49 
1913-14.] The Theory of Bigradients from 1859 to 1880. 
and it having been found that the first determinants of the arrays 
I (l,...,4)l 
(1 . • • • . 4), 
<1 ..... 4), 
! (5, . . . , 8) 2 
(5, . . .,8), 
1 ? 
(5 , • . . , 8) 4 
5 
(5 , . . . , 8) s 
have the values 
54, 0, 0, 0, 
the proposition states that the number of distinct roots of the equation 
A = 0 is 2, i.e. 5 — 3, and that the equation of these two roots is 
11-5-5. 
. 1 1 -5 . 
. . 5 4 1 
5 4-15 x 
5 4 -15 -2 x 2 
0 . 
As a matter of fact, this equation is 54}(x+.2)(x—l) = 0, and A=(x + 2) 2 
{x-lf. 
Lastly, Trudi takes up Sturm’s theorem for determining the number of 
roots of an equation which lie between two given values. In dealing with 
it he brings forward a new series of functions as a substitute for Sturm’s 
series, namely, 
% 
• 
> 
a 0 a Y 
«2 
% 
K 
1 
' « 0 
a l 
« 2 
\ 
X 
1 
■ \ 
X 
\ b 1 b 2 b s x 2 
Further, he points out that the individual members of this series can be 
lowered in grade by the use of his condensation-theorem, thus providing 
a variant of the series. He also notes that by means of the theorem which 
we have extended above into another condensation-theorem they can be 
transformed into 
s o 
1 A 
H ^ 
> a o 
s o 
S 1 
1 
S 1 
X 
S 1 
S 2 
X 
S 2 
h 
X 2 
and so he arrives by a different route at Joachimsthal’s series of 1854 
{Hist., ii. p. 171). 
Trudi’s work on bigradients, extending to 94 pages if both Teoria and 
Applicazioni be included, has suffered undeserved neglect. Why this 
should have been the case it is a little difficult to understand, its only 
demerits being an occasional wordiness, a not very acceptable notation, 
and a paucity of concrete examples. In his preface (p. vii) he tells us 
VOL. xxxiv. 4 
