1911-12.] The Theory of Circulants from 1861 to 1880. 137 
Zehfuss, G. (1862). 
[Anwendungen einer besonderen Determinante. Zeitschrift f. Math . 
u. Phys., vii. pp. 439-445.] 
Zehfuss proves Spottiswoode’s result by multiplying the rows in 
order by Or , Op 1 , . . . , 0 r respectively, and the columns in order by 
1 , 0 r , 0 ^. , . . . , Or -1 , a procedure which amounts to multiplying the 
determinant by 0”(1 . 0 r . 0 2 r . . . Op 1 ) 2 , that is, by 1. Addition of the rows 
is then all that is wanted to reach the desired result. 
The rest of the article (pp. 441-445) is devoted to the “ Anwendungen,” 
namely, (1) to Eisenstein’s expression 
x 2, + DD'y 3 + DU¥ - 3D xyz , 
where the letters denote complex numbers and D = D'D" ; and (2) to a letter 
of Jacobi’s on cyclotomy and the theory of integers. 
Baltzer, R. (1864). 
[Theorie und Anwendung der Determinanten. . . . 2 te vermehrte 
Aufl. viii + 224 pp., Leipzig.] 
With Baltzer (§ 11, i, 2 . s) the determinant 
>5 
1 
0 
e 
°i 
% 
%-l 
%- 1 
%-y 
cq 
. . . %-2 
%- 
a n- 1 
%-y 
• • • 
eq 
% 
% 
... a Q -y 
Cl 
iT' 
0 
1 
. %-i) say,— 
which, of course, is not more general than 
C(a 0 , % 5 % 5 • • • > %— 1) 5 
— is openly reached (p. 92) by eliminating x dialytically * from the equations 
y = a Q + a Y x + a 2 x 2 + . . . + a n _$! l ~ x and 1 = x n > 
* Namely, by using on the first equation the multipliers x , x 2 , . . . , Xn- 1 , and substitut- 
ing 1 for x n wherever the latter turns up, exactly as Beltrami did. The same result, however, 
is reached by following Bezout’s “ abridged method.” On the other hand, the application 
of Sylvester’s dialytic method unmodified entails, as we know, the performance of multipli- 
cation on both equations, and gives an eliminant of a higher order. For example, in the 
case of n = 3 it gives 
a 2 a i % ■ 
• CL 2 CL-^ CLq • 
• • 6^2 CLq 
1 . . Hi 
. 1 . . -1 , 
where we have to increase the 4th column by the 1st, and the 5th by the 2nd, before we 
can reach C(a 0 , a 2 , eq). 
