154 
Proceedings of the Royal Society of Edinburgh. [Sess. 
effected in Cayley’s manner of 1852 (Hist., ii, pp. 124-126), the full set 
of equations being 
ayl + byl + cyl + d =0 
ay + byl + cyl + dy$ = 0 
ay + by -f cyl + dyl = 0 
ay . yl + by . yi + cy + dyl = 0, 
and the eliminants y % , y * , yK 
It is also noted that the same procedure is effective when the roots of 
the new equation are to be the p th powers of the roots of the given equation. 
Thus, p being 3, we substitute y* for x, and multiply twice by y$, the set 
of equations now being 
ay + byl + cyl + d =0 
ay .yi + by + cy% + dyi = 0 
ay • + by . y\ + cy + dy% = 0 
and the desired equation 
b c d + ay 
c d + ay by 
d + ay by cy 
= 0 . 
Here, it may be remarked, the sum of the coefficients is still a circulant, 
namely, C (d + a , b , c). 
Weihrauch, K. (1880). 
[Werth einiger doppelt-orthosymmetrischer Determinanten. Zeitschrift 
f. Math. u. Phys., xxvi, pp. 132-133.] 
Three of the four results are Scott’s of the }^ear 1878, and the remaining 
one, namely, 
c(l ,3,6,..., \n(n + 1)) = ( - 1)“ _1 — * '"^ + 2) j (* + 3 )’‘ -(» + !)“}• 
resembles Scott’s third. 
Muir, T. (1881). 
[On new and recently discovered properties of certain symmetric 
determinants. Quart. Journ. of Math., xviii, pp. 166-177.] 
To start with, Glaisher’s theorem regarding a circulant of the (2 n) th 
order is shown to be readily obtained by multiplying together two slightly 
altered forms of the circulant. The first form is got by advancing the 
odd-numbered rows in order to the first n places, and then altering the 
signs in the even-numbered columns, the result being 
