1910-11.] The Less Common Special Forms of Determinants. 315 
Rubini, R. (1857, May). 
[Applicazione della teorica dei determinanti. Annali di Sc. Mat. e 
Fis., viii. pp. 179-200.] 
In treating of determinants with binomial elements Rubini’s most 
interesting example is that in which the element in the (r,s) th place is 
Urs + brs J — 1. By substitution in his general result he readily obtains the 
expansion of the determinant in the form C + D^-l, which is seen to 
alter into C — D J — 1 on changing the signs of the 6’s. The product of 
| Uin+bm J — 1 | and | a ln — b ln J — l | is consequently expressible as the sum 
of two squares. His next point is that on using the ordinary multiplication- 
theorem the same product is got in the form 
a ll 
a 12 “ J ~ 1 • • • 
• ®]n fi\n 1 
a 12 + 12 J ~ 1 
a 22 
• a 2 n An s! ^ 
a ln + An J ~ 1 
l 
1 
> 
8 
+ 
• a wn 
and that a comparison of the two forms may be fruitful of results. When 
n — 2, the identity resulting from such comparison is 
(ad -be - a8 + /3y) 2 + (aS -by + ad- /3c) 2 
= ( a 2 + a 2 + b 2 + (3 2 )(e 2 + y 2 + d 2 + 8 2 ) — ( ac + ay + bd + (38) 2 — (ay — a c + b8 - /3d) 2 , 
— a result which gives the product of two sums of four squares as a 
like sum. 
In connection with this special example, however, note should be taken 
that Hermite in a letter to Jacobi published in 1850 (see Crelle’s Journ., 
xl. p. 297), had pointed out that it followed from the row-by-row 
multiplication of 
a -r a J — \ b + /3 J — 1 
/3 + /3 J - 1 a — a J — 1 
by 
-c+yj-l 
d+8J - 1 
d+8J - 1 
Clebsch, A. (1859). 
[Theorie der circularpolarisirenden Medien. 
pp. 319-358.] 
Crelles Journ., lvii. 
In § 3 (pp. 324—330) Clebsch is led to consider the nature of the roots of 
the equation dealt with by Hermite in 1855, not knowing, apparently, what 
the latter had done. Unfortunately the proof given of the reality of the 
