696 
Transactions of the Royal Society of South Africa. 
Fontaine's is the case of this for ;z = 2 ; Bezout's is the case for n — Z \ 
and Monge's is obtainable from Bezout's by putting the /'s equal to the 
corresponding a's. It is important to note, however, that Monge's is with 
at least equal interest viewable as what is technically called an extensional " 
of Fontaine's. Thus, turning to (y) and dropping the ai in it which begins 
every one of the six minors, we find remaining 
• (^263 — Ife2'^3|' c.^eg + \h.e.;^ c^d^ = 0, 
which is the equivalent of (a). 
(3) It would appear that for long this theorem of Sylvester's formed the 
usual means of investigating relations between the primary minors of an 
oblong array. All such investigations therefore dealt with relations that 
were quadratic ; any others, like Monge's cubic relations, may be passed 
over, because they were directly deducible from those of lower degree. 
In the year 1893, however, Yahlen published his paper professedly 
devoted to the subject, the title being TJeber die Belationen zwischen den 
Determinant en einer Matrix (Crelle's Journal, cxii, 306-310*), and he struck 
out a new course. The theorem which he used was one regarding the 
multiplication of one n-line determinant by the (n — 1)^^ power of another, the 
product being given in the form of a compound determinant. It is most 
easily established by substituting for the multiplier the adjugate of the 
second determinant, and noting that every row-multiplication gives an n-lme 
determinant for result. Thus, the two determinants being \a-J)^c^\ and 
I'^iV^h ' have 
W'l h 'Vs h\ 1^1 
\x.^ %I \x-^ \Xo 
y., z.^\ \x^ a., z.^\ \x^ a^\ 
\h, y.^ z.^ \x^ &2 % 2/2 h\ 
2/2 h\ 1^1 ^2 ^3! 1^1 V'z 
where the element in the place (r, s) of the right-hand member is the result 
of putting the r*^ row of lajfeg^sl place of the s*^ row of \x-^yoZ^\- The 
theorem, or rather an extensional of it, seems to have been first put on record 
by Bazin in 1851 f ; since then, and prior to Vahlen, it has been restated by 
several writers — for example, by Zehfuss in 1862. 
(4) Yahlen' s mode of using it in connection with the subject under 
* Pascal in his " Deter minanti " gives a full account of it. (See pp. 148-152; or 
in Leitzmann's translation, pp. 115-117.) 
t Bazin, H., " Sur une question relative aux determinants/' Journ. (de Liouville) 
de Math., xvi, pp. 145-160. 
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