1909-10.] The Theory of Persymmetric Determinants. 429 
and which, if I , J , K be the other invariants of the octavic, is equal to 
- 2592A 5 + 1 8IA 3 + 3 JA 2 + 2KA + L. 
The expansions of I , J , K are those numbered 39, 43, 44 in Cayley’s 
collection. 
Bruno, Faa di (1856, August). 
[Sopra i resti di Sturm. Annali di sci. mat. efis., vii. pp. 313-317.] 
Beginning with two unrelated functions, P , Q, of the n tYi and (n — l) th 
degrees, Bruno gives an expression for any one of the Sturmian series of 
functions thence derived, the coefficients of x in this expression being 
determinants which resemble in outward form those of Cayley’s analogous 
expression of 1846 (August), but which have for elements the coefficients 
of x~\ x~ 2 , . ... in Q/P. He says that the expression “ salvo qualche 
modificazione ” was found by Cauchy, but gives no reference. 
Bellavitis, G. (1857, June). 
[Sposizione elementare della teorica dei determinant!. Mem. . . . 
Istituto Veneto .... vii. pp. 67-144.] 
Although the persymmetric determinant 
where s r is the sum of the rth powers of the roots of the equation 
x n + a 1 x n ~ 1 + . . . +d w = 0 has repeatedly come before us, it has always 
been with the understanding that no two of the roots were equal, and the 
order of the determinant has never been greater than the ^ th . Bellavitis 
takes up the subject (§§ 45-50) with these conditions removed. He 
affirms that when the number of rows exceeds the number of different 
roots, the persymmetric determinant 
So+r ^2+r .... 
S l+r S 2+r S 3+r .... 
S 2+r S 3+r S i+r .... 
vanishes, and when the numbers are the same the determinant is a 
multiple of the square of the difference-product of the said roots. By way 
of proof of the second part of the proposition the special case is taken 
