Dr T. Muir on the Theory of Determinants. 
509 
(A) . When the factors of each product are of different orders, as in 
(B) , (D), (F), the identity is an “ extensional ” of something still 
simpler than (A), viz., 
In exactly the same manner and at quite as great length the 
identity 
lcl\ncr\ /k l \/ k r\ /k I r\/k\ 
— already known to us from Lagrange — is made the source of a 
numerous progeny. By putting figures for h, Z, . . . and at the 
same time writing them as suffixes, these identities, original and 
derived, take the form 
!«l/ 2 il«l^/(>l 
l«l/ 2 il«]^ 2 ^ 6 l 
kh^ 2 / 3 iI^ 1 ^ 2 ^Z 6 l ~ 
Kh^ 2 / 3 !l^h^ 2 ^' 3 ^Z 6 i “ 
l^l^ 2 %/ 4 ll^l^ 2 ^ 3 ^ 4 ^ 6 l — 
1^15^211^1^2/61 ~ 
|«/ 25 ' 3 ll«/ 2 / 6 l 
1 ^ 1 ^ 25 ' 3 ll^l^ 2 *^ 3 / 6 l ~ 
|«/ 2 ^ 35 ^ 4 lI«lV 3 / 6 l = 
1 ^ 1 ^ 2 ^W 4 ll^l^ 2 ^ 3 ^^ 4 / 6 l ~ 
1 ^ 2 %^ 45 ' 5 ll^l^ 2 %^ 4 / 6 l ~ 
ki/ 25 'elKl : (A') 
I^l/ 25 ^ 6 ll^h^ 2 l J ) 
1^1^2/3^Z6'l‘^1^2l 5 ) 
l^l^2/35'6ll^l^2^3! 5 ) 
l«/ 2 ^ 3 / 45 ' 6 ll«/ 2 ^ 3 l ’ (^') 
klV3/45'6l|^^2^3^^4l 5 (F') 
1^2C3^4/55'6ll^/2^3^^4i * (^') 
Of these (O'), (E'), (G') deserve to be noted, being along with the 
original (A') extensionals of the manifest identity 
f 29 e - 92/6 = 1/29^1 • (™ i . 5 ). 
On the other hand (B'), (D'), (F') are essentially the same as (B), 
(I)), (F) already obtained — a fact which Desnanot overlooks. 
As the source of a third series of results, obtained in still the 
same way, the identity 
/kl\/kl\ /kl.fkl\ /kl\/kl\ ^ ^ 
is next taken. In reality, however, this does not differ from the 
first identity so treated, viz.. 
k l\fmp\ /kp\/ml\ /km\/lp\ 
a h j[a h) ~ ^ah hj — yah j[a h J • 
• (A). 
In (A) the letters ah remain unchanged throughout, and the indices 
vary ; while in (A") the indices remain the same, and the letters 
