698 Transactions of the Royal Society of South Africa. 
so that we end exactly as we started. Of the remaining 105 considered to 
be effective and independent we may select that in which the multiplicand 
is there being in this only one column that occurs also in \aih2C^d^\. 
The result of multiplying by lA^^B.^C^D^i now comes out in the form 
a-J).)C^d^\ 
!/i6,C3riJ \aj,c.^d^\ \a^h,f^d^\ l^^VsAl 
:giKc^d^\ \a^g.c.^d^\ \a^Kg.;^d^\ \a^h.,c^g^\ 
\hih.2C^d^\ la^h^c^^d^l {a^h^h^d^l [a^tgCgA^ | 
whence, of course, it is concluded that 
laj^c^d^l it''i?>/at/J I -^1^2^3/4 
la^f^gsh] • l^lVs^J' = \(^l9->^-A\ i«^l%3^4l l^lV3^4! 
\a^h.)C.^d^\ \a-J)2h.^d^\ {a-JjoC^h^l 
and there Vahlen allows the matter to rest. If, however, we expand the 
compound determinant here in terms of the elements of the first row and 
their complementary minors, we find by what we have called the extensional 
of Fontaine's identity that the said minors all have \a-J).2C.^d^\ for a factor, 
being equal to 
l^i^2^/3^4'-'^i^2^3'^4!' —\ciib-2^'S^h^-^^i^-29s^4}^ ' a-J).2C.^d^\ .\a-^d2ggh ^\ ; 
so that there results the simple equality 
a^f,g^^h^\.\a^\c^d^\ = 'aj,f^d^\.\a^h.jg^h^\ ~ la^h^f^d^l.la^c^g.^h^l + la^h^c^-f^Ua^d^g^h^l 
in agreement with Sylvester's theorem. 
(7) The first to note and to establish the reducibility of Bazin's theorem, 
to Sylvester's was Rubini in 1878.* The theorem, in fact, found in Eubini 
another discoverer, one too who was not content merely to state and prove 
his theorem, but to note special instances of it, draw deductions from it, 
view it from more than one standpoint, and indicate its affinities. The 
connected question, however, of the relations between the minors of a matrix 
he did not specifically refer to. This was left for Pascal, who in 1896 drew 
pointed attention to it in a paper whose title could leave no doubt as to its 
object.! In the first section of this he establishes what he calls his 
*' fundamental identity A " — not recognised, strange to say, as Sylvester's — 
then shows how the so-called Vahlen relations are dependent on it, and 
ends with the words ''in general all the possible relations between the 
* Eubini, I?., " Formole di trasformazioni nella teorica dei determinanti." 
Giornale di Mat.,' xvi, pp. 198-208. 
t Pascal, E., " Sulle varie forme che possono darsi alle relazioni fra i determinanti 
di una matriee rettangolare. Annali di Mat., xxiv, pp. 241-253. 
