d, or D@,,), 
and the given expression 
Giteetegt ... thi, 
then the 2 determinants referred to are 
Ry ee E.On | leer cepa aloe Oin 
Mg “Waar. sy Cy, Sp Neopet EiCon 
| ; 
Tn pe MS Oi) ere Grane | 3 
Now the coefficient of €, in the first of them is evidently a,,A,,, in the second 
@,A,,, and the third @,,A;,, and so on: therefore in the sum of the x dcter- 
minants the coefficient of €, is 
Qy, Ay + Gq, Aoi +45, Agy t+ . . - +Qn1 An or D(Gin) 
Similarly the coefficient of €, is seen to be 
yy Ayo t+ Ag Ago t Gyo Agot - - - +On2 Ane or D(a,) 
