1904-5.] Dr Muir on General Determinants. 919 
his problem being to find under what conditions, as regards the 
arbitrarily introduced symbols a , , y , the product 
4- a 2 B "b H~ b 2 /? + b^y^ic-yx + c 2 fi + Cg y) 
will be identical with the determinant ; and he is led to the need 
for imposing laws essentially the same as those of “ outer multi- 
plication.” Grassmann’s factors are each the sum of the elements 
of a column of the determinant, and, according to his quasi- 
demonstration, could not be anything else : Cauchy’s factors are 
each formed from the elements of a row ■ but had they been 
formed from the elements of a column, the result would not have 
been different. The root idea, viz., the expression of a determinant 
as a product of factors subject to multiplication of a special hind , 
was certainly first reached by Grassmann : Cauchy attained the 
same result, adding somewhat to its width, and presenting it in a 
fresh and more reasonable form. 
JOACHIMSTHAL (1849, Nov.). 
[Sur quelques applications des determinants a la geometrie. 
Crelle’s Journ ., xi. pp. 21-47.] 
Joachimsthal’s interesting series of ‘applications’ being mainly 
connected with the multiplication of determinants, he introduces 
them by enunciating the multiplication-theorem, and indicating 
a mode of proof “ pour eviter aux lectures la peine de le chercher 
ailleurs.” 
The enunciation is 
x y z 
I 
[ f V { ) 
det.- 
x 1 Vl Zl 
- x det.J 
fi Vi (1 
b X 2 V2 Z 2 , 
' 1 
{ £2 V 2 £2 J 
x£ +yrj +z£ xf x + y Vl +zC 1 + yrj 2 + z £ 2 1 
x i^+Viy + hC x i£i + ypii + Z]£i 4 + 2 /i% + ^ 2 r 
L X 2^1 4" 2 / 2^1 "b zffi x 2^2 z 2^2 ) ; 
= det. 
