47 
1913-14.] The Theory of Bigradients from 1859 to 1880. 
are then equal to 
^0^1 ’ ^0^2 "h ^2 ,l? 0 ’ .... 
a.Sr 
a 0 S n-l + • • • + a n-l S (i 
and that in addition we have 
(cKq , C 8 j , . . . , d n $ S n , S n _ i , 
K J 5 • • • 5 a n \ S Jl+l J S n 5 
(«0 J ^1 J • - • 5 $ S n+2 > S w+1 5 
® 0 ) = °» 
*i) = 0, 
s 9 ) = 0, 
The results arrived at are 
D, = a * 
°o °i 
Sn So 
«r-l 
S r 
s r+1 
°r+l 
S r+2 
= 0 when r>n - 1 
V,. — tto 
tl = 
. *1 
$9 
s r-l 
1 
S 2 
S 3 
. s r 
X 
! S 3 
*4 
S r+ 1 
X 2 
r+1 5 
S 0 
*1 
s 2 . . 
. . S r _! 
S l 
tS> 2 
s 3 . . 
. . 
s o 
S 2 
S 3 
S 4 • * 
. . s r+1 
S 0 X + Sj 
S 3 
S 4 
% • • 
• . S r _|_ 2 
S 0 X 2 + Sj# + s 2 
r+1- 
Here again, however, Trudi loses his opportunity from not being acquainted 
with Cayley’s multiplication-theorem of 1845, the use of which enables us 
to transform not only D r , but the whole bigradient array of which D r is the 
first determinant. In fact, it gives us for the case under consideration 
another condensation-theorem. For example, when 
A = a^x 5 + cqa? 4 + * • - + a 5 
and we consequently have to consider the four “ simplified remainders ” 
(®o > • • 
•» “5)1 1 
(x*, x\ X, 1 ), 
I (“o." 
•5 ^5)2 
(x 2 , X , 1 ), 
•> ^5)3 
|(*, 1 ). 
(«<>>•• 
•.“5)4 
•> ^4)2 ' 
II ( 6 o,.. 
•? ^4)3 
•^4)4 
!(»«,•• 
•>*4)5 
we find the condensation-results 
