306 Proceedings of the Royal Society of Edinburgh. [Sess. 
Souillart, C. (1860, Sept.). 
[Note sur la question 405 et sur une composition de carres. Nouv . 
Annales de Math., xix. pp. 320-322.] 
Souillart’s subject is the skew determinant 
abed 
-b a - d c 
- c d a —b 
- d — c b a , 
and his observations are (1) that it is equal to 
(a 2 + b 2 + c 2 + d 2 ) 2 , 
and (2) that if it be multiplied by the similar determinant which is equal 
to 
(p 2 + q 2 + r 2 + s 2 ) 2 
the result is a determinant of the same form, whether the multiplication be 
row-by-row or column-by-column. The object, of course, is to prove Euler’s 
theorem* that the product of two sums of four squares is a sum of four 
squares. 
Cayley, A. (1860, Dec.). 
[Note on the theory of determinants. Philos. Magazine, xxi. pp. 180- 
185 : or Collected Math. Papers, v. pp. 45-49.] 
After expounding his, or rather Cauchy’s last, mode of partitioning the 
ordinary expansion of a determinant, and giving his own diagrammatic 
representation of the partition, Cayley applies it to the expansion of a 
zero-axial skew determinant, showing, of course, that when of odd order 
it vanishes, and that when of even order it is expressible as a rational 
integral function of the elements. 
Trudi, N. ( 1862 ). 
[Teoria de’ Determinant^ e loro applicazioni, di Nicola Trudi : 
xii. + 268 pp., Napoli.] 
To “ determinanti gobbi ” Trudi devotes sixteen pages (pp. 78-94) of his 
text-book, the exposition, which is not a little influenced by Brioschi and 
Baltzer, being full and simple. There are only one or two points in it 
* Novi Commentarii Acad. Petropolitanae, xv. (1770), pp. 75-106. For the conclusion reached 
see Nouv. Annales de Math., xv. pp. 403-407. 
