1902-3.] Dr Muir on Axisymmetric Determinants. 557 
of the coefficients of x v x 2 , .... , x n in the first equation of that 
set : in other words, the first column of coefficients in the derived 
set of equations corresponds to the first row of coefficients in the 
original set. Then taking another set of n equations having the 
same coefficients 11, 12, . . . . differently disposed, viz., 
1 l*2/i + 21‘y 2 + •••• + nYy n = 
12't/j + 22 -y 2 + •••• + n'2'y n = v 2 
\n'iy 1 + 2 n% + ••••+ nn'y n = v n , 
hut where of course the determinant of the coefficients is in 
substance the same as before, and therefore denotable by N, and 
where consequently the cofactors of the elements of which the 
determinant is composed are also the same as before, he proves, 
rather unnecessarily, that 
fll-^ + f 1 2 ’v 2 + .... + fl n-v n = 
f 2 1 •«7 1 + f22‘v 2 + ••••+ f2 n'v n — N-y 2 
f n{v Y + in2'v 2 + •••• + inn'v n = N’y nj 
In this way it is made to appear that the coefficients of the one 
set of derived equations are the same as the coefficients of the 
other set of derived equations, the difference in the arrangement 
of them being exactly the difference observable in regard to the 
primitive sets. 
From this he passes to the case where the array of coefficients 
of the primitive set of equations possesses the property of axisym- 
metry, his words being (p. 301) — 
“1st endlich fur jedes p und q, pq = qp i oder ist bey den 
gegebenen Gleichungen, fur jedes m die rate Horizontalreihe 
der Coefficienten, mit der mten V erticalreihe derselben 
einerley ; die Horizontalreihen nehmlich von oben herab, 
und die Yerticalreihen, von der Linken nach der Rechten 
zu gerechnet, so ist auch allgemein {pq^iqp, oder die rate 
Horizontalreihe der Coefficienten, mit der raten Verticalreihe 
derselben, auch bey den Auflosungsgleichungen einerley.” 
It may be noticed in passing that as the determinant of the 
coefficients in the derived set of equations is the conjugate of the 
