308 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Bruno, Faa di (1856, February). 
[Sulle funzioni isobariche. Annali di Sci. Mat. e Fis., vii. pp. 76-89.] 
On p. 81 Bruno enunciates, as having been recently (“ ultimamente ”) 
discovered by him, a theorem which is essentially Faure’s, the A r of Faure 
being given by Bruno as the coefficient of x r in the expansion of 
c 0 + Cjtf + c 2 a 2 + .... 
a 0 + cl x x + a. 2 x 2 + .... 
The reason for A r being the same in both is evident on putting m = n = 0 in 
Faure’s. 
Allegret, A. (1857). 
[Solutions de quelques problemes curieux d’arithmetique. Nouv. 
Annates de Math., xvi. pp. 136-139.] 
In the course of Allegret’s work, the determinant 
1 ... 1 
1 oq 
1 1 . 
i i a, . 
i . ,i « 4 
appears, which he says 
1 
«1 
= a x a 2 a z a± - 
1 
1 
1 
1 
a 2 . 
1 
. 
1 
1 a 3 
1 
1 
a 2 . 
1 
. 1 
1 
1 « 3 
and, the four-line determinant now reached being similar in form to the 
original, he concludes that the final expansion of the latter must be 
a^a 2 a z a± - + a 1 a 2 - cq + 1. 
By passing the first row over the others to occupy the last place the 
determinant is recognisable as a special case of recurrent, and it is seen that 
the expansion in terms of the elements of the last row and their cofactors 
leads at once to Allegret’s result. 
Brioschi, Fr. (1857). 
[Solution de la question 350 (Wronski). Nonv. Annates de Math., xvi. 
pp. 248-249.] 
The problem having been set to find what Wronski called the “ Aleph ” 
functions * of the roots x 1} x 2 , ... ,x n of the equation 
a^x n + <q# n_1 + a 2 aj w-2 + ... +a n = 0 
* Wronski, H. Introduction a la Philosophic des Mathematiques . . . (pp. 65, . . .) 
vi + 270 pp., Paris, 1811. 
