54 
Proceedings of the Royal Society of Edinburgh. [Sess. 
a n 
1 
1 
1 
1 . 
1 
2 
3 
4 . 
1 
3 
6 
10 . 
1 
4 
10 
20 . 
The evaluation of the cofactor of a Caldarera effects with much trouble 
(pp. 226-231) by means of his (or, rather, Ohio’s) condensation-theorem, not 
observing that a series of similar steps can be made by simply diminishing 
each row by the row immediately preceding : for example, when n = 4, 
we have 
D(a,0) 
1 2 3 
1 3 6 
1 4 10 
1 3 
1 4 
= 1 , 
the theorem repeatedly used in performing the subtractions being 
0 n—l,r ' 0 Vi— l,r— 1‘ 
The same process applied directly to D(a,d) gives the value a n with equal 
ease, showing at one and the same time that D(«,c>) = D(a,0), and that the 
axisym metric cofactor of a in the latter is equal to 1. 
Schultze, E. (1871, Sept.). 
[Ueber die aus einer symmetrischen Determinante A n = 2±a u . . . a nn 
gebildete Reihe A„, A n _ x . . . , A 0 . Sch. Progr. 22 pp. Berlin.] 
The title is misleading, the subject really being the linear transformation 
of a quadric, whose discriminant is A,„ into an aggregate of multiples of 
squares. As, however, one of the modes of transformation referred to is that 
resuscitated by Brioschi in 1856,* but originally due to Lagrange (1759 j*), 
namely, that which changes 
a n x \ + « 22^2 +....+ a nn x n 2 + 2a l2 x 1 x 2 + . . . . 
into 
A2/i 2 + fv + §V + • • ■ +~yj 
by a transformation of the form 
* Brioschi, F., “ Sur les series qui donnent le nombre de racines reelles des equations 
algebriques a une ou a plusieurs inconnues,” Nouv. Annales de Math., xv. pp. 264-286 ; or 
Zeitschrift f. Math. u. Phys., ii. pp. 209-222 ; or Opere Mat., v. pp. 127-143. 
t Lagrange, J. L., “ Recherche sur la methode de maximis et minimis,” Miscell. . . . 
Taurinensis , i. p. 18 ; or (Euvres , i. pp. 3-20. See also Gauss (1823), TVerke, iv. pp. 27-53, 
and Jacobi (1847), Crellds Journ., liii. pp. 265-270. 
