1915-16.] The Theory of Circulants from 1880 to 1900. 
163 
reference to the corresponding results regarding shew circulants (“gobbi 
circolanti ”). 
Mum, T. (1882). 
[A proposed general method for the solutions of equations. Proceed. 
Philos. Soc. Glasgow, xiii, p. 616 : Educ. Times, xlvi, p. 442 : Math, 
from Educ. Times (2), v, pp. 106-107.] 
The property proposed to be utilised is the resolvability of C(x, a, b,c, . . .) 
into linear factors and the consequent solubility of the equation 
C(x , a , b , c, . . . ) = 0 . 
The application to the quartic x 4: -\-px 2 -\-qx-\-r — 0 is carried out by 
E. Nesbitt in the Educ. Times, and, in effect, to a specialised quintic 
by Cayley in the Quart. Journ. of Math., xviii, pp. 154-167. 
Bhut, A. B. (1882). 
[Question 7072. Educ. Times, xxxv, p. 155 : Math, from Educ. Times, 
xl, p. 48.] 
The problem is to find x, y, z, u from the equations 
X 
y 
z 
y 
z 
u 
z 
u 
X 
u 
X 
y 
u 
X 
y 
= a 3 , 
X 
y 
z 
ii 
O ' 1 
CO 
y 
z 
u 
= c 3 , 
z 
u 
X 
z 
u 
X 
u 
X 
y 
X 
y 
z 
y 
z 
u 
and it is not observed that this is the same as to express x, y, z, u in 
terms of their complementary minors in C(x, y , z, u). Calling these 
when signed X, Y, Z, U, we have from the theorem regarding a minor 
of the adjugate 
and therefore 
x — 
X 
Y 
z 
u 
X 
Y 
= x 
z 
u 
X 
X 
Y 
Z 
u 
X 
Y 
-r 
z 
u 
X 
y z u 2 
x y z 
u x y 
z u x , 
Y Z U i I 
X Y Z 
U X Y 
Z U X . 
Cesaro, E. (1883): Neuberg, J. (1883). 
[Question 245. Mathesis, iii, pp. 118-119; vi, pp. 60-62.] 
[Question 273. Mathesis, iii, p. 192; viii, p. 215 ; x, pp. 117-119.] 
These concern special circulants which have already been fully 
dealt with. 
