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NOTE ON AN OVEELOOKED THEOEEM EEGAEDING THE 
PEODUCT OF TWO DETEEMINANTS OF DIFFEEENT 
OEDEES. 
By Thomas Mum, LL.D. 
(Eeceived January 8, 1913.) 
(Eead April 16, 1913.) 
1. Hidden away in an investigation on the common roots of two 
equations (Comptes Bendus . . . Acad, des Sci. (Paris), Ixxxviii, pp. 223- 
224), there occurs the following theorem : — 
Soient A = S + a„a22 a^^, et B = 2: + b„b22 ... bnn(ni>n) deux de- 
terminants : si Von designe ixir B^g le resultat de la substitution des n 
premiers elements de la k'""'' ligne de A d la place de la s'^™*" ligne de B, par 
ttir les mineurs de A par rapport au r^'"'" colonne et par /3tr le mineur de B 
par rapport au t'^^^ element de la k^^^^ ligne, on a identiquement — 
871 considerant /3]jr coinme nul quand r est plus grand que n." 
No proof is given, and it is consequently a little difficult to see how 
the author came to reach a result of such importance without obtaining a 
much more extensive generalisation. 
2. A brief scrutiny suffices to convince one that what the theorem 
really gives is an expression for the product of any two determinants of 
the p\h and ^th orders (p>q) in the form of a sum of products of two 
determinants of the (p - l)th and {q + l)th orders. It is at once clear, for 
example, that the kth row of B in the identity is a fiction, for on the left- 
hand side it is explicitly supplanted by rows obtained from A, and on the 
