198 Proceedings of the Koyal Society of Edinburgh. [Sess. 
of a special type were considered by Jacobi in 1841, namely, those whose 
elements are functional determinants, his main theorem being 
V+T ( 1 , r. T (w) = i ^± d A. d A.JA l"’ 1 . 
^J“jl J2 ‘ Jm ( X dx 2 ^ x n f dx 2 dx n+m ’ 
where f v f v . . . , f n+m are functions of x v x 2 , ... , x n+m , and 
T (S) = WnVn+S' 
tP r- X 1 “ 'dx 1 dx 2 dx n cx n+ f 
This theorem and certain deductions therefrom have been already dealt 
with in another connection (see pp. 381-385 of History). 
Sylvester (1850). 
[On the intersections, contacts, and other correlations of two conics 
expressed by indeterminate co-ordinates. Cambridge and Bub. 
Math. Journ., v. pp. 262-282: or Collected Math. Papers, i. pp. 
119-137.] 
In a footnote to this paper the name “ Compound Determinants ” first 
appears. The passage is (p. 270): “. . . a theorem given by M. Cauchy, 
and which is included as a particular case in a theorem of my own relating 
to Compound Determinants, i.e. Determinants of Determinants, which will 
take its place as an immediate consequence of my fundamental theorem 
given in a memoir about to appear. The well-known rule for the Multipli- 
cation of Determinants is also a direct and simple consequence from my 
theorem on Compound Determinants, which indeed comprises, I believe, in 
one glance all the heretofore existing doctrine of determinants.” 
It will be of interest as we advance to try to identify the theorems of 
this perf ervid statement, namely, ( a ) Sylvester’s “ fundamental theorem ” . 
(6) his widely general “ theorem on compound determinants ” deduced 
therefrom, and including as a particular case a theorem of Cauchys, and 
giving rise to the multiplication-theorem and many others as corollaries. 
Sylvester (1851, March). 
[On the relation between the minor determinants of linearly equivalent 
quadratic functions. Philos. Magazine (4), i. pp. 295-305, 415 : or 
Collected Math. Papers, i. pp. 241-250, 251.] 
As we have already had occasion to note,* there is here given, by way 
of illustrating the power of the umbral notation, a theorem regarding a 
* Proc. Roy. Soc. Edinburgh , xxv., p. 929. 
