34 
Proceedings of the Royal Society of Edinburgh. [S 
ess. 
a special sense while dealing with such determinants ; for example, scala , 
scala, di grado r, scala diretta, scala inversa, scala completa, etc. He next 
gives an account of the simple properties of bigradient arrays, or, as he calls 
them, two-scale matrices, and introduces a notation for them, writing 
j { a o)r 
I (\)s 
to denote the bigradient array in which the elements a 0 , a v . . ., a m are 
repeated r times, and the elements b 0 , b v . . ., b n are repeated s times, a 0 and 
b 0 when furthest to the left being in the first column, and a m and b n when 
furthest to the right being in the last column. Clearly, the notation would 
have been less imperfect if written 
(“o> • 
• • ) ^"m)r 1 
! (&», • 
. . , u ! 
For example, the bi gradient array 
°0 
cq 
«8 
<q 
a h 
“6 
a 7 
• 
a o 
«1 
a 2 
a 3 
a i 
«5 
«6 
a 7 
• 
a 0 
<q 
a 2 
«3 
a i 
°5 
a 6 
a 7 
b o 
h 
K 
h 
\ 
• 
K 
\ 
h 
h 
\ 
h 
h 
h 
^2 
h 
h 
K 
h 
h 
K 
might with fair appropriateness be denoted by 
(«o> • 
• • j a r)s 
1 (K ■ 
• • > ^4)6 
the only weak point then being that the introduction of the 6 is uncalled 
for, on account of the necessary equality of m + r and n + s, either of which 
specifies the number of columns in the array. It is a convenience, however, 
to have both the outside suffixes 3, 6 in front of us, because their sum 
gives the number of the rows, a sum we should otherwise have to know 
from m + 2r — n. Instead of all the determinants of such an array being 
viewed, as hitherto, of equal prominence, Trudi only concerns himself with 
the first two of the ten, namely, those which have in common the first eight 
columns of the array. These n — r+ 1 determinants he designates not very 
happily “ the successive determinants of the array.” The name “ principal ” 
which he gives to the first determinant of all may be advantageously 
translated “ leading.” 
