10 THOMAS MUIR ON SOME TRANSFORMATIONS CONNECTING 
for 1, 4 respectively, and then proceed with (3’) and (4), as we have just done 
with (1’) and (2), we find 
[@bs| ej 1rd | Vande | 7 
- | %¢5| 1B, __|dyes| [bes (7) 
ied €| Cds | 
dy 
| ce; | = 
This is the complementary of (6), and might so have been obtained. As it is 
easily verified, we can therefore readily have by means of it a verification of the 
more important theorem with which it is related. 
§ 8. In the continuants of §§ 2-5 the zeros were introduced by operating 
only with rows upon rows, or with columns upon columns. If now, however, 
we introduce those on the one side of the principal diagonal by operating with 
rows upon rows, and those on the other side by operating with columns upon 
columns, we obtain a result quite distinct in character, and not less interesting, 
viz., we have 
y is 0 0 0 
Bi “4; Be 0 0 
co) Ys De 0 
Re Ys av, 
0 % Ys 
| %b,¢,7,f, |= : (8) 
Gy | ybz| | dgbse, | b, | Dye | | dyegds | 
where 
£i==O, 2 =Uye 
Lox | abs \} Z| DiC, ls 
Lyd, | ADC, | ? @3—= Og | biC.d | , 
T= iGy Abel Apa xen ea  aalls 
and 
Nn = Oy ? 
Yg=b, ? 
Y3=a4| DCs |—az| byes | , 
Y4=| Aad, | | yey, |—| aab4 | | dead, | , 
Y5=| qdge, | | bends f's |—| aabses | | byeoe f, | « 
The process of transformation is not given, because to do so would unneces- 
sarily lengthen the paper ; the reader, however, will find it worthy of attention, 
one or two little-known identities turning up in the course of it. 
The corresponding expression for |@,b,¢,d,| is got by merely deleting the 
last row and column of the numerator, and the last two factors of the deno- 
minator. 
