772 Proceedings of Royal Society of Edinburgh. [sess. 
Produkte wegfallen, welclie zwei gleiche Factoren enthalten, so 
erhalt man 
P1P2P3 • • • 1 = P0P2P3 • • • Pn- 
Also da beide Produkte, als demselben System n - ter Stufe 
angehorig einander gleichartig sind, so hat man 
x = P 0 P 2 P 3 ' • • Pn » 
1 P1P2P3 •••/>»' 
The method is thus seen to consist in the deduction of a new equa- 
tion by addition, and in the elimination of all the unknowns, except 
one, from the equation, by multiplying both sides by the product of 
the coefficients of the other unknowns, — the multiplication in 
question being “ outer,” and for the purposes of the multiplication, 
any two coefficients of one and the same equation being considered 
as “ like,” and any two belonging to different equations as “ unlike.” 
For example, in the case of n = 3 we have 
x _ 0 0 4-a o + Co).(g 2 + & 2 + <; 2 ).(a 3 + 6 3 + C3 ) 
1 («! + + cj . (a 2 + b 2 + c 2 ) . (a, + b 3 + c 3 ) ’ 
{a^a 2 + af> 2 d* ^ 0^2 d~ byi 2 d- bfo 2 d - . . .) . (a 3 + bo + c 3 ) 
— (eqa 2 + af> 2 + a x c 2 + \a 2 + bf 2 + . . .) . (n 3 + b 3 + c 3 ) ’ 
_ (a 0 b 2 + a 0 c 2 + b 0 a 2 + \c 2 + c 0 a 2 + c 0 b 2 ) . (a 3 + b 3 + c 3 ) 
(a 1 b 2 + a 1 c 2 + b x a 2 + b x c 2 + c Y a 2 + c x 6 2 ) . (a 3 + b 3 + c 3 ) ’ 
since a 0 a 2 = b 0 b 2 = ... = c Y c 2 — 0 ; and finally 
x _ a f ) 2 c 3 ~ a (fis c 2 d* a 2^3, C 0 ~ a 2^0 C 3 d~ a f ) 0 C 2 ~ a S^2 C 0 
1 af) 2 c 3 - af> 3 c 2 4- af) 3 c x - ajb 1 c 3 - 1 - af) Y c 2 - af) 2 c Y ’ 
“ worin wir, da alles entsprechend geordnet ist, wieder die gewohn- 
liche Multiplicationsbezeichnung einfuhren konnten.” (in. 39) 
All this semblance of demonstration is of little moment compared 
with the fact sought to be demonstrated, viz., that a determinant is 
expressible as the outer product of the sums of the elements of its 
columns. Grassmann, however, makes no reference to determinants. 
In a paragraph of a subsequent chapter (p. 129), he takes up the 
problem of elimination between two equations of the m th and w th 
degrees. What it contains is a reproduction of Sylvester’s dialytic 
method, without any reference to the author of the method. 
